{"paper_id":"7a0941ce-0759-4bcc-8aae-811a3cd63e7f","body_text":"Spatial Dynamics of Malaria Transmission\nAuthors\n• Sean L. Wu† (seanwu89@uw.edu), Institute for Health Metrics and Evaluation,\nUniversity of Washington\n• John M. Henry† (henry529@uw.edu), Quantitative Ecology and Resource\nManagement and Institute for Health Metrics and Evaluation, University of\nWashington\n• Daniel T Citron† (daniel.citron@nyulangone.org), Department of Population\nHealth, Grossman School of Medicine, New York University\n• Doreen Mbabazi Ssebuliba (doreenresty@gmail.com), Department of Mathematics\nand Statistics, Kyambogo University.\n• Juliet Nakakawa Nsumba (juliet.nakakawa@mak.ac.ug), Department of\nMathematics, Makerere University Department of Mathematics, School of\nPhysical Sciences, College of Natural Science, Makerere University, P.0. Box 7062,\nKampala, Uganda.\n• H´ ector M. S´ anchez C. (sanchez.hmsc@berkeley.edu), Divisions of Epidemiology\nand Biostatistics, School of Public Health, University of California Berkeley\n• Oliver J. Brady (Oliver.Brady@lshtm.ac.uk) Centre for Mathematical Modelling\nof Infectious Diseases, London School of Hygiene & Tropical Medicine, London,\nUK; Department of Infectious Disease Epidemiology, Faculty of Epidemiology and\nPopulation Health, London School of Hygiene & Tropical Medicine, London, UK;\n• Carlos A. Guerra (cguerra@mcd.org), MCD Global Health\n• Guillermo A. Garc´ ıa (ggarcia@mcd.org), MCD Global Health\n• Austin R. Carter, aucarter@uw.edu, Institute for Health Metrics and Evaluation,\nUniversity of Washington\n• Heather M. Ferguson (Heather.Ferguson@glasgow.ac.uk), Faculty of Biomedical\nand Life Sciences, University of Glasgow\n•\nBakare Emmanuel Afolabi (emmanuel.bakare@fuoye.edu.ng); International Centre\nfor Applied Mathematical Modelling and Data Analytics, Federal University Oye\nEkiti, Ekiti State, Nigeria; Department of Mathematics, Federal University Oye\nEkiti, Ekiti State, Nigeria\n• Simon I. Hay (sihay@uw.edu), Department of Health Metrics Science and\nInstitute for Health Metrics and Evaluation, University of Washington\n1\n . CC-BY 4.0 International licenseIt is made available under a \n is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity.(which was not certified by peer review)preprint \nThe copyright holder for thisthis version posted November 15, 2022. ; https://doi.org/10.1101/2022.11.07.22282044doi: medRxiv preprint \nNOTE: This preprint reports new research that has not been certified by peer review and should not be used to guide clinical practice.\n\n• Robert C. Reiner Jr. (bcreiner@uw.edu), Department of Health Metrics Science\nand Institute for Health Metrics and Evaluation, University of Washington\n• Samson Kiware (skiware@ihi.or.tz), Ifakara Health Institute\n• David L Smith‡ (smitdave@uw.edu), Department of Health Metrics Science and\nInstitute for Health Metrics and Evaluation, University of Washington\n† authors contributed equally; ‡ corresponding author\nT arget Journal PLoS Computational Biology\n2\n . CC-BY 4.0 International licenseIt is made available under a \n is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity.(which was not certified by peer review)preprint \nThe copyright holder for thisthis version posted November 15, 2022. ; https://doi.org/10.1101/2022.11.07.22282044doi: medRxiv preprint \n\nAuthor summary\nThe Ross-Macdonald model, a simple mathematical model of malaria transmission\nbased on the parasite life-cycle, established basic theory and a set of metrics to describe\nand measure transmission. Here, we extend the Ross-Macdonald model so it has the\nskill to study, simulate, and analyze parasite dispersal and heterogeneous malaria\nspatial transmission dynamics in a defined geographical area with malaria importation.\nThis extended framework was designed to build models with complexity that scales to\nsuit the needs of a study, including models with enough realism to support monitoring,\nevaluation, and national strategic planning. Heterogeneity in human epidemiology or\nbehaviors – differences in age, immunity, travel, mobility, care seeking, vaccine status,\nbed net use, or any trait affecting transmission – can be handled by stratifying\npopulations. Mosquito spatial ecology and behaviors are responding to heterogeneous\nresource availability and weather, which affects adult mosquito dispersal, blood feeding,\nand egg laying in a structured set of aquatic habitats. We propose new formulas for\nhuman biting rates and entomological inoculation rates that integrate exposure as\nhumans move around. We rigorously define parasite dispersal, and we develop matrices\ndescribing the spatial dimensions of vectorial capacity and parasite dispersal in mobile\nhumans. We relate these to the parasite’s overall reproductive success, local\nreproductive numbers and thresholds for endemic transmission. [185/200 words]\n3\n . CC-BY 4.0 International licenseIt is made available under a \n is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity.(which was not certified by peer review)preprint \nThe copyright holder for thisthis version posted November 15, 2022. ; https://doi.org/10.1101/2022.11.07.22282044doi: medRxiv preprint \n\nAbstract\nThe Ross-Macdonald model has exerted enormous influence over the study of malaria\ntransmission dynamics and control, but it lacked features to describe parasite dispersal,\ntravel, and other important aspects of heterogeneous transmission. Here, we present a\npatch-based differential equation modeling framework that extends the Ross-Macdonald\nmodel with sufficient skill and complexity to support planning, monitoring and\nevaluation for Plasmodium falciparum malaria control. We designed a generic interface\nfor building structured, spatial models of malaria transmission based on a new\nalgorithm for mosquito blood feeding. We developed new algorithms to simulate adult\nmosquito demography, dispersal, and egg laying in response to resource availability. The\ncore dynamical components describing mosquito ecology and malaria transmission were\ndecomposed, redesigned and reassembled into a modular framework. Structural\nelements in the framework – human population strata, patches, and aquatic habitats –\ninteract through a flexible design that facilitates construction of ensembles of models\nwith scalable complexity to support robust analytics for malaria policy and adaptive\nmalaria control. We propose updated definitions for the human biting rate and\nentomological inoculation rates. We present new formulas to describe parasite dispersal\nand spatial dynamics under steady state conditions, including the human biting rates,\nparasite dispersal, the “vectorial capacity matrix,” a human transmitting capacity\ndistribution matrix, and threshold conditions. An R package that implements the\nframework, solves the differential equations, and computes spatial metrics for models\ndeveloped in this framework has been developed. Development of the model and metrics\nhave focused on malaria, but since the framework is modular, the same ideas and\nsoftware can be applied to other mosquito-borne pathogen systems.\n4\n . CC-BY 4.0 International licenseIt is made available under a \n is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity.(which was not certified by peer review)preprint \nThe copyright holder for thisthis version posted November 15, 2022. ; https://doi.org/10.1101/2022.11.07.22282044doi: medRxiv preprint \n\nIntroduction 1\nPlasmodium falciparum transmission dynamics are complex: they involve 2\nmultiple-agents, non-linear dynamics, localized spatial interactions, spatial, temporal 3\nand behavioral heterogeneity, stochasticity, and exogenous forcing by weather, 4\nhydrology, and malaria control. Over time, these processes can be modified by economic 5\ndevelopment; by changing socioeconomic status, human incentives and social norms; 6\nand by the evolution of resistance. Every one of these features of malaria transmission 7\ndynamics and control presents its own set of challenges to the quantitative study of 8\nmalaria for scientific research and for analytics to support policy. An important 9\npractical problem is how to quantify and synthesize all of the factors affecting 10\ntransmission at some particular place and time to support malaria control programs in 11\nvarious ways, including monitoring and evaluation of malaria control. The study of 12\ncomplex spatial processes are best addressed using some sort of mathematical model. 13\nHere, to fill a need to give robust policy advice, we have developed a modular 14\nframework with accompanying software to build and analyze suites of models with 15\nscalable complexity for malaria spatial transmission dynamics and control. 16\nA starting point for the quantitative study of malaria transmission dynamics has 17\nbeen the Ross-Macdonald model, which played a central role in developing basic theory 18\nand metrics for malaria [1,2]. That model is simple, general, and conceptually useful, 19\nbut it is not realistic enough to describe many important features of transmission [3]. 20\nThe model’s lack of realism has also limited its applicability: simple models support 21\ngeneric policy advice, but specific advice – tailored to context – must be based on 22\nmodels that can quantify and weigh the effects of locally relevant details [4]. A basic 23\nlimitation of the Ross-Macdonald model was that it lacked features required to describe 24\nspatial transmission dynamics and control. Mathematical models for spatial dynamics 25\nof mosquito-borne pathogens have been developed [5–18], but there is a need for a 26\ngeneralized synthetic framework to develop and use spatial dynamic models, to extend 27\nthe Ross-Macdonald model to define and analyze parasite dispersal, to define and 28\nmeasure malaria connectivity [19], and to link spatial dynamics to spatial data. The 29\nRoss-Macdonald model is also missing other features that are relevant for malaria 30\ndynamics and control, which can be identified from a survey of studies that have 31\nmodeled mosquito-borne diseases (see Box #1) [2]. Modeling and analyzing real 32\nsystems can become overwhelming because of computational, parametric, or conceptual 33\nchallenges that arise from combining all the factors, dimensions, interactions, features, 34\nand processes. Individual-based models (IBMs) have been developed around algorithms 35\nthat make it possible to deal with the complexity by simulating individual states and 36\ntransitions in silico [20], but these high-dimensional computational approaches have 37\nsome limitations that limit their use and applicability. IBMs require intensive 38\ncomputation, are challenging to parameterize, are difficult to critically evaluate, and 39\ntheir output that is often as difficult to analyze and understand as malaria itself. Using 40\na modular framework, we present an alternative way of dealing with the complexity that 41\nis analytically tractable, including some new algorithms to understand mosquito ecology, 42\nparasite transmission by mosquitoes, and parasite dispersal on spatial landscapes. 43\nIn most places, malaria transmission has been modified by control. The extent of 44\neffect modification by malaria control is occasionally revealed when health systems are 45\n5\n . CC-BY 4.0 International licenseIt is made available under a \n is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity.(which was not certified by peer review)preprint \nThe copyright holder for thisthis version posted November 15, 2022. ; https://doi.org/10.1101/2022.11.07.22282044doi: medRxiv preprint \n\ndisrupted (e.g., [21,22]), when malaria control is relaxed or abandoned [23], or when 46\nresistance evolves to drugs or insecticides (e.g., [24, 25]). Programs must weigh evidence 47\nand make decisions through analysis of counterfactuals, rather than through direct 48\nestimation of control effect sizes, since there would be drastic consequences to 49\nexperimentally disrupting control. A predominant need in most contexts is thus a set of 50\nmethods to quantify transmission in its local context as a baseline that has been 51\nmodified by control. A challenge to achieving this has been that the responses to 52\ncontrol efforts are context dependent and have been highly variable across settings. 53\nRelevant factors affecting responses to control include details about blood feeding, 54\nmosquito ecology, and mosquito behaviors that affect contact with interventions ( e.g., 55\nresting indoors and IRS). To reconstruct the counterfactual baseline, transmission must 56\nbe understood in terms of innate mosquito behaviors responding to local resources, 57\nvector control, and other contextual factors that have been modified by control. All 58\nthese have been characterized as being notoriously context dependent and 59\nheterogeneous [26–28]. What are the local factors that determine baseline malaria 60\ntransmission, effect modification, and differences in effect modification at some 61\nparticular place and time? Basic concerns about the heterogeneous impacts of vector 62\ncontrol raise a larger set of questions about how to study and quantify transmission in a 63\nway that is relevant for planning malaria control. 64\nThis new framework is thus an attempt to bridge two well-established but somewhat 65\ncontradictory views of malaria. One view is that human malaria transmission dynamics 66\nand control are so moulded by local ecology and other conditions that the factors 67\ndriving transmission or responses to control at one time and place are unlikely to hold 68\nelsewhere [27]. Another view – encouraged by the rigorous analysis of the 69\nRoss-Macdonald model and extensions of it – is that malaria transmission intensity can 70\nbe quantified using a small set of bionomic parameters to compute basic reproductive 71\nnumbers, which also provide a basis for computing threshold conditions for endemic 72\nmalaria. To build a bridge, the contextual factors affecting basic bionomic parameters 73\nmust be identified and integrated with new theory describing spatial extensions of the 74\nbasic metrics, including rigorous, quantitative description of parasite dispersal, and 75\nsome estimates of the appropriate spatial scales to measure malaria transmission [3]. 76\nContext-dependency is an uncomfortable but unavoidable fact of malaria ecology. 77\nThe heterogeneous nature of transmission and the causes and consequences of variable 78\nresponses to control have been a difficult and sometimes contentious problem for 79\nscientists studying malaria, for national malaria programs and funding agencies making 80\nmalaria policy, and for malaria advocates. Historical trends in malaria and the outcomes 81\nof malaria control have been so variable that case studies can be found to support rosy 82\nprojections, alarmist warnings, or contradictory claims about the underlying causes of 83\ntrends or patterns. To be useful, studies of malaria and programmatic evaluations must 84\nacknowledge the important role of context, the multi-factorial nature of causation in 85\nthese complex systems, non-linear responses to control, the difficulty of measuring 86\nheterogeneous systems, and the resulting uncertainty. A consequence of context 87\ndependency is the difficulty in drawing conclusions that generalize across systems. 88\nThe framework is designed to support development of robust malaria policy advice 89\nand to find practical ways of dealing with uncertainty. While scientific research and 90\npolicy analytics grapple with the same issues and use similar methods, they often put 91\n6\n . CC-BY 4.0 International licenseIt is made available under a \n is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity.