An age-stratified mathematical model to inform optimal measles vaccination strategies

An age-stratified mathematical model to inform optimal measles vaccination strategies | medRxiv /* */ /* */ <!-- <!-- /*! * yepnope1.5.4 * (c) WTFPL, GPLv2 */ (function(a,b,c){function d(a){return"[object Function]"==o.call(a)}function e(a){return"string"==typeof a}function f(){}function g(a){return!a||"loaded"==a||"complete"==a||"uninitialized"==a}function h(){var a=p.shift();q=1,a?a.t?m(function(){("c"==a.t?B.injectCss:B.injectJs)(a.s,0,a.a,a.x,a.e,1)},0):(a(),h()):q=0}function i(a,c,d,e,f,i,j){function k(b){if(!o&&g(l.readyState)&&(u.r=o=1,!q&&h(),l.onload=l.onreadystatechange=null,b)){"img"!=a&&m(function(){t.removeChild(l)},50);for(var d in y[c])y[c].hasOwnProperty(d)&&y[c][d].onload()}}var j=j||B.errorTimeout,l=b.createElement(a),o=0,r=0,u={t:d,s:c,e:f,a:i,x:j};1===y[c]&&(r=1,y[c]=[]),"object"==a?l.data=c:(l.src=c,l.type=a),l.width=l.height="0",l.onerror=l.onload=l.onreadystatechange=function(){k.call(this,r)},p.splice(e,0,u),"img"!=a&&(r||2===y[c]?(t.insertBefore(l,s?null:n),m(k,j)):y[c].push(l))}function j(a,b,c,d,f){return q=0,b=b||"j",e(a)?i("c"==b?v:u,a,b,this.i++,c,d,f):(p.splice(this.i++,0,a),1==p.length&&h()),this}function k(){var a=B;return a.loader={load:j,i:0},a}var l=b.documentElement,m=a.setTimeout,n=b.getElementsByTagName("script")[0],o={}.toString,p=[],q=0,r="MozAppearance"in l.style,s=r&&!!b.createRange().compareNode,t=s?l:n.parentNode,l=a.opera&&"[object Opera]"==o.call(a.opera),l=!!b.attachEvent&&!l,u=r?"object":l?"script":"img",v=l?"script":u,w=Array.isArray||function(a){return"[object Array]"==o.call(a)},x=[],y={},z={timeout:function(a,b){return b.length&&(a.timeout=b[0]),a}},A,B;B=function(a){function b(a){var a=a.split("!"),b=x.length,c=a.pop(),d=a.length,c={url:c,origUrl:c,prefixes:a},e,f,g;for(f=0;f<d;f++)g=a[f].split("="),(e=z[g.shift()])&&(c=e(c,g));for(f=0;f<b;f++)c=x[f](c);return c}function g(a,e,f,g,h){var i=b(a),j=i.autoCallback;i.url.split(".").pop().split("?").shift(),i.bypass||(e&&(e=d(e)?e:e[a]||e[g]||e[a.split("/").pop().split("?")[0]]),i.instead?i.instead(a,e,f,g,h):(y[i.url]?i.noexec=!0:y[i.url]=1,f.load(i.url,i.forceCSS||!i.forceJS&&"css"==i.url.split(".").pop().split("?").shift()?"c":c,i.noexec,i.attrs,i.timeout),(d(e)||d(j))&&f.load(function(){k(),e&&e(i.origUrl,h,g),j&&j(i.origUrl,h,g),y[i.url]=2})))}function h(a,b){function c(a,c){if(a){if(e(a))c||(j=function(){var a=[].slice.call(arguments);k.apply(this,a),l()}),g(a,j,b,0,h);else if(Object(a)===a)for(n in m=function(){var b=0,c;for(c in a)a.hasOwnProperty(c)&&b++;return b}(),a)a.hasOwnProperty(n)&&(!c&&!--m&&(d(j)?j=function(){var a=[].slice.call(arguments);k.apply(this,a),l()}:j[n]=function(a){return function(){var b=[].slice.call(arguments);a&&a.apply(this,b),l()}}(k[n])),g(a[n],j,b,n,h))}else!c&&l()}var h=!!a.test,i=a.load||a.both,j=a.callback||f,k=j,l=a.complete||f,m,n;c(h?a.yep:a.nope,!!i),i&&c(i)}var i,j,l=this.yepnope.loader;if(e(a))g(a,0,l,0);else if(w(a))for(i=0;i (function(w,d,s,l,i){w[l]=w[l]||[];w[l].push({'gtm.start':new Date().getTime(),event:'gtm.js'});var f=d.getElementsByTagName(s)[0];var j=d.createElement(s);var dl=l!='dataLayer'?'&l='+l:'';j.src='//www.googletagmanager.com/gtm.js?id='+i+dl;j.type='text/javascript';j.async=true;f.parentNode.insertBefore(j,f);})(window,document,'script','dataLayer','GTM-P4HH5NV'); Skip to main content Home About Submit ALERTS / RSS Search for this keyword Advanced Search An age-stratified mathematical model to inform optimal measles vaccination strategies Samiran Ghosh , View ORCID Profile Indrajit Ghosh , Siuli Mukhopadhyay doi: https://doi.org/10.1101/2025.04.01.25325066 Samiran Ghosh a Department of Mathematics, Indian Institute of Technology Bombay , Maharashtra, India Find this author on Google Scholar Find this author on PubMed Search for this author on this site Indrajit Ghosh b National Disease Modelling Consortium, Indian Institute of Technology Bombay , Maharashtra, India Find this author on Google Scholar Find this author on PubMed Search for this author on this site ORCID record for Indrajit Ghosh Siuli Mukhopadhyay a Department of Mathematics, Indian Institute of Technology Bombay , Maharashtra, India Find this author on Google Scholar Find this author on PubMed Search for this author on this site For correspondence: siuli{at}math.iitb.ac.in Abstract Full Text Info/History Metrics Data/Code Preview PDF Abstract Measles a highly transmissible viral disease continues to pose a major public health threat globally, despite the availability of an effective vaccine. The World Health Organisation recommends that the first dose of the measles vaccine be administered between 9–12 months of age, based on the risk of maternal antibody interference and the timing of an infant’s immune system maturity. However, infants younger than 9 months remain vulnerable to measles, particularly in areas with high transmission rates and low herd immunity. To address the interplay between maternal immunity decay, early infection risk, and vaccination effectiveness, this study considers a new age-stratified, multi-compartmental model of measles transmission that explicitly accounts for infections in children too young to be vaccinated. The novelty of the proposed model lies in incorporating the dynamics of maternal immunity loss and infection progression within the maternal immune compartment, addressing a crucial factor influencing disease susceptibility in very young age groups. The positivity of the solutions of the age-stratified model is proved, and an analytical expression is derived for the effective reproduction