(which was not certified by peer review)preprint \nThe copyright holder for thisthis version posted November 15, 2022. ; https://doi.org/10.1101/2022.11.07.22282044doi: medRxiv preprint \n\nvery different weights on uncertainty. Uncertainty affects the ability to do effective 92\ninference for scientific research versus policy analytics – questions about what is known 93\nversus what should be done. To address these concerns and give policy advice despite 94\nuncertainty, an integrated inferential framework is needed to weigh evidence, integrate 95\nthe effects of multiple exogenous factors (often involving experts from distinct 96\nspecialties), estimate their effect sizes, quantify uncertainty, and identify critical gaps. 97\nStatistical theory and inferential methods have been developed around the principle of 98\nparsimony for scientific inference, but substantially less attention has been given to 99\nappropriate designs for analyses that can give advice that is robust to uncertainty. Are 100\nthe conclusions of an analysis robust to reasonable alternative formulations of a model, 101\nand how well are policy recommendations really supported by the evidence? Concerns 102\nabout robustness could lead to study designs that make different tradeoffs between 103\nrealism and abstraction. For example, compared with parsimonious models, models 104\nwith a high degree of realism might be more useful for identifying critical missing data 105\nand prioritizing studies to collect it. Robust analytics requires having a modeling 106\nframework to build suites of models that are realistic enough to weigh the importance of 107\nthe major drivers of transmission despite major knowledge gaps. 108\nTo address these needs, we have developed a new, modular framework designed to 109\nsupport development of models for robust, simulation-based analytics and adaptive 110\nmalaria control with scalable complexity. With scalable complexity in model building, 111\nmembers of a model ensemble could range from very simple to very complex, and that 112\nmodels along that spectrum are related to one another through a logical sequence of 113\nstructural or parametric changes. At one extreme, this framework includes the 114\nRoss-Macdonald model, a simple system of differential equations describing the parasite 115\nlife-cycle in mosquito and vertebrate host populations linked by transmission during 116\nblood feeding [1,29,30]. By extending the Ross-Macdonald model, simple models can be 117\nextended step by step to add complexity or heterogeneity that could be important – 118\nbased on a priori considerations – yet difficult to quantify or poorly informed by 119\nexisting data (Box #1). With modularity, it is possible to develop new dynamical 120\nsystems models describing some parts of the system, add or modify components, or add 121\na set of exogenous factors that force a system. It is also relatively straightforward to 122\nmodify functional responses, or to modify some basic parameters affecting the outcome. 123\nSwarms of models can thus be developed to analyze data and to test the robustness of 124\nany conclusions. To demonstrate scalable complexity, we here present a complicated 125\nfamily of models that has terms and variables anticipating modification by weather or 126\nmalaria control. For practical reasons, the model family we present here was scaled back 127\nto include a limited set of elements describing transmission, but leaving in place the 128\nelements that facilitate modeling control (Box #1). The resulting extensible framework 129\nthat is capable of describing and analyzing malaria spatial transmission dynamics and 130\ncontrol with a high degree of realism in any particular setting. An R package which 131\nimplements the modular differential equations and spatial metrics presented in the 132\narticle is available with documentation (Supplement 1 - 133\nhttps://dd-harp.github.io/exDE/). Despite being programmed in R, the 134\nimplementation of the mathematical framework into code should be easily adapted to 135\nany high-level programming language. 136\nIn F ramework, we first present the modular concepts and structural elements, 137\n7\n . CC-BY 4.0 International licenseIt is made available under a \n is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity.(which was not certified by peer review)preprint \nThe copyright holder for thisthis version posted November 15, 2022. ; https://doi.org/10.1101/2022.11.07.22282044doi: medRxiv preprint \n\nincluding a new blood feeding model. Next we present one exemplar model family for 138\neach dynamical component. In Spatial Metrics, we develop a set of metrics that 139\ndescribe various aspects of parasite spatial dynamics, including metrics for parasite 140\ndispersal, connectivity, and the parasite’s reproductive success. Finally, in 141\nQuantifying T ransmission in a Place , we discuss the application of these models to 142\nthe investigation of malaria transmission dynamics and control in a particular place. 143\n8\n . CC-BY 4.0 International licenseIt is made available under a \n is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity.(which was not certified by peer review)preprint \nThe copyright holder for thisthis version posted November 15, 2022. ; https://doi.org/10.1101/2022.11.07.22282044doi: medRxiv preprint \n\nBox 1: F eatures This generalized, modular framework presents equations\nintegrating multiple agents and interacting processes. Many of these innovations\nappeared first elsewhere, but here they are integrated into a single framework:\n• Immature mosquito population dynamics structured in distinct aquatic\nhabitats linked to adult populations through egg laying and emergence\n[31,32];\n• Spatially heterogeneous blood feeding and parasite mixing on vertebrate\npopulations (i.e., blood hosts) with dynamically changing availability, such\nthat feeding rates and the human fraction change adjust to changing\nconditions [33–36];\n• Heterogeneous adult mosquito behaviors, including dispersal, survival,\nblood feeding, egg laying, mating, and sugar feeding on landscapes in\nresponse to spatially heterogeneous resource availability ( e.g., mating sites,\nsugar sources, blood hosts, aquatic habitats) [37–39];\n• Multiple vector species or types with different host preferences, daily\nactivity patterns, habitats, etc. [40], and potentially with inter-specific\nresource-based competition in habitats;\n• Human mobility based on a concept of time at risk, which combines time\nspent by humans in places where they are at risk with mosquito blood\nfeeding activity, preferences and other factors [9,18];\n• The capability to model indoor and outdoor spaces for blood feeding,\nexposure, and vector control;\n• A non-linear relationship between the daily entomological inoculation\nrate (EIR) and the daily force of infection (FoI) due to heterogeneous\nexposure [41].\n• Malaria importation through multiple routes [42];\n• An exogenously forced, time-varying extrinsic incubation period (EIP) to\nmodel effects of temperature on parasite development;\nThe model has flexible structural elements to stratify an area into patches, to\nmodel any distribution of aquatic habitats, and to stratify a human population\ninto sub-populations by age, immunity, or any heterogeneous, epidemiologically\nrelevant trait. The software also includes time-dependent terms and structures\nto model exogenous forcing by weather, modification of exposure or transmission\nby vector control in relation to coverage, including effects of spatial repellents\nand mosquito behaviors that result in heterogeneous local contact patterns with\nvector-based interventions.\n144\n9\n . CC-BY 4.0 International licenseIt is made available under a \n is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity.(which was not certified by peer review)preprint \nThe copyright holder for thisthis version posted November 15, 2022. ; https://doi.org/10.1101/2022.11.07.22282044doi: medRxiv preprint \n\nFramework 145\nTo describe malaria spatial dynamics with scalable complexity, we designed a modular 146\nframework for model building around four core dynamical components, each one a 147\n(potentially non-linear) state-space model. An interface rigidly defines interactions 148\namong those components, based on passing terms we call dynamical quantities. All state 149\nvariables are vectors of arbitrary length, to accommodate models with different 150\nstructure or spatial granularity. 151\nTo model mosquito ecology, we consider immature mosquitoes in a set of aquatic 152\nhabitats, and adult mosquitoes in a set of patches. A state space model describes 153\naquatic immature mosquito populations (L) with dynamics dL/dt requiring an input 154\nterm from adult mosquito populations: the daily rate eggs are laid in each habitat (η ). 155\nA coupled state space model describes mature adult female mosquito populations ( M) 156\nwith dynamics dM/dt requiring an input term from the aquatic mosquito populations: 157\nthe rate adults emerge from all the habitats in each patch (Λ). A state space model for 158\nparasite infection dynamics in mosquitoes (Y, which extendsM) with dynamics dY/dt, 159\nrequires an input term from human malaria epidemiology: the net infectiousness of 160\nhumans (NI), the probability a mosquito becomes infected after blood feeding on a 161\nhuman (denoted κ). A state space model describing parasite infection dynamics in 162\nhumans, immunity, and disease (X ) with dynamics dX /dt, requires an input term from 163\nadult mosquito infection dynamics: the daily EIR ( E). The inputs to one component 164\ncan be passed as trace functions or as the outputs of another coupled component, which 165\nis called the interface of each dynamical component; a generic interface is coded for 166\neach term and if needed specialized methods can be written for particular models. 167\nModels in the framework have the following form: 168\ndL/dt = FL (η,L)\ndM/dt = FM (Λ,M)\ndY/dt = FY (κ,M,Y)\ndX /dt = FX (E,X )\n. (1)\nThe interactions among these dynamical components are thus defined by four input 169\nterms (η, Λ, κ, and E), which may be computed as outputs of another component or 170\nprovided as an external forcing term (Fig. 1). Because these terms can be computed 171\nfrom the state of the model and are used to couple different model components together, 172\nwe call these dynamical quantities. These terms are rates which determine how 173\ncomponents interact (e.g., flows between components). Because construction of these 174\ndynamical quantities can be done in a generic way, computation of these quantities in 175\ncode can be done for any model which fulfills the interface of its dynamical component. 176\nThe dynamical quantities responsible for transfer of pathogens between hosts and 177\nvectors are E and κ, the EIR and NI of humans, respectively. These quantities couple 178\nthe dynamics between the human X and mosquitoY dynamical components. To allow 179\ncomputation of E and κ to be highly generic across various types of models of human 180\nand mosquito infection, we developed a new model of blood feeding which produces β, 181\nthe biting distribution matrix describing how bites arising from mosquitoes at patches 182\nare taken on human population strata. 183\n10\n . CC-BY 4.0 International licenseIt is made available under a \n is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity.(which was not certified by peer review)preprint \nThe copyright holder for thisthis version posted November 15, 2022. ; https://doi.org/10.1101/2022.11.07.22282044doi: medRxiv preprint \n\nSimilarly, the adultM and aquaticL mosquito components are coupled via egg 184\nlaying from adults in aquatic habitats, and emergence of new adults from those aquatic 185\nhabitats. Because the patches where adult mosquitoes are found may contain many (or 186\nno) aquatic habitats, another matrix translates the rate of egg production from adults 187\ninto egg deposition in each aquatic habitat η. Likewise, each aquatic habitat produces 188\nnewly emerging adult mosquitoes at some rate α, which in general depends on the 189\ncurrent aquatic population, and therefore on lagged adult densities. Another matrix 190\nmaps this into the rate at which new adults are added to each mosquito population, Λ. 191\nIn addition to reformulating blood feeding and egg laying, the framework includes 192\nmathematical descriptions of survival, search for blood hosts or habitats, and dispersal. 193\nThese new models of adult mosquito behaviors have all been reformulated around the 194\nconcept of heterogeneous resource availability and functional responses to available 195\nresources. 196\nFig 1. Models for malaria transmission dynamics are naturally modular (see Eq. 1).\nThe dynamic modules describe a stratified human population (purple) that interacts\nthrough blood feeding (red) with adult mosquito populations in a discrete spatial\ndomain; each patch could contain a set of aquatic habitats. Two components, L andM,\ndescribe mosquito ecology: dynamics of immature mosquitoes (blue) in aquatic habitats\nare described by a system of equations dL/dt; and dynamics of adult mosquitoes (green)\nare described by dM/dt. Habitat locations within patches are described by a\nmembership matrix,N . Eggs hatch into larval mosquitoes, that develop, pupate, and\nlater emerge from habitats as mature adults ( α) and added to the adult populations in\neach patch (Λ). Adults lay eggs ( ν), which are distributed spatially according to which\npatch habitats belong (N ). Egg deposition rates at the habitats are ( η). Two additional\ncomponents,Y andX , describe parasite infection dynamics and transmission: that for\nmosquitoes, described by dY/dt and in humans, described by dX /dt, are linked through\nparasite transmission. A new model for blood feeding describes how blood meals are\nallocated among humans ( β) and associated parasite transmission rates: the density of\ninfectious humans by strata ( X) is used to compute net infectiousness (NI) of humans\nto mosquitoes in patches ( κ); and the density of infectious blood feeding mosquitoes ( Z)\nis used to compute the entomological inoculation rate (EIR) on each strata ( E).\nThe modular framework was implemented as a software package in R [43] 197\n(Supplement 1 is the website https://dd-harp.github.io/exDE/). The software 198\nbuilds dynamical models of malaria in a modular way using method dispatch to define 199\ngeneric code which implements the framework described here. The dynamical models 200\nare functions which return arrays of derivatives of state variables, and can be solved 201\nusing the integrators available in deSolve, or other tools in R [43,44]. The software also 202\n11\n . CC-BY 4.0 International licenseIt is made available under a \n is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity.(which was not certified by peer review)preprint \nThe copyright holder for thisthis version posted November 15, 2022. ; https://doi.org/10.1101/2022.11.07.22282044doi: medRxiv preprint \n\nincludes routines that compute steady state conditions and spatial metrics (see Spatial 203\nMetrics, below). Because each component has an interface – the generic functions that 204\ncompute and pass of dynamical quantities between components – any new model can be 205\nimplemented which fulfills a specific interface, independent of the rest of the framework. 206\nIn this way, building and testing new models of particular components is straightforward, 207\nand the framework is flexible and extensible. As new models are required, they will be 208\nadded to the package, increasing its applicability and scope over time. 209\nWe have developed a glossary of terms (Supplement 2). In the equations that follow, 210\nfor each dynamical component, we describe the model structure in detail, and we 211\npresent one family of models describing transmission dynamics in a single vector species. 212\nIn Supplement 3, we formulate a model using both conventional notation and the 213\nmodular notation of this framework. In Supplement 1, we have implemented a 214\npreviously published model of malaria transmission on Bioko Island [45]. In Supplement 215\n4, we extend the discussion of vector dynamics, including a discussion of models with 216\nmultiple vector species. All the terms and parameters may be time dependent to 217\naccommodate seasonality or modification by exogenous factors: seasonal travel, 218\nexogenous forcing by weather, and parameter modification by vector control. Analysis 219\nof temporal heterogeneity in this same framework is outside the scope of this study, it 220\nbut would be straightforward extension following approaches analogous to those shown 221\nin the supplements. 222\nBox 2: Notation Equations describing spatial processes include terms\ndescribing scalar quantities, vectors of scalars, vectors of functions, and\nmatrices. We have avoided using any notation to designate a vector or\nindicate it could be time-dependent, in part, because it would be ubiquitous;\nmost parameters could vary by space and time. The most general form of a\nterm or parameter is usually described when it is first presented, but most\nterms describing a vector or matrix should be assumed to be modifiable.\nIn writing out the equations, we consistently use the center dot, “\n·”, in\nequations to denote the dot product of two matrices, or a matrix and a\nvector. The juxtaposition of two vectors denotes element-wise product,\nand 1/∗ denotes the vector of the inverses of each element. The symbol ⊙\ndenotes the Hadamard product (i.e., element-wise multiplication) of two\nmatrices. When\nx is a vector, diag(x) is a matrix with the elements of x\non the main diagonal. The identity matrix is denoted I, and 1 denotes a\nrow or column vector with each element equal to 1. When F is a functional\nresponse, we assume it accepts vector arguments and returns a vector of\nthe same length, i.e.,|F (X)| =|X|. The glossary (Supplement 1) discusses\nthe dimensions of each term.\n223\n224\n12\n . CC-BY 4.0 International licenseIt is made available under a \n is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity.(which was not certified by peer review)preprint \nThe copyright holder for thisthis version posted November 15, 2022. ; https://doi.org/10.1101/2022.11.07.22282044doi: medRxiv preprint \n\nModel Structure 225\nThe following describes, in detail, the structural elements and the algorithms that 226\nconnect them. Adult mosquito and human population strata are connected through 227\nblood feeding and transmission, and adult and aquatic mosquito populations are 228\nconnected through egg laying and emergence. 229\nStructural Elements The framework has been designed to build model ensembles 230\nwith the goal of studying the spatial transmission dynamics of malaria in a defined 231\ngeographical area, called the spatial domain. An important part of this framework is 232\nhaving flexibility in defining the model structure to describe spatial and population 233\nheterogeneity at the appropriate level of detail, depending on the needs of a study and 234\nthe available data. The structural elements – the patches, the aquatic habitats, and the 235\npopulation strata – were designed to handle arbitrary patch definitions, arbitrary 236\nhuman population residency patterns and stratification, and arbitrary numbers and 237\nlocations of aquatic habitats. 238\nTo deal with spatial heterogeneity in transmission, we subdivide the spatial domain 239\nand identify a set of p patches that includes all locations relevant for studying and 240\nquantifying mosquito ecology or transmission: places where people live; places where 241\nmosquitoes blood feed; or places with aquatic habitats where mosquitoes lay eggs. We 242\nassume that there are l aquatic habitats with actual physical locations that are nested 243\nwithin the patches. To deal with heterogeneity in the human population, the model 244\naccommodates stratification. The human population is sub-divided into a set of n 245\npopulation strata by residency, immunity, behaviors affecting risk, or any other 246\nepidemiologically relevant factors (Supplement 5). Human populations are assigned a 247\nsingle residency patch, where they live and spend most of their nights. Other 248\nsubdivisions of the human population could take into account age, sex, travel patterns, 249\nITN usage, or any trait that is heterogeneous and epidemiologically relevant. The total 250\ncensus population size, the number of people who reside in each patch in the spatial 251\ndomain, is given by a vector denoted P (of length p). The number of people in each 252\nstratum is given by a vector H (of length n). In this model, it is not necessary for every 253\npatch to have some residents. 254\nTo manage terms for interactions among structural elements, we create two 255\nmathematical objects called membership matrices that aggregate quantities to patches 256\n(Supplement 3). Since the l aquatic habitats are scattered among the patches, we define 257\nthe habitat membership matrix N , a p× l matrix, that aggregates quantities from the l 258\naquatic habitats to p patches where they are found. Similarly, we define the strata 259\nmembership matrixJ , a p× n matrix, that aggregates the n human population strata 260\nto the p patches where they reside. The census population size, for example, is 261\nP =J· H. If a human population were stratified by other traits, such as frequent travel 262\nor age, a membership matrix could be created to aggregate model output by trait. 263\nThe framework has also been designed to accommodate models with multiple 264\nmosquito vector species or types (see Supplement 4). Most of the following discussion 265\nassumes there is just one vector species, but we point out where the framework has can 266\ngeneralize to multiple vector species. 267\n13\n . CC-BY 4.0 International licenseIt is made available under a \n is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity.(which was not certified by peer review)preprint \nThe copyright holder for thisthis version posted November 15, 2022. ; https://doi.org/10.1101/2022.11.07.22282044doi: medRxiv preprint \n\nHuman Mobility After defining the model structure (i.e., the patches and 268\npopulation strata), the next challenge is to construct the algorithms describing local 269\nhuman mobility and travel. Local mobility determines where and when humans are 270\navailable and exposed to blood feeding mosquitoes within the spatial domain. We define 271\ntravel in this model by time spent outside the spatial domain; travel and mobility are 272\nthus different modalities and handled with different constructs. 273\nTo model local human mobility patterns within the patches, we develop a model 274\ndescribing the fraction of time spent by humans in each stratum among the 275\npatches [9,18]. The information is summarized in a time-dependent p× n matrix Θ(t), 276\ncalled the Time Spent (TiSp) matrix (Supplement 5). Each column in a TiSp matrix 277\ndescribes the fraction of time spent in each patch by an individual from a single 278\nstratum. In formulating the TiSp matrix, we account for time spent by time of day in 279\nthe patches where mosquitoes are blood feeding. Total time spent should subtract time 280\nspent traveling and and time spent in the spatial domain in places where there is no risk 281\n(e.g., in office buildings). 282\nBlood feeding combines human and mosquito behaviors. Since mosquito blood 283\nfeeding has a daily rhythm [46], time at risk modifies time spent to account for 284\ndifferences in mosquito daily blood feeding activity rates. We let ξ(t) denote a 285\nspecies-specific circadian weighting function for blood feeding rates, constrained such 286\nthat\n∫1\n0 ξ(t)dt = 1, which appropriately assigns a weight to time spent by time of day 287\n(Supplement 5). Using ξ, we compute the Time At Risk (TaR) matrix as time spent 288\nweighted by mosquito activity: Ψ( t) = diag (ξ(t))· Θ(t). 289\nThis distinction between TiSp and TaR matrices makes it possible to study human 290\nmosquito contact in detail, to quantify differential transmission by multiple vectors with 291\nthe same human mobility patterns, and to quantify other aspects of mosquito-human 292\ncontact [47,48]. A model could have two or more vector species, each with different 293\nblood feeding patterns ( ξ1 and ξ2), so that one TiSp matrix would be transformed into 294\ntwo different TaR matrices (Ψ1 = ξ1Θ and Ψ 2 = ξ2Θ). 295\nDenominators and Availability After defining host population movement, it is 296\nnecessary to compute appropriate denominators to model blood feeding, based on the 297\nmodels for time spent and time at risk. Because of mobility, mosquito preferences, and 298\nhuman behaviors, the denominators for blood feeding are different from the resident 299\npopulation size – the number that would be used by most studies (Fig. 2). 300\nAn important intermediate quantity is ambient population density, which describes 301\nthe population present in patches at a point in time. In a mobile population, the 302\nambient population density will tend to be different from resident population density. 303\nFrom the time spent matrix, the ambient population density is a vector of length p 304\ngiven by: 305\nA(t) = Θ(t)· H. (2)\nSimilarly, ambient population density at risk is given by: Ψ( t)· H. One way to 306\nunderstand what the TiSp matrix means is by taking ratios of ambient to resident 307\npopulations. The ambient density of residents is Ar = (J⊙ Θ)· H, where⊙ denotes the 308\nHadamard (element-wise) product. The non-resident, non-visitor, ambient population is 309\nA− Ar. The ratios of various census and ambient population densities (e.g., the ratio of 310\n14\n . CC-BY 4.0 International licenseIt is made available under a \n is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity.(which was not certified by peer review)preprint \nThe copyright holder for thisthis version posted November 15, 2022. ; https://doi.org/10.1101/2022.11.07.22282044doi: medRxiv preprint \n\nFig 2. Denominators and Mixing A schematic diagram relating various concepts of\npopulation density under a model of human mobility, resulting in a biting distribution\nmatrix, β. Here, and and in Figures 3-6, rounded rectangles denote endogenous state\nvariables, sharp rectangles denote endogenous dynamical quantities, and parallelograms\nrepresent exogenous data or factors. Purple indicates the element is related to human\npopulations, green for mosquitoes, and red for biting and transmission. Population\nstrata (H ) describe how persons are allocated across demographic characteristics. The\nmatrixJ distributes these strata across space (patch), according to place of residency.\nBy combining information on how people spend their time across space (Θ( t)) and\nmosquito activity (ξ(t)) a time at risk (TaR) matrix Ψ is generated describing how\nperson-time at risk is distributed across space. Because blood feeding can be modified\nby human and mosquito factors ( e.g., net use and biting preferences), search weights\n(wf(t)) may further weight person-time at risk. The final result is a biting distribution\nmatrix β, which is the fraction of each bite in each patch that would arise on an\nindividual in each stratum, so diag(H )· β = 1.\nresidents to ambient population P/A, defined wherever A > 0), can be used to 311\nunderstand and diagnose unrealistic terms in a TiSp or TaR matrix. The ambient 312\npopulation thus provides one easy statistic to understand TiSp or TaR matrices. 313\nTo model the denominators for blood feeding, we also consider other factors – 314\nmosquito preferences or human behaviors or traits such as ITN usage – that affect host 315\navailability to mosquitoes and relative biting rates on the strata [33]. We assign biting 316\nweights, wf, to each strata, where we think of wf = 1 as the value that would be 317\nassigned to an average person under baseline conditions ( e.g., without a net). These 318\nweights affect both the total biting rates and the relative biting rates on the ambient 319\npopulation. We define the availability of the host populations to mosquitoes for blood 320\nfeeding as: 321\nW = Ψ· wf H. (3)\nAvailability is thus defined in units of weighted person-days at risk, and W is a vector 322\nof length p describing total human availability in each patch. 323\nWe also consider the presence of a population of visitors, a non-resident population 324\nspending time in the spatial domain (Supplement 5). We assume that some visitors 325\ncould be present, and that some of them could be infectious. We can let Aδ denote the 326\nambient density of visitors, but we let Wδ denote their availability by patch. The 327\nresident fraction or fraction of human blood meals taken on a resident in each patch, a 328\nvector of length p denoted υ, is: 329\nυ = W\nW + Wδ\n. (4)\n15\n . CC-BY 4.0 International licenseIt is made available under a \n is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity.(which was not certified by peer review)preprint \nThe copyright holder for thisthis version posted November 15, 2022. ; https://doi.org/10.1101/2022.11.07.22282044doi: medRxiv preprint \n\nThe total availability of humans for blood feeding, in each patch, is thus W + Wδ. 330\nFig 3. Blood F eeding and Human Biting Rates The daily human biting rates\n(HBR) for the resident population strata are defined as the expected number of bites by\nvectors, per person, per day. To compute the HBR, we count up exposure over all the\npatches where residents spend time. We also consider the presence of visitors and other\nbloodhosts (yellow input), which increases the total available hosts.\nBlood F eeding With a well-defined population denominator, we can compute the 331\nfrequency of blood feeding rates and the human fraction (i.e., the fraction of human 332\nblood meals among all blood meals) in each patch in response to the availability of 333\nhumans and other available vertebrate hosts. To do so, we use functional responses to 334\nmodel blood feeding rates and habits [33–36]. 335\nHuman availability, W , is often highly variable among patches and over time, which 336\ncould affect the rate mosquitoes blood feed (Fig. 3). Mosquitoes could also feed on other 337\nvertebrate hosts. To model blood feeding, we supply a vector of functions describing the 338\navailability of non-human vertebrate hosts in each patch over time, denoted O(t). We 339\nassume that mosquito preferences could scale with host densities, so we assign a shape 340\nparameter, ζ, that modifies how preferences scale with host densities. Total availability 341\nof all vertebrate hosts for blood feeding is B = W + Wδ + Oζ (Supplement 5). 342\nLet f(t) denote the blood feeding rate, the number of blood meals, per mosquito, 343\nper day. To guarantee mathematical consistency in computing blood feeding rates (e.g., 344\nif B = 0, then it should be true that f = 0), we can model time-dependent blood 345\nfeeding rates, where f(t) is a vector of length p, as: 346\nf(t) = Ff(B) = fx\nsf B\n1 + sf B . (5)\nDepending on a shape parameter(s), sf, blood feeding rates increase with host 347\navailability up to a maximum (or maxima) fx, which is limited by the time it takes to 348\nsearch, process the blood meal, lay eggs, and perhaps to sugar feed. The fraction of 349\nblood meals taken on humans at a point in time, a vector of length p denoted q(t), is 350\ncalled the human blood feeding fraction or human fraction: 351\nq(t) = W + Wδ\nB . (6)\nThe local human fraction, the fraction feeding on resident humans, is thus υq = W/B. 352\nThe functional forms guarantee that when no humans are present, it must be true that 353\nf q = 0; and when only humans are available, it must be true that q = 1. 354\n16\n . CC-BY 4.0 International licenseIt is made available under a \n is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity.(which was not certified by peer review)preprint \nThe copyright holder for thisthis version posted November 15, 2022. ; https://doi.org/10.1101/2022.11.07.22282044doi: medRxiv preprint \n\nMixing and Parasite T ransmission The model for mixing is an answer to the 355\nquestion: How are blood meals in a patch allocated among humans in the strata? The 356\ntime at risk matrix and the factors affecting blood feeding rates and habits in each 357\npatch must be consistent with the algorithm that computes the distribution of biting 358\nand parasite mixing. 359\nTo allocate mosquito bites in patches among the resident strata, we let β denote a 360\nn× p biting distribution matrix: 361\nβ(t) = diag(wf)· ΨT· diag\n( 1\nW (t)\n)\n. (7)\nEach column of β describes the fraction of a bite in a patch that lands on an individual 362\nin each strata, so the matrix diag(H )· β gives the fraction of bites that land on each 363\nstratum, and its columns sum to unity. 364\nIn the models for mosquito ecology and infection dynamics, we define variables 365\n(vectors of length p) for the density of mosquitoes ( M) and infectious mosquitoes (Z ). 366\nFrom these, we derive an expression for the daily human biting rate (HBR) and 367\nentomological inoculation rate (EIR) for all the strata. The sporozoite rate (SR) in each 368\npatch is given by: 369\nz = Z\nM . (8)\nThe net per-capita human blood feeding rates in each patch, or f qM/W, are hereafter 370\ncalled the patch HBR (pHBR), and f qZ/W is hereafter called the patch EIR (pEIR) 371\nfor infectious mosquitoes. By way of contrast, exposure risk for the strata – the HBR 372\nand EIR – are defined as the number of bites / infectious bites by vectors, per person, 373\nper day. The HBR is β· f qυM, and the EIR is the product of the HBR and the SR, or 374\nE = β· f qυZ. (9)\nTo draw a sharp contrast between the terms, the pHBR and pEIR describe the number 375\nof bites / infectious bites, per person, in patches. They are stratified by location, so 376\nthey are vectors of length p. The HBR and the EIR are stratified quantities that sum 377\nexposure over all locations for the strata, so they are vectors of length n. 378\nEach model for parasite infection dynamics in humans defines a quantity, x, the 379\nprobability a mosquito becomes infected after biting a human in each stratum. The 380\nquantity X = xH, a vector of length n, is herein called the infective density of 381\ninfectious human residents. We can also specify the infective density of visitors, Xδ 382\nwhere Xδ = xδWδ is intrinsically using the availability of visitors. The net 383\ninfectiousness (NI) for the mosquito populations in all the patches, denoted κ, is: 384\nκ = υβT· X + (1− υ)Xδ (10)\nThe force of infection for the mosquito population is thus f qκ. 385\nEgg Laying To compute quantities affecting mosquito ecology and population 386\ndynamics, we need to formulate algorithms to compute egg laying rates and egg laying 387\ndistributions: how many eggs are laid by adult mosquitoes in a patch, and how are they 388\n17\n . CC-BY 4.0 International licenseIt is made available under a \n is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity.(which was not certified by peer review)preprint \nThe copyright holder for thisthis version posted November 15, 2022. ; https://doi.org/10.1101/2022.11.07.22282044doi: medRxiv preprint \n\nFig 4. Egg Laying and Egg Deposition The availability of aquatic habitats (Q)\nthe patch sum of habitat search weights ( Q =N· wν), and the egg distribution matrix\n(U) describes the locally normalized search weights. Available habitat determines\nper-capita oviposition rates ( ν) by the population of gravid mosquitoes ( G) in a patch\nthrough a functional response to availability, Fν(Q). The net egg laying rate, per-patch,\nis Γ = χνG. The eggs are distributed among the aquatic habitats (U ) so that the egg\ndeposition rates in habitats is η =U· Γ.\ndistributed among the aquatic habitats in that patch? To do so, we develop the concept 389\nof habitat availability. We assign a search weight to each aquatic habitat, wν. Using the 390\npatch membership matrix,N , we define aquatic habitat availability as: 391\nQ(t) =N· wν(t) (11)\nFor each patch, total habitat availability is the sum of the search weights for habitats in 392\nthat patch. 393\nDaily, per-capita oviposition rates of gravid mosquitoes are computed using a 394\nfunctional response to habitat availability, such as: 395\nν = Fν(Q) = νx\nsνQ\n1 + sνQ . (12)\nwhere νx is the highest possible egg-laying rate for a gravid female, and sν is a shape 396\nparameter. We note that if Q = 0, then ν = Fν(0) = 0. We let G = FG(M) denote the 397\ndensity of gravid mosquitoes, and we let χ denote the number of eggs laid, per batch. 398\nThe net egg laying rate, per patch, per day, is: 399\nΓ = χνG (13)\nTo model egg distribution among habitats, we formulate an egg distribution matrix ( U) 400\nthat allocates eggs to habitats in proportion to local habitat availability. To compute U, 401\nfor computational reasons we first create Q∗ by setting any zero entries to an arbitrary 402\npositive value (if Q = 0, then ν = 0, so associated products will later be multiplied by 403\nzero), and the egg deposition rate, η, is computed by: 404\nU (N , wν) = diag(wν)·N T· diag\n( 1\nQ∗\n)\n. (14)\nFinally, we can compute egg deposition rates in the habitats: 405\nη =U· Γ (15)\n18\n . CC-BY 4.0 International licenseIt is made available under a \n is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity.(which was not certified by peer review)preprint \nThe copyright holder for thisthis version posted November 15, 2022. ; https://doi.org/10.1101/2022.11.07.22282044doi: medRxiv preprint \n\nWhile Γ (a vector of length p) describes the net egg-laying rate of the adult mosquito 406\npopulation in each patch, per day η (a vector of length l) describes the number of eggs 407\nlaid, in each habitat, per day. 408\nCore Dynamical Components 409\nThe dynamical quantities whose computation was described above, are configurable 410\nelements that connect the four dynamical components: aquatic mosquito ecology; adult 411\nmosquito ecology and infection dynamics; and infection and immunity, including human 412\ndemography. In the following, we describe one model family for each component, 413\nincluding functions that compute terms required for the dynamical quantities; in code 414\nthese are the generic interface of each dynamical component. In Supplement 1 415\n(https://dd-harp.github.io/exDE/), we have formulated alternative model families for 416\nsome of the components. 417\nAquatic Mosquito Ecology The first core dynamical component describes aquatic 418\nmosquito population dynamics; the algorithm computes mosquito survival and 419\ndevelopment from eggs laid through adults emerging. For aquatic population dynamics, 420\nwe here adapt a previously published model [31,32]. 421\nLet L(t) denote the total density of immature mosquitoes. We let ψ(t) denote 422\nmaturation rates, φ(t) the density independent mortality rate, and θ(t)L(t) describes 423\nincreased per-capita mortality due to mean crowding. The aquatic dynamics are thus: 424\ndL\ndt = η− (ψ + φ + θL)L (16)\nThe total emergence rate of female mosquitoes in this model, per aquatic habitat, is: 425\nα(t) = Fα (L (t)) = ψ(t)L(t)\n2 . (17)\nThese are recruited into the adult population in the patch, so that the net emergence 426\nrate per patch is: 427\nΛ(t) =N· α (18)\nWhile α is a vector of length l, Λ is a vector of length p. This is passed as input to the 428\nequations describing adult populations (below). 429\nGiven uncertainty about the factors affecting immature mosquito populations, we 430\nassume studies might choose to formulate and analyze alternative dynamics. Other 431\ndynamical systems models for aquatic ecology in the framework are defined by state 432\nvariables,L, with dynamics defined by a system of equations dL/dt = η− FL(L), and a 433\nfunction such that α = FΛ(L), such that Λ =N· α (Supplement 4). 434\nAdult Mosquito Ecology The second core dynamical component describes adult 435\nmosquito ecology. Given all the functions, terms and parameters above, we have 436\nformulated a set of algorithms describing adult mosquito mortality and dispersal that 437\nare internally consistent. All this is embodied in the mosquito demographic matrix, 438\ncalled Ω(t). 439\n19\n . CC-BY 4.0 International licenseIt is made available under a \n is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity.(which was not certified by peer review)preprint \nThe copyright holder for thisthis version posted November 15, 2022. ; https://doi.org/10.1101/2022.11.07.22282044doi: medRxiv preprint \n\nFig 5. Adult mosquito demography is defined by survival and dispersal. Mobility\nrates and dispersal are determined by the available of resources: aquatic habitats (Q),\navailable humans (W + Wδ) and other blood hosts ( Oζ), and sugar ( S). The emigration\nrate is a functional response ( Fσ) that increases if any one of the resources is missing.\nResource availability and distance also play a role in computing the dispersal kernel, K,\nthat determines where mosquitoes land if they leave a patch. When combined with\nmortality, a matrix Ω is produced which describes the behavior of adult mosquitoes\nafter emergence.\nWe assume mosquito mobility is driven by a search for resources. We have already 440\ndefined total blood host availability B, and aquatic habitat availability Q. We also 441\nconsider sugar availability, S(t), which is passed to the model as a function vector of 442\nlength p. We assume mosquitoes leave a patch while searching for resources, and that 443\nthey leave a patch more frequently if the resources are less available. Patch-specific 444\nemigration rates, σ(t), are a functional response to resource availability: 445\nσ = Fσ(B, Q, S) = σx\n( σB\n1 + sBB + σQ\n1 + sQQ + σS\n1 + sSS\n)\n(19)\nThe parameters σB, σQ, and σS determine the rate that mosquitoes leave a patch if no 446\nresources are available, and the shape parameters sB, sQ, and sS determine how the 447\nrate of patch leaving is reduced by the availability of resources. The shape parameter σx 448\nis a scaling parameter that can be used to adjust models with differing patch sizes. 449\nSimilarly, we formulate a mosquito dispersal matrix, K(t) that describes where 450\nmosquitoes land after they leave each patch (the diagonal elements of K are constrained 451\nto be equal to zero, Supplement 4). 452\nWe let g(t) denote the local per-capita mortality rate of mosquitoes in each patch. 453\nThe matrix Ω(t) describes adult mosquito survival and dispersal: 454\nΩ = diag(g) + (I−K )· diag(σ) (20)\nwhere I is the identity matrix. 455\nWe let Λ(t) be the net emergence rate of mosquitoes into the patches from aquatic 456\nhabitats (see Eq. 18, above). The dynamics of adult mosquitoes are described by the 457\nequation: 458\ndM\ndt = Λ− Ω· M (21)\n20\n . CC-BY 4.0 International licenseIt is made available under a \n is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity.(which was not certified by peer review)preprint \nThe copyright holder for thisthis version posted November 15, 2022. ; https://doi.org/10.1101/2022.11.07.22282044doi: medRxiv preprint \n\nUnder the assumptions of this model, the density of gravid mosquitoes, G, is: 459\ndG\ndt = f(M− G)− νG− Ω· G (22)\nThis model thus assumes that only gravid mosquitoes can lay eggs (Eq. 13), but that all 460\nmosquitoes (including gravid mosquitoes) can blood feed. 461\nOther models for adult mosquito ecology, denoted dM/dt, could be formulated that 462\ndescribe separate functions for mosquito survival and dispersal, depending on their 463\nbehavioral states (possibly including sugar feeding, mating and maturation), or that 464\ndescribe a mosquito’s reproductive states, or its chronological age or reproductive age. 465\nAll models developed in this framework must accept the adult emergence rates, Λ, and 466\nthey must be formulated in enough detail to specify a population of egg-laying 467\nmosquitoes, G, to compute ν (see Eq. 13). 468\nParasite Infection Dynamics in Mosquitoes The third core dynamical 469\ncomponent describes parasite infection dynamics in adult mosquito populations. Here, 470\nwe extend a previously published delay differential equation for the density of infectious 471\nmosquitoes to include space and a time-varying extrinsic incubation period (EIP) [49]. 472\nLet Y (t) denote the density of infected mosquitoes. Using κ from Eq. 10, the 473\ndynamics of infection in mosquitoes are described by: 474\ndY\ndt = f qκ(M− Y )− Ω· Y (23)\nWe include a time-dependent EIP so that parasite development can be modulated by 475\ntemperature or other factors exogenous to the system: let τ(t) denote the EIP for a 476\nmosquito that becomes infected at time t (i.e., it becomes infectious at time t + τ(t), 477\nSupplement 4). We must also define the inverse τ−1(t), the delay for a mosquito that 478\nbecame infectious at time, t. Let Υτ(t) denote a matrix describing survival and dispersal 479\nof a cohort from time t− τ−1(t) through the EIP to become infectious at time t: 480\n− ln Υτ(t) =\n∫ t\nt−τ −1(t)\nΩ(s)ds. (24)\nWhen Ω and τ are constant, survival and dispersal through the EIP is Υ τ = e−Ωτ. 481\nOtherwise, let the τ-subscript denote the value of a variable or parameter at time 482\nt− τ−1(t). 483\nTo model the density of infectious mosquitoes, let Z(t) denote the density of 484\ninfectious mosquitoes. The dynamics of infectious mosquitoes are: 485\ndZ\ndt = Υτ· fτ qτ κτ(Mτ− Yτ)− Ω· Z (25)\nThe number of human blood meals per patch, called the net infectious biting rate, is 486\nf qZ. 487\nModels for infection dynamics, generically denoted dY/dt are nested within the 488\nmodel for adult mosquito population dynamics dM/dt (for example, see [50]). These 489\nmodels accept the net infectiousness (κ), they must define a variable describing the 490\n21\n . CC-BY 4.0 International licenseIt is made available under a \n is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity.(which was not certified by peer review)preprint \nThe copyright holder for thisthis version posted November 15, 2022. ; https://doi.org/10.1101/2022.11.07.22282044doi: medRxiv preprint \n\ndensity of infectious, blood feeding mosquitoes, Z, in order to compute the EIR (see 491\nEq. 9). 492\nEpidemiology The fourth core dynamical component describes parasite infection 493\ndynamics in human populations. Models for malaria infection, immunity, disease, and 494\ninfectiousness in humans, denoted dX /dt, can become quite complicated, depending on 495\nthe needs of a study. Studies of malaria epidemiology could consider the complex time 496\ncourse of infections, superinfection, disease, detection, infectiousness, and immunity. 497\nThe state space describing malaria infection and immunity X can be modified to suit 498\nthe needs of a study, and the framework also has enormous flexibility to model 499\nheterogeneity in populations through stratification. The following is one model family 500\nthat is complex enough to illustrate the generic features of the framework. 501\nLet h = fh(E) denote the local daily force of infection (FoI) and δ(t) the FoI during 502\ntravel. In general, fh(E) could be modified to include heterogeneous biting [41], but in 503\nthis model, we assume h = bE. Both terms are defined for each sub-population. In 504\nthese models, we stratify on variables relevant for the epidemiology, including immunity, 505\nand we model the effects by assigning different parameter values to each stratum. 506\nTo model infection dynamics, we modify a hybrid model for the multiplicity of 507\ninfection (MoI). The dynamics are based on a queuing model, in which new infections 508\noccur at the rate h, and each parasite clears at the rate r, where we track apparent and 509\nactual clearance as linked but distinct processes. The variables m1 and m2 track the 510\nmean MoI for present and detectable parasites in each strata, which fully describe the 511\nepidemiological state space in a simple model with superinfection [51]. We assume that 512\nparasites clear at the per-capita rate, r1, so that: 513\ndm1\ndt = h + δ− r1m1 (26)\nIn this model, the true prevalence is: 514\nx1 = 1− e−m1 (27)\nWe also formulate a model for the MoI of apparent infections. We assume parasite 515\ninfections are detectable for a shorter time so they appear to clear at a higher rate, r2, 516\nand 517\ndm2\ndt = h + δ− r2m2 (28)\nSimilarly, we let x2 denote the apparent prevalence 518\nx2 = 1− e−m2 (29)\nWe assume that if the infection is patent, a bite infects a mosquito with a higher 519\nprobability, c2, and c1 if it is not. A bite on a person in each stratum infects a mosquito 520\nwith probability: 521\nx = c2x2 + c1(x1− x2) (30)\nTo compute κ, the infective density of infectious resident hosts by strata is X = xH. 522\nThe vector X is passed to Eq. 10 to compute a vector of patch-specific net 523\n22\n . CC-BY 4.0 International licenseIt is made available under a \n is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity.(which was not certified by peer review)preprint \nThe copyright holder for thisthis version posted November 15, 2022. ; https://doi.org/10.1101/2022.11.07.22282044doi: medRxiv preprint \n\ninfectiousness, κ. 524\nTo compute some of the spatial transmission metrics, including basic reproductive 525\nnumbers (see below), a model must compute the human transmitting capacity 526\n(HTC) [52]. In this model, the number of days infecting mosquitoes at the higher 527\nprobability, c2 is 1/r2. The remaining days, are spent infecting mosquitoes at the lower 528\nprobability. Expressed as the equivalent number of perfectly infectious days, the HTC is: 529\nD = c2\nr2\n+ c1\n( 1\nr1\n− 1\nr2\n)\n(31)\nThis framework can accommodate other systems of equations describing parasite 530\ninfection and immune dynamics in humans. This particular model was designed to 531\nillustrate some basic features of the modular design. These particular equations were 532\ndesigned to incorporate the effects of immunity on transmission through stratification, 533\nallowing parameters describing the duration of infections or detection and the 534\ninfectiousness to vary among strata ( e.g., r1, r2, c1 and c2). New models for human 535\nepidemiology can use any epidemiological state space, X , and any system of equations, 536\ndX /dt, including models with dynamical changes in the host population size. While the 537\ntravel FoI is recommended, it is not required. The modules should accept the EIR, and 538\nto interact with other components, they must provide a function to compute the 539\ninfective density of infectious hosts, X. 540\nSpatial Metrics 541\nThe Ross-Macdonald model defined a set of concepts and metrics that have formed a 542\nbasis for measuring and understanding malaria transmission, including vectorial capacity 543\nand the basic reproductive number R0, but that model and associated metrics did not 544\ninclude metrics for spatial dynamics, parasite dispersal, or malaria importation [3]. 545\nHere, we define parasite dispersal by the set of locations ( i.e. patches) where 546\ninfecting bites occurred in continuous chains of transmission stretching back in time. 547\nDispersal for any parasite transmission chain is thus defined by locations of the bites 548\nthat caused each infection, and dispersal alternating between moving humans and 549\nmosquitoes between bites. We acknowledge that, due to an observational process, there 550\nis an important difference between where an infection occurred and where an infectious 551\nperson or mosquito is found. There is also an important difference between the formulas 552\ndefining dispersal and those used to compute reproductive numbers, which count from 553\nafter a host becomes infectious. Using this definition of parasite dispersal in the context 554\nof a model, we have developed formulas and metrics to compute and study parasite 555\ndispersal and reproductive success. 556\nTo develop these metrics, we assume steady state conditions. This is done for 557\nconvenience to avoid discussing the complications of understanding spatial dispersal 558\nunder dynamically changing conditions, and it is a necessary first step to understanding 559\nsuch models. Analysis of malaria transmission dynamics under temporally varying 560\nconditions are being developed in a subsequent manuscript. 561\nThe formulation of this static model helps to clarify the role of some of the 562\nintermediate terms – if all parameters in a model were constant, the transmission model 563\n23\n . CC-BY 4.0 International licenseIt is made available under a \n is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity.(which was not certified by peer review)preprint \nThe copyright holder for thisthis version posted November 15, 2022. ; https://doi.org/10.1101/2022.11.07.22282044doi: medRxiv preprint \n\ncould be fully defined by a much smaller set of parameters, but it may not be clear why 564\nthe parameters take on those values. Some of the terms that appear in the static 565\nanalysis correspond to parameters or variables in some Ross-Macdonald models, while 566\nothers are new: net emergence rates (Λ) or adult mosquito density (M ), scaled to the 567\nappropriate human population density denominator of host availability ( W ), mosquito 568\nbionomics (f, q, and Ω), and epidemiological parameters ( r1, r2, c1, and c2). New terms 569\ndescribe the spatial biting distribution matrix ( β) and parameters describing malaria 570\nimportation (δ, υ, and xδ). 571\nIn models where the context is changing dynamically – due possibly to weather, land 572\nuse changes, or vector control – exogenous forcing functions can be passed to the model 573\nthat change resource availability or that perturb the dynamics; the functional forms and 574\nintermediate terms (e.g. availability) are used to describe changes in the local 575\nparameter values and guarantee mathematical consistency. In these static models, the 576\nfunctions and terms are used to set up the model, but after setting parameter values, 577\nthey need not be called again. 578\nFig 6. To model malaria importation, we define a travel FoI for each stratum, δ(t), and\ntwo set of terms to model the role of visitors in mosquito blood feeding and parasite\ntransmission: the available visitor population Wδ and the NI for the visitor population,\nby patch xδ. To model blood feeding and transmission, we compute a patch-specific\nresident fraction for blood feeding, υ, the fraction of all biting that occurs on a resident\nof the spatial domain. From this, we can compute the visitor reservoir fraction, γ, the\ntravel fraction for incidence, and other measures of malaria importation.\nNet Malaria Importation and Travel Fractions 579\nTerms describing the travel FoI (δ) and visitor populations were defined above and 580\nintegrated into the models for blood feeding and human epidemiology. We define an 581\nimported malaria case as a human infection that traces back to a location outside of the 582\nspatial domain in the parasite’s previous generation, i.e., the mosquito and human host 583\npreceding this one in a chain of infections [42]. Net malaria importation rates describe 584\nthe number of imported malaria cases, per day. 585\nThe fraction of all cases that were imported called the travel fraction can be defined 586\nas either: 1) the fraction of incident infections that were imported; or 2) the fraction of 587\nprevalent infections that were imported [45,53]. To compute these travel fractions, we 588\n24\n . CC-BY 4.0 International licenseIt is made available under a \n is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity.(which was not certified by peer review)preprint \nThe copyright holder for thisthis version posted November 15, 2022. ; https://doi.org/10.1101/2022.11.07.22282044doi: medRxiv preprint \n\nlet γ = (1− υ)xδ/κ denote the visitor fraction, the fraction of infectious mosquitoes that 589\nwere infected by visitors. We let h denote the FoI. The travel fraction for incidence is: 590\nhγ + δ\nh + δ (32)\nThe travel fraction for true prevalence is: 591\n1− e−(δ+hγ)/r1\n1− e−(δ+h)/r1\n(33)\nWe note that these are per-capita terms defined for the strata. The net malaria 592\nimportation rate, the number of imported malaria incidence per day for each patch is: 593\nJ· (hγ + δ) H (34)\nso the travel fraction for incidence for the patches would be: 594\nJ· (hγ + δ) H\nJ· (h + δ) H (35)\nFormulas for the travel fraction for prevalence are formulated in the same way. 595\nParasite Dispersal 596\nTo compute quantities related to parasite dispersal, from bite to bite, we focus on local 597\ntransmission, and we need some formulas that describe how mosquitoes move around in 598\nhumans and in mosquitoes. 599\nMosquito Dispersal and Steady States In these models, we can compute steady 600\nstate mosquito population density, assuming Λ is constant over time. At the steady 601\nstate of Eq. 21, 602\nM = Ω−1· Λ (36)\nHere, the inverse Ω−1 can be understood as a measure of time spent alive in each patch 603\nby mosquitoes emerging habitats in each patch. In other Markov chain models with 604\nfinite state space, it has also been shown that the elements of the matrix inverse can be 605\ninterpreted as residence times [54,55]. In the simpler Ross-Macdonald model, the 606\ninverse of a mortality rate, g, is a measure of time spent alive or the average mosquito 607\nlifespan [56,57]. The time spent alive interpretation of Ω−1 is more apparent if there is 608\nno movement: if we set σ = 0, then Ω−1 = diag(1/g). 609\nIn spatial models, the matrix Ω accounts for both survival and movement. To 610\nillustrate – and to demonstrate that if Ω is a sensible description of mosquito 611\ndemography, then the matrix inverse must exist – we construct a tracking matrix. Let 612\nΞ(t) denote a matrix that tracks cohorts of mosquitoes: 613\nΞ(t, M0) = e−Ωt· diag (M0) (37)\nIt describes the density of mosquitoes left from an initial cohort in each patch M0 that 614\nis found in each location at each point in time. There is a duality between the 615\n25\n . CC-BY 4.0 International licenseIt is made available under a \n is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity.(which was not certified by peer review)preprint \nThe copyright holder for thisthis version posted November 15, 2022. ; https://doi.org/10.1101/2022.11.07.22282044doi: medRxiv preprint \n\nequilibrium population density from Eq. 21 and time spent alive by a cohort, computed 616\nby integrating Eq. 37 (i.e. orbits of the related equation dM/dt =−Ω· M). Just as we 617\ncan compute g−1 =\n∫∞\n0 e−gtdt, we can compute: 618\nM = Ω−1· Λ =\n∫ ∞\n0\ne−Ωtdt· Λ (38)\nso that the steady state can be found by simply adding up the time spent alive in each 619\npatch by a cohort emerging from every other patch. Under generalized static conditions 620\n(i.e. σ > 0), Ω−1 can thus be interpreted as the average time spent alive in every patch 621\nby cohorts of mosquitoes initially found in each patch. 622\nParasite Dispersal in Mosquitoes Using mosquito tracking matrices, we can also 623\ntrack parasite dispersal in mosquitoes to derive a matrix that has the same 624\ninterpretation as the formula for vectorial capacity [57,58]. 625\nTo transmit, mosquitoes must blood feed on a human to become infected: the net 626\ninfection rate in each patch, per available human, is f qκM/W. After becoming infected, 627\na mosquito must survive while dispersing through the EIP (Υ = e−Ωτ). After becoming 628\ninfectious, a mosquito must blood feed to transmit parasites, so we use the matrix 629\ninverse Ω−1 which describes where the mosquitoes are for each infectious human blood 630\nmeal as long as they remain alive; after becoming infectious, the distribution of 631\ninfectious bites is given by f qΩ−1. We can describe parasite transmission by mosquitoes 632\nby following the story of infection in mosquitoes: after emerging ( diag(Λ)), a mosquito 633\nmust blood feed on a human to become infected ( f qΩ−1/W ); then survive the EIP 634\n(e−Ωτ); and then blood feed to transmit ( f qΩ−1). 635\nIn the Ross-Macdonald model, the formula for vectorial capacity can be derived from 636\nthe formula for the daily EIR as a limit [57]. In spatial models, a vectorial capacity 637\nmatrix can be derived as the limit of a tracking matrix describing the number of 638\ninfectious bites arising, per available person (i.e., the denominator is W ), per day at the 639\nsteady state (Supplement 4): 640\nV = lim\nκ→0\nf qZ\nW = f qΩ−1· e−Ωτ· diag\n(f qM\nW\n)\n(39)\nElements in the matrix V are the expected number of infectious bites eventually arising 641\nin every patch from all the mosquitoes in a single patch blood feeding on a single human 642\non a single day, computed as if each human were perfectly infectious. The derivation 643\nassumes that no mosquitoes are already infected, and the assumption that humans are 644\nperfectly infectious ( κ = 1) is made so that the formula deals only with phenomena 645\nrelated to mosquitoes. In models with multiple vector species, the notion of what it 646\nmeans to be “perfectly infectious” is not as simple because of differences among vector 647\nspecies in their capacity to be a host for the parasites, or vector competence 648\n(Supplement 4). 649\nParasite Dispersal by Humans To quantify parasite dispersal by humans, we 650\ncompute the human transmitting capacity distribution (HTCD) matrix. We let human 651\ntransmitting capacity (HTC) describe the net number of perfectly infectious days for 652\n26\n . CC-BY 4.0 International licenseIt is made available under a \n is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity.(which was not certified by peer review)preprint \nThe copyright holder for thisthis version posted November 15, 2022. ; https://doi.org/10.1101/2022.11.07.22282044doi: medRxiv preprint \n\neach stratum: since infectiousness varies over the time-course of infections, we sum 653\npartially infectious days and interpret the HTC as an equivalent number of days spent 654\nperfectly infectious [52]. For the population strata in this model, the HTC ( D) is 655\ndefined by Eq. 31. Since transmission requires two bites, we use the TaR matrix to 656\ndetermine both where a human becomes infected and where it infects a mosquito. Using 657\nthe transposed TaR matrix, we can describe where infectious days at risk are spent, 658\nΨT· D. Parasite dispersion by mosquitoes for the sub-populations also accounts for 659\nwhere a mosquito becomes infected, or bΨ. 660\nThe HTCD matrix uses the biting distribution matrix, β, to count from the 661\ninfectious bite and weight biting appropriately for subsequent blood feeding by all the 662\npopulation strata. The HTCD, a p× p matrix (D), is: 663\nD = diag (W)· βT· diag (bDH)· β. (40)\nWe note thatD in spatial models is analogous to bD in models with a single patch. 664\n(The equivalency ofD and bD is most apparent if no humans move, and if there is one 665\nstratum per patch, and if all search weights are 1, in which case H = W and 666\nβ = diag(1/H).) Like bD,D describes days spent infectious by an individual human, 667\nbut inD, describes both where a human got infected and where the mosquitoes were 668\nsubsequently infected. 669\nThe definition ofD as a time-dependent matrix is substantially more complicated if 670\nlocal human mobility patterns change dynamically. 671\nParasite Dispersal through one Parasite Generation Parasite dispersal is 672\ndefined by the locations where infecting bites occurred, alternatively moving in infected 673\nmosquitoes and humans. The equations for D andV describe the expected movement 674\nfor a parasite among patches in humans or mosquitoes, respectively, counting from bite 675\nto bite. Notably, the formulas are defined for a parasite in either a mosquito or a 676\nhuman. We can also define parasite dispersal through one parasite generation ( i.e., from 677\nhuman to human, or from mosquito to mosquito) but the formula depends on where we 678\nstart counting. If we started from all the mosquitoes blood feeding on a single human 679\n(averaged appropriately) on a single day in every patch, then we would get a matrix 680\ndescribing dispersal from every patch to every patch: 681\nD·V . (41)\nIf we started counting from a typical human infected in a patch on a single day, we 682\nwould get a different dispersal matrix: 683\nV·D . (42)\nImportantly, these formulas follow the same process in the same order, and thus closely 684\nresemble the reproductive numbers for malaria (described below), which measure 685\nreproductive success for a single parasite. These formulas are two among many that 686\ncould be developed to count events through a parasite’s life-cycle starting at different 687\npoints. 688\nFormulas that describe the parasite’s per-capita reproductive success, such as 689\n27\n . CC-BY 4.0 International licenseIt is made available under a \n is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity.(which was not certified by peer review)preprint \nThe copyright holder for thisthis version posted November 15, 2022. ; https://doi.org/10.1101/2022.11.07.22282044doi: medRxiv preprint \n\nEqs. 41-42, counting events arising from a single host. In some cases, we might wish to 690\ncount the total number of events arising from a patch. To measure the contribution of a 691\npatch to overall transmission, we must have a measure of connectivity, or total parasite 692\nflows. A tracking matrix describing all of the infections arising from each patch on a 693\nday, is: 694\ndiag (W)·D·V (43)\nIf we started counting infections occurring on humans in a patch, we would get an 695\nalternative patch-based tracking matrix. The number of infections arising from a patch 696\nis thus tracked by: 697\ndiag (W)·V·D (44)\nThese measures emphasize the role of places with larger available populations. 698\nThe same sort of formulas can be devised to describe transmission from human 699\nstrata to human strata, but the resulting formulas are only spatial insofar as the human 700\nstrata are anchored to a residency. If we focused instead on parasite reproductive 701\nsuccess starting with an infection in humans, regardless of location, we would get 702\nR = bβ·V· diag (W)· βT· diag (DH) . (45)\nor we could also count bulk transmission from humans as diag(H)·R . Notably, Eq. 45 703\nis a stratum-based measure. To make it quasi-spatial, we would need to assign events to 704\npatches by stratum residency using the membership mapping operator J·R·J T . 705\nDistances Dispersed To get a measure of the distribution of distances travelled by 706\nparasites, we match a measure of transmission intensity with the corresponding element 707\nin a patch distance matrix describing the distance. We take the couplet (distance and 708\nintensity) and sort by distance, then compute the cumulative distribution function 709\n(CDF). From the CDF, we derive a probability mass function [39]. These dispersal 710\nkernels provide a simple way of visualizing distances dispersed by mosquitoes, humans, 711\nor parasites. 712\nThese formulas and algorithms draw attention to the differences in metrics 713\ndescribing parasite transmission dynamics and dispersal. Because of spatial 714\nheterogeneity in mosquito and human population densities, there are many sensible 715\nformulas for counting dispersal, some of which correspond to describing rates, ratios, 716\nproportions, and numbers. Careful thought should be given to choosing or developing a 717\nmetric that fits the analysis. 718\nReproductive Numbers 719\nReproductive numbers are a measure of the parasite’s average reproductive success. 720\nWhen transmission is spatially heterogeneous, reproductive success will vary for 721\nparasites, depending on where they are. As parasites spread over several generations, 722\nthe expected success of its progeny will change. To calculate threshold criteria for 723\npersistence (in the absence of malaria importation), we want a reproductive number to 724\nbe a measure of average success taken over the whole system, but we want to use an 725\naverage that does not change across generations. Doing so requires that we compute the 726\n28\n . CC-BY 4.0 International licenseIt is made available under a \n is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity.(which was not certified by peer review)preprint \nThe copyright holder for thisthis version posted November 15, 2022. ; https://doi.org/10.1101/2022.11.07.22282044doi: medRxiv preprint \n\nspectral average, which is computed as the dominant eigenvalue of the parasite’s next 727\ngeneration matrix. 728\nFor many reasons, it is useful to formulate local reproductive numbers that describe 729\na parasite’s average reproductive success at a particular place and time – an arithmetic 730\naverage. These local reproductive numbers could ignore differences across generations, 731\nso they would not serve as thresholds for parasite persistence. In this section, we define 732\nlocal reproductive numbers at the steady state, but the formulas could also serve as 733\npoint estimates. 734\nReproductive numbers describe malaria transmission under a range of different 735\nconditions that are relevant for understanding malaria transmission dynamics and 736\ncontrol or for national strategic planning. Baseline conditions are described by the basic 737\nreproductive number, R0, which is defined for a population with no acquired immunity 738\nand no malaria control. The adjusted reproductive number, RC, describes a family of 739\nnumbers defined for a population with no acquired immunity adjusted by malaria 740\ncontrol, at a fixed level of control denoted C. In other words, R0 is defined as a special 741\ncase of RC, but in the absence of control. The total effect size of malaria control on 742\ntransmission is R0/RC. Here, we also describe the endemic reproductive number, RE, 743\nwhich describes potential transmission modified by immunity. The total effect size of 744\nimmunity on transmission is RC/RE. In computing RE, as with R0 and RC, we ignore 745\nthe fact that some hosts are already infected. In this way, RE is defined differently than 746\nthe effective reproductive number, denoted Re, which is lower than RE because it does 747\nnot count infections occurring in someone who is already infected. We note that, by 748\ndefinition, at an endemic steady state Re = 1. By way of contrast, RE counts the 749\nnumber of infections that would occur after one generation, which is useful for planning 750\nbecause it helps to clarify how success in malaria control can be assisted by immunity 751\nthat will eventually wane. 752\nBoth R0 and RC are computed as if there were no acquired immunity. In this model, 753\nthe effects of acquired immunity on transmission are quantified through the stratified 754\nvalues of b, r1, r2, c1 and c2. These parameters determine the HTC for all the strata ( D, 755\nsee Eq. 31). If D were computed using values that have been tuned to a stratum with 756\nsome level of immunity, we would be computing RE. To compute RC, we would need to 757\nreplace D with values set to a non-immune baseline ( i.e., D0), and then recompute the 758\nnext-generation matrix. Next generation matrices computed with values of D that 759\ninclude the effects of acquired immunity are thus describing an endemic reproductive 760\nnumber. Depending on how D is computed, and whether the bionomic parameters 761\nincorporate effects of vector control, we may thus be computing R0, RC or RE. 762\nLocal Reproductive Numbers One way to define local reproductive numbers is to 763\nmodify Macdonald’s formula using the local values of parameters, as if there was no 764\nmovement of mosquitoes or humans. To write the formula using some models in this 765\nframework, we may need to modify HTC (which is defined for the strata, of length n) to 766\ntake a patch average. To compute a patch average HTC, ˘D (a vector of length p), we 767\ntake the population weighted average, 768\n˘D = ΨT· wf DH\nΨT· wf H (46)\n29\n . CC-BY 4.0 International licenseIt is made available under a \n is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity.(which was not certified by peer review)preprint \nThe copyright holder for thisthis version posted November 15, 2022. ; https://doi.org/10.1101/2022.11.07.22282044doi: medRxiv preprint \n\nWe can then describe a local reproductive number, ˘RC (or possibly ˘RE, depending on 769\nhow the interpret parameters are defined in D): 770\n˘RC = Λ\nW\nf2q2\ng2 e−gτ ˘D (47)\nThis local measure is similar to Macdonald’s formula [59]. While useful in some 771\ncontexts, the formula should be applied with caution. 772\nAn alternative way to compute local reproductive numbers uses V andD (perhaps 773\nmodified to remove the effects of immunity on transmission). Since the matrices count 774\ninfections arising from each patch, and we add all infections arising to the patch where 775\nthe bite originates. We let 1 be a row vector of ones of length p, and we can count 776\ninfections arising starting from all the humans infected in a patch on a single day: 777\nˆRC = 1·V·D (48)\nthat counts infections occurring on humans, or we can start from all the mosquitoes 778\nblood feeding on humans on a single day, and: 779\n˜RC = 1·D·V (49)\nthat counts infected mosquitoes. These patch reproductive numbers could provide 780\nvaluable information about whether to target the mosquitoes or humans in some patch 781\nfor enhanced interventions. We could also consider the equivalent formulas for total 782\npatch outputs: 783\nWT·V·D or WT·D·V (50)\nwhere WT is a row vector. Alternatively, we can also weigh transmission from strata 784\nusing Eq. 45: 785\n1·R (51)\nor the equivalent scaled by stratum size: 786\nHT·R, (52)\nwhere HT is a row vector, which gives us valuable information about infections arising 787\nfrom every stratum on every strata, a way of identifying the relative importance of 788\nvarious population strata. 789\nNext Generation Matrix In the Ross-Macdonald model, a parasite’s reproductive 790\nsuccess in the next generation is described by a single number. It is computed by 791\ncounting forward from the moment a mosquito or human becomes infectious. Since 792\nparasites move in infected mosquitoes and humans, parasite reproductive success – 793\nmeasured as the number of infections in the next generation – varies across generations 794\nas the parasite distributions evolve across generations among strata and among patches. 795\nThe matricesV andD describe parasite transmission and dispersal in mosquitoes and 796\nhumans, respectively. While the product of these formulas does describe net 797\nreproductive success, the computation of threshold conditions has been developed 798\n30\n . CC-BY 4.0 International licenseIt is made available under a \n is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity.(which was not certified by peer review)preprint \nThe copyright holder for thisthis version posted November 15, 2022. ; https://doi.org/10.1101/2022.11.07.22282044doi: medRxiv preprint \n\naround the concept of a next generation matrix [60,61], which traces the same process 799\nin the same sequence but that start counting at a different point in the parasite’s life 800\ncycle (Fig. 7). A threshold condition is found by taking the spectral average of the next 801\ngeneration matrix. 802\nFig 7. A Spatial Life-Cycle Model . A diagram that illustrates how the parameters\ndescribing each stage in the parasite’s life-cycle translate into a parasite’s reproductive\nsuccess spatially, when mosquitoes and hosts move. The right half of the circle\nrepresents mosquitoes and the left half humans. The flow of events is clockwise.\nMosquitoes must blood feed to become infected ( f qM), and then survive and disperse\nthrough the EIP ( e−Ωτ). infectious bites are distributed as long as a mosquito survives,\nwhile it blood feeds and disperses ( f qΩ−1). The bites are distributed among humans ( β)\nand some of them cause an infection (b). Parasites are transmitted for as long as\nhumans remain infectious, measured in terms of the human transmitting capacity (HTC,\nor D days). Infectious humans are distributed wherever humans spend time at risk\n(affecting β). These processes are summarized differently to model parasite dispersal\nand parasite reproductive success. Dispersal counts from bite to bite using the VC\nmatrix (V) and the HTC matrix (D). Threshold computations count from when a host\nbecomes infectious to measure a parasite’s reproductive success in infectious mosquitoes\n(RZ); in infectious humans (RX); from human to humans among strata after a human\nbecomes infectious (R); and from mosquito to mosquitoes (Z). R0 is the lead\neigenvalue ofR orZ. Under endemic conditions, we can also consider how frequently\nparasites are actually transmitted by including the probability a mosquito gets infected\nκ, and the probability a mosquito is infectious, given by the sporozoite rate z.\nIn computing next generation matrices, we focus on transmission within a defined 803\nspatial domain. For mathematical convenience here, we thus set υ = 1, though we could 804\neasily develop matrices leaving υ undetermined to discount exported malaria cases. 805\nWe first compute offspring transmitted from a single infectious mosquito to humans 806\nor from a single infectious human to mosquitoes, each of which defines a stage in the 807\nparasite’s next-generation [60]. After a mosquito has become infectious, how many 808\nhumans (in each stratum) would it infect? In these models, the answer to that question 809\nis n× p matrix, denoted RZ, describing transmission from an infectious mosquito in 810\neach patch to humans in each strata: 811\nRZ = bβ· f qΩ−1. (53)\n31\n . CC-BY 4.0 International licenseIt is made available under a \n is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity.(which was not certified by peer review)preprint \nThe copyright holder for thisthis version posted November 15, 2022. ; https://doi.org/10.1101/2022.11.07.22282044doi: medRxiv preprint \n\nHow many infectious mosquitoes would arise from each human infection? The answer is 812\na p× n matrix, denoted RX, describing transmission from a human in each stratum to 813\nmosquitoes: 814\nRX = e−Ωτ· f qM·\n(\nβT· diag (DH)\n)\n. (54)\nThe next-generation matrix by type is: 815\nG =\n[\n0 RZ\nRX 0\n]\n(55)\nTo describe reproductive success in terms of the parasite biology, we count reproductive 816\nsuccess through one full parasite generation, either from humans back to humans, or 817\nmosquitoes back to mosquitoes. For the parasites, reproductive success through one full 818\ngeneration requires two events, one of each type, so we square the matrix given by 819\nEq. 55 to get a new matrix in block form: 820\nG2 =\n[\nR 0\n0 Z\n]\n. (56)\nWe thus get two diagonal block sub-matrices describing reproductive success in the 821\nparasite’s next generation, denoted R andZ. Reproductive success from human 822\npopulation strata back to human strata is described by an n× n matrixR = RZ· RX: 823\nR = bβ·V· diag (W)· βT· diag (DH) . (57)\nReproductive success from mosquito through the population strata back to mosquitoes, 824\ndescribed patch-by-patch is described by the p× p matrixZ = RX· RZ: 825\nZ = e−Ωτ· diag\n(f qM\nW\n)\n·D· f qΩ−1 (58)\nWe have also formulated the next-generation matrix for systems with multiple vector 826\nspecies (Supplement 4). 827\nThe Spectral Average We can also compute RC as a spectral average through 828\nsimulation, which is one useful way of illustrating what a spectral average means. To do 829\nso, we choose a vector describing the distribution of parasites in a founding generation, 830\nX0 orY0, and iterate parasite infections across i successive parasite generations: 831\nYi+1 =ZYi or Xi+1 =RXi. (59)\nWe define the vector:\nEi =Xi+1\n∥Xi∥ or Ei =Yi+1\n∥Yi∥ .\nwhere∥X∥ or∥Y∥ is a scalar that denotes is magnitude. Over many generations, Ei 832\nconverges to the lead eigenvector, a scalar value also called the spectral average or RC: 833\nRC = lim\ni→∞\n∥Ei∥ (60)\n32\n . CC-BY 4.0 International licenseIt is made available under a \n is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity.(which was not certified by peer review)preprint \nThe copyright holder for thisthis version posted November 15, 2022. ; https://doi.org/10.1101/2022.11.07.22282044doi: medRxiv preprint \n\nand it is interpreted as the asymptotic average reproductive success expressed as a 834\nnumber of infected hosts per host, per generation. Note that it is asymptotic only for 835\nthe linearized system defined by Eq. 55 or Eq. 56. 836\nQuasi-Thresholds for Endemic Malaria Without malaria importation, RC > 1 is 837\na threshold criterion. Analysis of models without malaria importation have consistently 838\ndemonstrated that malaria is either absent or that there is a single globally, 839\nasymptotically stable equilibrium. When there is imported malaria, there are three 840\nsufficient criteria for some local parasite transmission to occur within the area: 841\n1. max{δ} > 0 and RC > 0; 842\n2. max{(1− υ)Xδ} > 0 and RC > 0; 843\n3. RC > 1. 844\nIf condition 1 or condition 2 is satisfied, then malaria will be present in an area, and if 845\nRC > 0 then there will be some local transmission. If RC > 1, malaria transmission 846\nwould be sustained in the absence of importation. We thus call RC > 1 a 847\nquasi-threshold for endemic transmission to occur within the spatial domain: endemic 848\ndescribes places where RC > 1, and pseudo-endemic places where 0 < RC < 1 with 849\nsignificant levels of transmission. 850\nQuantifying Transmission in a Place 851\nThe framework, models developed within it, and the associated spatial metrics were 852\ndesigned to have the skill required to describe and quantify heterogeneous spatial 853\ntransmission dynamics of malaria in a specific place at a particular time. We have not 854\nexplicitly defined algorithms for the observational processes that would map model 855\nstates onto observable quantities, which would be required to extend this mathematical 856\nmodeling framework into a state space modeling framework to rigorously fit models to 857\ndata. Instead, we have focused on the mathematics of these processes: time spent by 858\nhumans; other blood hosts; daily mosquito rhythms; mosquito host preferences, time at 859\nrisk; and mosquito mobility. Similarly, the models for mosquito ecology and population 860\ndynamics describe the mathematics of mosquito mobility, in terms of explicit 861\nassumptions about the locations of aquatic habitats, heterogeneous distributions of 862\nresources, and mosquito mobility patterns that emerge from a search for resources. By 863\nquantifying spatial patterns in terms of the underlying processes – including malaria 864\nimportation, mosquito ecology and spatial population dynamics, parasite transmission 865\ndynamics, human mobility, and malaria epidemiology – the equations point towards a 866\ngeneral inferential framework. 867\nModels developed within this framework involve substantially more parameters than 868\nthe Ross-Macdonald model. This is an inevitable consequence of a decision to model 869\ntransmission at a particular place and time. If any local features are important for 870\ntransmission, then a larger set of quantities must be estimated to understand and 871\nquantify those features. This gives rise to an important but difficult practical question: 872\nWhat is the relationship between the amount of local intelligence and the specificity of 873\n33\n . CC-BY 4.0 International licenseIt is made available under a \n is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity.(which was not certified by peer review)preprint \nThe copyright holder for thisthis version posted November 15, 2022. ; https://doi.org/10.1101/2022.11.07.22282044doi: medRxiv preprint \n\nthe policy advice that can be offered? With minimal local information, it is possible to 874\noffer generic policy advice, but it may not be necessary to know everything about a 875\nplace to tailor advice to context. With this framework, it is possible for models to 876\nevolve as the amount information increases, and the models may be used to look ahead 877\nto prioritize missing data: How can programs identify missing information that would 878\nmost rapidly improve the effectiveness of malaria control? These contextual factors and 879\nthe related questions are addressed below. 880\nMalaria Landscapes While the Ross-Macdonald model describes parasite 881\ntransmission between abstractly defined mosquito and human populations, the 882\nframework we have described was developed to understand and quantify malaria 883\nimportation and transmission among structured mosquito and human populations in a 884\nwell-defined geographical area. (Using a model dX /dt that describe the infection 885\ndynamics of other pathogens and immunity in vertebrate host populations, and making 886\nother appropriate choices, the framework could be used as a basis for modeling dengue, 887\nWest Nile virus, or other mosquito-borne pathogen transmission dynamics, as well.) 888\nSince the models are developed to approximate malaria transmission in an actual place, 889\nafter defining an observational process, the model outputs would be verifiable 890\nstatements about real quantities over some specific period of time. 891\nAs a practical first step, model building starts by defining a set of structural 892\nelements – patches, human population strata, and aquatic habits – that are appropriate 893\nfor the needs of a study (e.g. Fig 8 illustrates some options for simulating malaria on 894\nBioko Island, Equatorial Guinea). A geographical study area is usually defined by 895\nprojects, programs, or political boundaries. In planning interventions for a defined area, 896\nan important concern is connectivity to surrounding areas. How much malaria is 897\nimported by daily human movement or travel? Are the mosquito populations within the 898\narea strongly connected to others nearby? 899\nUsing spatial metrics to identify differences in transmission patterns and the flow of 900\nparasites across a landscape can help control programs prioritize drugs, outreach, and 901\nmedical attention to populations, and vector or larval control to places. Using our 902\ndifferential equation framework to reconstruct the equilibrium analysis presented in [45], 903\nwe have generated spatial bulk transmission matrices (diag(H )·R) among areas for 904\nBioko Island, Equatorial Guinea. In Fig. 9 different patterns of pathogen transport are 905\nreadily apparent between persons who live in Malabo (left), the densely populated 906\ncapitol of the island and a sink for travellers, and Luba (right), a small settlement in the 907\nSouthern half of the island. The pattern of travel seen in Luba typifies most of the areas 908\noutside of Malabo, where individuals most often travel to the capitol but not to the 909\nother outlying settlements. These patterns affect transmission, where we see parasites 910\noriginating in Malabo tend to stay in the city. Parasites originating in Luba either tend 911\nto stay highly local, or are transported to Malabo when those persons move. Because 912\nmalarial mosquitoes tend to fare less well in urban settings, these spatial metrics can 913\nhelp understand how high prevalence can be sustained in otherwise unsuitable locations. 914\nAn equally important question is about heterogeneity in mosquito population 915\ndensities within the area and heterogeneity in the risk of exposure, which should inform 916\nthe definition of patches and the choice of a patch size. Patches, in this model, are 917\ndefined around adult mosquito activities, and each “patch” has a geographical location. 918\n34\n . CC-BY 4.0 International licenseIt is made available under a \n is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity.(which was not certified by peer review)preprint \nThe copyright holder for thisthis version posted November 15, 2022. ; https://doi.org/10.1101/2022.11.07.22282044doi: medRxiv preprint \n\nThe patch is the spatial unit that defines the algorithms for time spent, blood feeding, 919\negg laying, adult mosquito survival and dispersal. The concept of a patch is flexible 920\nenough to model blood feeding indoors and outdoors at the same geographical locations, 921\nwhich may be useful to inform programmatic questions about the effectiveness of vector 922\ncontrol measures that target indoor biting (Fig. 10). Since the patch is the basis for 923\ncomputing most aspects of blood feeding, the patches define the structure for human 924\ntime spent at risk, including (if required) quantifying time spent indoors vs. outdoors, 925\nand mosquito movement rates from indoors to outdoors, from outdoors to indoors, or 926\nfrom outdoors to other outdoor patches. 927\nAn important basic concern is the spatial granularity of the patches used for 928\nsimulation (see Fig. 8). Some questions remain unresolved about the appropriate spatial 929\nscales and ways to define patches for describing and analyzing malaria transmission for 930\npolicy (e.g., to compute IRS coverage). One advantage of this framework is that it is 931\npossible to build nested models with different spatial grains and compare them. Smaller 932\npatches more accurately capture heterogeneity in a landscape while increasing the 933\nnumber of parameters that need to be inferred during calibration to data.k 934\nAquatic habitats are located in patches, but the model was designed to assign 935\npatches to habitats assuming the habitats had an actual location. Patches in this 936\nframework need not have any human residents or any available hosts, so that mosquito 937\nhabitats in the uninhabited areas around human households are contributing to 938\ntransmission. Mosquito population dynamics are coupled through related equations 939\ndescribing gravidity, egg laying and egg deposition. The framework thus does not 940\nimpose any constraints on either the method for constructing patches, or on the number 941\nor arrangement of aquatic habitats within the spatial domain. Given the modular 942\nnature of these models, the dynamics of immature mosquito populations in each aquatic 943\nhabitat depend only on its parameters and the egg deposition rates. The productivity of 944\nany one aquatic habitat in an area is, however, coupled to other habitats through egg 945\nlaying by adult mosquitoes that could have emerged anywhere. 946\nTo improve the accuracy of models, human populations can be segmented into strata 947\nto reduce heterogeneity in traits that affect malaria: the first segmentation is by 948\nresidency. In this framework, which is designed to quantify process affecting 949\ntransmission, heterogeneity in any trait affecting transmission is dealt with by 950\nsub-dividing the population into homogeneous (or less heterogeneous) strata, such as by 951\nage, travel habits or patterns, ITN usage, vaccination, care seeking, or any effects of 952\nimmunity affecting malaria epidemiology or transmission. 953\nNotably, all this structural flexibility is achieved through membership matrices and 954\nthrough the variables describing resource availability, which links search weights, 955\nfunctional responses, and other functional forms to guarantee mathematical consistency 956\n(e.g. avoiding problems when denominators are zero) despite structural changes. Suites 957\nof models can be developed to address concerns about data gaps and uncertainty that 958\nare appropriate for studies. Model complexity can be modified by changing dynamical 959\nmodules, by changing functional forms, by fixing or changing parameters, by splitting 960\nand joining patches, by splitting or joining strata, or by adding and subtracting aquatic 961\nhabitats. With the ability to split and join patches or strata, any model can be mapped 962\nonto simpler, nested models in a series of simple join operations until it is collapsed onto 963\na single-patch, single-stratum Ross-Macdonald model. This is functionally what is 964\n35\n . CC-BY 4.0 International licenseIt is made available under a \n is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity.(which was not certified by peer review)preprint \nThe copyright holder for thisthis version posted November 15, 2022. ; https://doi.org/10.1101/2022.11.07.22282044doi: medRxiv preprint \n\nmeant by scalable complexity. 965\nIt is thus as easy to modify and evaluate the effects of model structure ( e.g. the 966\nnumber of strata) as it is to vary parameters, to facilitate developing suites of models, 967\nincluding models with nested patches or nested strata, to explore tradeoffs in building 968\nand calibrating models at various levels of detail. 969\nMosquito Blood F eeding and Ecology Three constant parameters describing 970\nmosquito behavior are a standard part of the Ross-Macdonald model [56,57]: the daily 971\ndeath rate of mosquitoes ( g), the overall daily blood feeding rate ( f), and the human 972\nblood feeding fraction ( q). Incorporating the possibility of dynamical feedback between 973\nthe future emergence of adults and current population size means we have added the 974\npopulation egg-laying rate (Γ). Adding spatial complexity to the model means the daily 975\nemigration rate ( σ), mosquito dispersal (K ), distribution of habitats (N ) and the 976\ndistribution of eggs among patches (U) are additional parameters which define how 977\npopulations may interact in space. While our analysis has focused on steady states, the 978\nmodels were formulated with parameters that can vary over time in response to 979\nchanging availability of resources [33–36]. 980\nIn this framework, the values of all these parameters are computed with functional 981\nresponses based on resource availability, mosquito biology and innate preferences that 982\nconstrain the parameters within sensible ranges. This formulation emphasizes how 983\nbaseline mosquito bionomics for different species could respond to available resources 984\nand how those responses would be modified by control. In particular, the same human 985\nbehaviors can give rise to very different blood feeding patterns for different vector 986\nspecies, depending on the daily rhythms, host preferences, and aquatic ecology of 987\ndifferent vector species (Supplement 4). We thus have a basis for understanding 988\nmosquito behaviors and ecology as a baseline that may have been modified by vector 989\ncontrol or weather. 990\nBlood feeding in this model thus makes an important distinction between 991\nanthropophily, or innate mosquito preferences for hosts of different types, and 992\nanthropophagy, summarized by the human blood feeding rate ( f q). Models can also 993\nconsider a difference between the time of day when mosquitoes are actively searching for 994\nblood (ξ ) and the blood feeding rates by time of day (f ), which vary with host 995\navailability. Innate, species-specific host preferences are embodied in functional forms 996\nand parameters, while the rates describing what has happened also depend on context. 997\nSimilarly, mosquito population dynamics are an emergent feature of a resource 998\nlandscape. Since searching for resources is also associated with resource availability, 999\nadult mosquitoes will tend to aggregate in patches that have habitats and other 1000\nrequired resources. In these models, egg-deposition rates in habitats by volant adult 1001\npopulations are spatially heterogeneous and only partially determined by the emergence 1002\nrates of adults from a single habitat. The concept of a carrying capacity is, perhaps, not 1003\nas useful as the concept of habitat productivity and the functional forms that determine 1004\nhow the number of adults emerging is related to the number of eggs laid [31]. A 1005\nhabitat’s carrying capacity only makes sense in the abstract – if adult mosquitoes 1006\nemerging from a single habitat only laid eggs in that natal habitat. In this framework, 1007\nthe aquatic population dynamic module determines how adult mosquito emergence 1008\nrates respond to egg laying by the adult population. 1009\n36\n . CC-BY 4.0 International licenseIt is made available under a \n is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity.(which was not certified by peer review)preprint \nThe copyright holder for thisthis version posted November 15, 2022. ; https://doi.org/10.1101/2022.11.07.22282044doi: medRxiv preprint \n\nThe parameters describing these processes are both habitat-specific and 1010\ntime-dependent: density-independent mortality, density-dependent mortality, the 1011\nresponse to crowding, maturation rates, and search weights could vary for every habitat. 1012\nA habitat can thus disappear seasonally (which occurs when wν = 0), or weather could 1013\naffect immature mosquito maturation and mortality rates. If a study called for 1014\nmodeling resource-based competition or stage-structured mosquito populations, the 1015\nequations describing aquatic populations (dL/dt) can be modified as needed (Fig 1). 1016\nThe framework thus facilitates the construction of realistic models of mosquito ecology, 1017\ninsofar as it is justified by data available and the needs of a study. 1018\nLocal Exposure, Human Biting Rates and Mixing In defining the algorithms 1019\nfor blood feeding, we also developed a new model for the human biting rate (HBR) and 1020\nby extension, the entomological inoculation rate (EIR), two basic metrics used to 1021\nmeasure malaria transmission entomologically. 1022\nThe model emphasizes that for any population stratum, the risk of exposure to 1023\nbiting mosquitoes is distributed spatially. In these models, this is determined by a biting 1024\ndistribution matrix (β ). A similar matrix has appeared in other models for the spatial 1025\ndynamics of mosquito-borne diseases for which human mobility is based on a concept of 1026\n”visitation” or time spent – classified as Lagrangian movement [7,8,10,12 –15,17,18,45]. 1027\nHere, β is based on a concept of availability, the weighted, ambient population at risk. 1028\nAvailability is computed from observable quantities, and it is computed dynamically for 1029\narbitrarily defined human strata and changing availability (the denominator). The 1030\nformulas guarantee consistency in blood feeding: the number of human blood meals 1031\ntaken by mosquitoes is equal to the number of blood meals received by the humans. 1032\nIn the new model, the HBR is defined as β· f qM and the EIR is β· f qZ, so that the 1033\nnumber of bites received by each stratum depends on how they spend their time at risk. 1034\nIn studies that have reported a value for the HBR or EIR, the quantity reported is 1035\nbased on catch counts by a person or device in a place. In this model, the quantity that 1036\nis closest to the quantities being estimated is pHBR or pEIR, the number human blood 1037\nmeals, or infectious human blood meals in a patch, per available person, per day 1038\n(f qM/W or f qZ/W). A person who is in a patch at a particular time of day would 1039\nexperience the local biting rates at that time scaled by a search weight ( f qM ξ(t)ωf /W 1040\nor f qZξ(t)ωf /W ). The quantity being estimated by human landing catches is a 1041\nmeasure of the intensity of exposure in a place. 1042\nSince other hosts are also available, the number of mosquitoes caught also depends 1043\non the biases of the trapping method. In this model, each method for trapping 1044\nmosquitoes can be thought of as having its own “availability,” and it is competing for 1045\nthe attention of mosquitoes. Each method for catching mosquitoes is biased in some 1046\nunknown way. We thus suggest that field methods designed to estimate the EIR are 1047\nbest interpreted as a location-specific measure of risk in a place, and that 1048\nepidemiologically relevant measures of risk must acknowledge exposure occurring for a 1049\nperiod of time, including all the places where a person spends time. The pEIR, weighted 1050\nby total availability, is a good approximation of the EIR only if a person spends most of 1051\ntheir time at risk in that place. The formulas presented here are useful to quantify how 1052\nlocal measures of mosquito blood feeding in a place could differ from what the humans 1053\nliving in that place would experience. What is the difference between risk for a human 1054\n37\n . CC-BY 4.0 International licenseIt is made available under a \n is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity.(which was not certified by peer review)preprint \nThe copyright holder for thisthis version posted November 15, 2022. ; https://doi.org/10.1101/2022.11.07.22282044doi: medRxiv preprint \n\nwho moves around compared to their counterfactual self who never leaves home? 1055\nThe Spatial Scales of T ransmission Important considerations for planning, 1056\nmonitoring, and evaluating malaria control are the spatial scales that characterize 1057\ntransmission, as defined by parasite dispersal in mosquitoes or humans. We have 1058\ndefined parasite dispersal rigorously in terms of the locations where blood meals 1059\noccurred that transmitted parasites in dispersal chains. While these definitions are 1060\ncompelling, the distribution of distances separating every pair of infectious bites in a 1061\nchain of malaria infections can only be approximated using other data. In practice, the 1062\nframework we have described makes a distinction between local transmission and 1063\nimported or exported malaria. The framework makes the most sense mathematically if 1064\nmost transmission is local, but the framework also defines quantities for malaria 1065\nimportation and exportation, making it possible to study connectivity using a frame 1066\nthat shifts among spatial domains and across spatial scales. 1067\nAfter drawing a bounding box to define a spatial domain and a set of patches, we 1068\nclassify any pair of bites in a transmission chain where at least one occurred in the 1069\npatch: either both bites occurred somewhere in the spatial domain, called local 1070\ntransmission; or the first bite occurred outside the spatial domain, called imported 1071\nmalaria; or the second bite occurred outside the spatial domain, called exported malaria. 1072\nThese measures of imported and exported malaria thus provide a basis for 1073\nunderstanding and quantifying dispersal within and among defined geographical areas. 1074\nThese models weigh the consequences of imported malaria, but as a practical matter, 1075\nthe importance of exported malaria is difficult to quantify because the expected number 1076\nof subsequent bites depends on conditions somewhere else. Importantly, the fraction 1077\nthat stays local may differ depending on whether the parasite is moving in a mosquito 1078\nor a human. Similar definitions and arguments would apply to transmission through a 1079\nparasite, a full parasite generation encompassing three bites and two jumps. The 1080\nmetrics we have developed describe transmission within a defined geographical domain, 1081\nbut if there is a need, the models can be reformulated for a larger spatial domain. 1082\nThe models and metrics provide a way of characterizing the spatial scales of 1083\ntransmission by computing the cumulative fraction of all transmission occurring within 1084\na circle of a given radius. Sensible points on that curve can be compared by patch: 1085\nWhat distances contain 80%, 90%, 95%, or 98% of all transmission? These estimates 1086\nare, out of necessity, based on estimated quantities – models of mosquito mobility, 1087\nhuman mobility, and modeled mosquito population density – about which there is 1088\nsubstantial uncertainty. 1089\nDespite the overall uncertainty, these spatial scales are constrained by limits on time 1090\nand travel. Some quantities are known from census data ( e.g. population distributions). 1091\nMost mosquito dispersal distances are short. Mosquitoes can move large distances, but 1092\nmost stay within 1 km of a natal habitat [62]. For humans, the fraction of time spent 1093\ndeclines sharply with distance away from home. A large fraction of time is spent at 1094\nhome, especially at night, and a larger fraction of the time is spent within roughly 10 1095\nkm of home. The fraction of time spent drops off sharply with log 10 distance. The 1096\nspatial scales also depend on transmission intensity. In places with highly heterogeneous 1097\ntransmission, places with the highest transmission intensity, will have the greater the 1098\nfraction of transmission that occurs at short distances. 1099\n38\n . CC-BY 4.0 International licenseIt is made available under a \n is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity.(which was not certified by peer review)preprint \nThe copyright holder for thisthis version posted November 15, 2022. ; https://doi.org/10.1101/2022.11.07.22282044doi: medRxiv preprint \n\nMosquitoes, T ravel, and T ransmission Highly spatially resolved data describing 1100\nthe EIR is rarely available. It is often cheaper, albeit less accurate, to use cross-sectional 1101\nblood survey data describing malaria prevalence ( i.e. the parasite rate, PR) to estimate 1102\nlocal transmission. Spatial models and spatial metrics described herein provide some 1103\nguidelines about how patterns in the PR can be used to identify areas with the most 1104\nmosquitoes, particularly given the enormous heterogeneity in human population density. 1105\nIt is commonly assumed that local clustering of cases implies that there is local 1106\ntransmission. For models developed in this framework, the vectorial capacity matrix 1107\n(Eq. 39) describes parasite dispersion by mosquitoes, and evidence suggests that the 1108\nspatial scales describing parasite dispersal by mosquitoes could vary by context [62]. 1109\nImportantly, imported malaria can confound the relationship between local 1110\ntransmission by mosquitoes and prevalence. Travel habits and other traits describing 1111\nhumans often cluster spatially, partly because human neighborhoods are organized by 1112\nsocio-economic status. Spatial clustering of cases could arise if travel habits and thus 1113\nmalaria importation rates are spatially clustered, giving the appearance of local 1114\ntransmission. 1115\nMeasuring Reproductive Success The most complete measure of transmission in 1116\nan area is a reproductive number – the number of malaria cases arising from each 1117\nmalaria case after one complete parasite generation. We have defined reproductive 1118\nmatrices in several ways as matrices describing reproductive success among patches 1119\nwithin a spatial domain, which can be used to define local reproductive numbers as 1120\ncases arising from a patch. These reproductive matrices form a basis for investigating 1121\nthe appropriate spatial scales to measure and model transmission, for estimating 1122\ncontamination in randomized control trials, and for understanding the spatial effect 1123\nsizes of control. These can put other data into a context that is relevant for 1124\ntransmission. For example, mosquito counts data and measures of malaria can vary over 1125\nvery short distances [28,62]. The functional relevance of local heterogeneity in mosquito 1126\ncatch counts or in malaria prevalence can be critically examined by examining a matrix 1127\nthat integrates the effects of parasite movement in both mosquitoes and humans. After 1128\nfully considering the uncertainty, it may be possible to determine the relevant spatial 1129\nscales of transmission and thus the relevant spatial units for estimating reproductive 1130\nnumbers for malaria dynamics and control. 1131\nDiscussion 1132\nThe simplicity of the Ross-Macdonald model can be contrasted with Hackett’s 1133\ndescription of the elaborate and context-dependent nature of malaria that he observed 1134\nin the field [27]: 1135\n. . .malaria is so moulded and altered by local conditions that it becomes a 1136\nthousand different diseases and epidemiological puzzles. Like chess, it is 1137\nplayed with a few pieces, but is capable of an infinite variety of situations. 1138\nThe Ross-Macdonald model clearly identified enough chess pieces to develop basic 1139\nconcepts and theory to describe and measure malaria transmission [1], such as vectorial 1140\n39\n . CC-BY 4.0 International licenseIt is made available under a \n is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity.(which was not certified by peer review)preprint \nThe copyright holder for thisthis version posted November 15, 2022. ; https://doi.org/10.1101/2022.11.07.22282044doi: medRxiv preprint \n\ncapacity, the basic reproductive numbers, daily human biting rates, sporozoite rates, 1141\nentomological inoculation rates, and malaria parasite rates ( i.e. prevalence). These 1142\nbasic metrics have formed the basis for quantitative studies of malaria transmission, but 1143\nthey ignored heterogeneity and complexity. In particular, the metrics and associated 1144\nconcepts describing parasite dispersal in infected mosquitoes and humans were missing. 1145\nParasite dispersal is defined by the locations where infecting bites occurred in chains 1146\nof transmission, tracing dispersal events backwards through alternating jumps in 1147\nmoving, infected humans and mosquitoes. It is practically impossible to study 1148\ntransmission directly, but this framework has established a quantitative basis for 1149\nstudying transmission through a set of constructs describing closely related processes 1150\nthat can be observed. We have established a basis for describing dispersal rigorously, 1151\nand for analyzing dispersal and simulating transmission. The metrics and concepts we 1152\nhave proposed here are designed to quantify transmission (and uncertainty about 1153\ntransmission) through the study of patterns and the processes that generated them. 1154\nThe metrics provide a rigorous way of quantifying parasite dispersal and spatial 1155\ntransmission intensity. 1156\nIn developing models of a specific place for monitoring and evaluating malaria, it is 1157\nimportant to understand where and when transmission occurs as well as the local 1158\ncontextual factors that shape transmission. In the Ross-Macdonald model, the basic 1159\nnotions of reproductive success, transmission, and community effect sizes of control were 1160\nbased on the abstract notion of a population, but it was never clear how to define a 1161\npopulation for purposes of quantifying malaria transmission dynamics: “What, if 1162\nanything, is a malaria population?” Focal transmission has been described [63], but 1163\nwithout a quantitative basis for quantifying malaria spatial heterogeneity and spatial 1164\ndynamics, there was no basis for a nuanced quantitative discussion about “What, if 1165\nanything, is a focus?” Without defining explicit boundary conditions, it was easy to 1166\nignore malaria importation: “What fraction of malaria in a defined area was 1167\nattributable to local transmission?” Without modeling structured populations, it was 1168\nimpossible to understand how differences in human behaviors would affect 1169\ntransmission [64]. Who is responsible for most local transmission or malaria 1170\nimportation? In malaria control, these discussions have focused on the issue of 1171\nstratification, but it remains unclear whether those strata should define sub-populations, 1172\nspatial areas, or both. Without a framework for understanding malaria transmission 1173\nspatially in heterogeneous populations, it was difficult to develop a consistent 1174\nmethodology for quantifying transmission in a specific place and time. 1175\nWe have synthesized a set of old, new, and revised models to fully develop concepts, 1176\nconstrain parameters, and update basic concepts and metrics in a spatial context. New 1177\nalgorithms have filled a need to connect model parameters with data and remove bias 1178\nwhile guaranteeing mathematical consistency. The new framework and spatial metrics 1179\nmake model complexity scalable, and it provides a way to study the role of context in 1180\nmosquito ecology and malaria transmission. How and why do bionomic parameters vary 1181\nover space and time? What spatial scales characterize mosquito populations? What are 1182\nthe appropriate spatial scales to measure transmission and intervention coverage as a 1183\nspatial average? What are some appropriate methods for dealing with population 1184\nheterogeneity, including heterogeneity arising from differences in behavior, exposure, or 1185\nimmunity? 1186\n40\n . CC-BY 4.0 International licenseIt is made available under a \n is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity.(which was not certified by peer review)preprint \nThe copyright holder for thisthis version posted November 15, 2022. ; https://doi.org/10.1101/2022.11.07.22282044doi: medRxiv preprint \n\nThe framework emphasizes the way we organize our knowledge about malaria into 1187\nbins of expertise. Given the complexity of the problem, this means modellers can build 1188\nmodels that adapt over time as more information about transmission in a place 1189\naccumulated. The first models can focus on components whose dynamics are better 1190\nknown, and use simpler, pragmatic approaches to parts of the model whose mechanistic 1191\nfoundations are more uncertain. Our framework can make building ensembles of 1192\nplausible models to cover this uncertainty easier. Model building and model comparison 1193\nmakes it possible to weigh the importance of various factors in context. In asking where 1194\ntransmission is occurring, we are concerned about mosquito populations, human 1195\nbehaviors, and human blood feeding. In asking who is responsible for malaria, we are 1196\nnot just concerned about differences in infectiousness, but also populations who import 1197\nmalaria, and strata who play an out-sized role in moving malaria around an area. These 1198\nare the basic quantities that play a role in spatial targeting and in tailoring 1199\ninterventions to context. 1200\nConclusion 1201\nThe goal of this study was to develop and present framework that could support model 1202\nbuilding for planning, monitoring and evaluating malaria control programs. Suites of 1203\nmodels developed in this framework can be used to synthesize data, to quantify the 1204\nmajor factors affecting transmission in a particular place, to identify critical data gaps 1205\nand prioritize new data collection, to propagate uncertainty through analyses, and to 1206\nsupport policy. We plan to use the framework to synthesize evidence and to give robust 1207\npolicy advice about malaria control on Bioko Island, and elsewhere, iteratively as part 1208\nof adaptive malaria control. The spatial metrics and concepts describe an important 1209\ndimension of malaria transmission that can help tailor interventions and spatially target 1210\ninterventions. In future studies, we plan to address concerns about the temporal 1211\ndimensions of transmission, including threshold conditions, forcing by weather, and the 1212\nspatial dimensions of malaria control. In adaptive management, the goal is to support 1213\nmonitoring and evaluation by developing rigorous methods that quantify malaria 1214\ntransmission as a changing baseline (e.g., forced by weather) that has been modified by 1215\ncontrol. In other settings, the framework can be used to enhance the design of 1216\nrandomized control trials or to help programs implement and interpret ad hoc 1217\nexperiments to fill local knowledge gaps. Simulation-based analytics in this framework 1218\ncan be updated using evidence collected by malaria programs to update models and 1219\nanalysis and revise policy recommendations, to target and tailor interventions, and to 1220\nuse evidence to adapt to changing local conditions. 1221\nAcknowledgments 1222\nDLS and SLW were supported by a grant from the National Science Foundation, 1223\nDirectorate for Technology, Innovation, and Partnerships (TIP) as part of the 1224\nConvergence Accelerator Program (NSF 2040688). DLS, JMH, and AC were supported 1225\nby a grant from the National Institute of Allergies and Infectious Diseases (R01 1226\nAI163398). DLS was supported by grant from the National Institutes of Health 1227\n41\n . CC-BY 4.0 International licenseIt is made available under a \n is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity.(which was not certified by peer review)preprint \nThe copyright holder for thisthis version posted November 15, 2022. ; https://doi.org/10.1101/2022.11.07.22282044doi: medRxiv preprint \n\n(2U19AI089674). DLS, DMS, and JNN, were supported by a grant from the Bill and 1228\nMelinda Gates Foundation (INV 030600). This project is partly a product of discussions 1229\nwith the mosquito working groups of working groups over several years RAPIDD 1230\n(Research and Policy for Infectious Disease Dynamics). Over that time, many of us 1231\nbenefitted from the unwavering support and inspiration of F. Ellis McKenzie. OJB was 1232\nsupported by a UK Medical Research Council Career Development Award 1233\n(MR/V031112/1). 1234\n42\n . CC-BY 4.0 International licenseIt is made available under a \n is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity.(which was not certified by peer review)preprint \nThe copyright holder for thisthis version posted November 15, 2022. ; https://doi.org/10.1101/2022.11.07.22282044doi: medRxiv preprint \n\nBioko Island: Sectors, Areas, Regions\nW. Malabo\nMalabo\nE. Malabo\nEast \nCoastWest\nCoast\nLuba\nRiaba\nSouth\nFig 8. An important practical concern is spatial granularity of patches for\nsimulation-based studies. For Bioko Island, Equatorial Guinea, for example, we could\ndefine patches at several scales: the whole island; or approximately 240 occupied areas\n(1km× 1km, the squares); approximately 4 , 400 occupied 100m× 100m sectors (points);\nor 8 distinct regions (the colors of the squares); or clusters of contiguous sectors (the\ncolors of the points); or approximately 70,000 individual households. An important\nconcern is that the weight of evidence – the number of observations per patch – declines\nsharply as granularity of the simulations increases. This framework makes it possible to\ndefine a set of nested (or partially nested) studies that modify the number and size of\npatches, which requires modifying the human and mosquito mobility sub-models, but\nthat holds other aspects of the model constant.\n43\n . CC-BY 4.0 International licenseIt is made available under a \n is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity.(which was not certified by peer review)preprint \nThe copyright holder for thisthis version posted November 15, 2022. ; https://doi.org/10.1101/2022.11.07.22282044doi: medRxiv preprint \n\nFig 9. (Left): bulk transmission metric describing transmission from the most densely\npopulated area in Malabo, the capitol city, seen as the bright cell in the Northern tip of\nthe island, to all other populated areas. (Right): bulk transmission from the most\nhighly populated area in the south of the island (Luba), seen as the bright cell in the\nsmall harbor on the Western coast of the island.\nFig 10. Structural Elements of the framework are flexible to facilitate building\nmodels that are appropriate for various settings. These diagrams illustrate two\nexamples. left) A forest malaria model with seven patches (including 3 villages and 2\ncampsites), 6 population strata, and 5 aquatic habitats. The village residents are\nstratified into loggers and other residents. Loggers from different villages spend time at\nhome or in campsites, which have no permanent residents. Aquatic habitats (the\nmoons) can be in villages, in campsites, or in patches near villages. Some villages ( e.g.\nvillage 3), could lack mosquitoes but still have populations at risk. Right) It is also\npossible to model indoor and outdoor blood feeding with indoor and outdoor patches\nthat share the same place. In these models, movement indoors vs. outdoors in the same\nplace is modeled differently from movement among outdoor patches.\n44\n . CC-BY 4.0 International licenseIt is made available under a \n is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity.(which was not certified by peer review)preprint \nThe copyright holder for thisthis version posted November 15, 2022. ; https://doi.org/10.1101/2022.11.07.22282044doi: medRxiv preprint \n\nSupplements 1235\nList of Supplements 1236\n1. Supplement 1 - a github repository https://dd-harp.github.io/exDE/ 1237\n2. Supplement 2 - Glossary 1238\n3. Supplement 3 - Modular Notation 1239\n4. Supplement 4 - Human Travel and Mobility 1240\n5. Supplement 5 - Vector Dynamics 1241\n45\n . 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CC-BY 4.0 International licenseIt is made available under a \n is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity.(which was not certified by peer review)preprint \nThe copyright holder for thisthis version posted November 15, 2022. ; https://doi.org/10.1101/2022.11.07.22282044doi: medRxiv preprint","source_license":"CC-BY-4.0","license_restricted":false}