number. Moreover, for the model without age stratification, the existence of equilibrium points and the local and global stability are studied. Global sensitivity analysis is used to identify critical parameters that affect the incidence of measles. Using a bootstrap algorithm, the model estimates age-specific transmission rates by fitting it to yearly incidence data from six countries where measles is prevalent. Different vaccination strategies are thoroughly examined, classified by the age of administration for routine and supplementary doses, as well as by the frequency of supplementary doses. Furthermore, the impact of an additional zero-dose measles vaccination for children aged 6 to 9 months is extensively analyzed. The estimated number of averted cases and the projected year of elimination are noted in different strategies for the six countries. In addition to routine measles immunization activities, combining conventional supplementary immunization activities with an additional zero-dose measles vaccination for children aged 6 to 9 months emerges as the most effective strategy. The findings aim to inform vaccine policy updates and contribute to enhanced global measles eradication efforts. 1. Introduction Measles, an extremely contagious viral disease, continues to pose a major global public health challenge despite the availability of a highly effective vaccine. Measles routine immunization (RI) activities as recommended by World Health Organization (WHO) consist of two doses of the measles vaccine, measles containing vaccine 1 (MCV1) and measles containing vaccine 2 (MCV2), respectively, at 9-12 months and 15-24 months of age with at least 95% coverage [ 1 ]. In addition, several measles-endemic countries conduct supplementary immunization activities (SIAs) through which vaccines are administered to specific age groups regardless of their RI status [ 2 ]. Although vaccination has been successful in preventing an estimated 60 million deaths between 2000 and 2023, the disease is still prevalent in large parts of Africa and Southeast Asia. From the RI data, we note that in Southeast Asia, vaccination coverage has steadily increased to 91% for MCV1 and 85% for MCV2, however, several regions in those parts of the world are still recording frequent measles outbreaks. Thus, there is an urgent need for identifying the factors responsible for these continued measles outbreaks and in investigating alternative measles vaccination strategies. WHO recommends that the MCV1 be administered at 9 to 12 months of age, based on the risk of maternal antibody interference and the time of maturity of the infant’s immune system [ 3 ]. However, infants younger than 9 months remain vulnerable to measles [ 4 ], particularly in areas with high transmission rates and low herd immunity. A surveillance study suggested that 12% of measles cases reported to WHO were too young to be vaccinated [ 5 ]. This is of great concern as [ 6 ] notes a higher risk of developing severe symptoms in measles-infected infants of less than 9 months of age. Also, the proportion of measles cases in infants is likely to increase as more mothers have vaccine-induced immunity, resulting in their children having lower maternal antibody levels as compared to those of naturally immune mothers [ 4 ]. Recent studies suggest that vaccinating infants aged 6-9 months could bridge this vulnerability gap and contribute to better measles control [ 4 ]. However, higher protection and seroconversion were seen in children who received MCV1 at a higher age. The optimal age of vaccination of MCV1 particularly, is still an active area of research which depends on various factors such as the maturity of the immune system to respond to MCV, the age at which children are at risk of infection and the objectives of national immunization programmes. A survey analysis based on case-based measles surveillance data from WHO headquarters, spanning 2011–2016, reveals a significant number of measles cases in the 6–8 month age group [ 7 ] (see Figure 1 ). Infants under one year of age are particularly vulnerable, as they are typically too young to receive immunization under most current vaccination guidelines and are no longer sufficiently protected by maternally transmitted antibodies. These maternal antibodies generally provide protection for an average of 3–6 months in infants born to naturally immune mothers, but this period is even shorter for infants of vaccinated mothers [ 8 ; 9 ; 10 ]. This evidence underscores the critical need to reduce measles cases among children younger than 9 months. In fact, countries like South Africa, Cambodia and Malaysia (Sabah state) have implemented MCV1 vaccination at 6 months whereas Sri Lanka and Colombia have administered SIA measles vaccination starting at 6 months age group [ 11 ; 12 ]. Although early vaccination offers a potential solution, its impact is limited by the reduced effectiveness of vaccines in younger age groups [ 13 ; 14 ]. This highlights the necessity of re-evaluating the balance between the benefits of early vaccination and the challenges posed by diminished vaccine effectiveness. Download figure Open in new tab Figure 1: Average annual age-specific incidence of measles during 2011-2016 (adapted from [ 7 ]). Age-stratified compartmental mathematical models have been widely used to study the dynamics of measles and evaluate the impact of vaccination strategies. Age stratification is particularly relevant for measles, as susceptibility, transmission, and immune responses vary significantly across age groups due to factors, namely waning maternal antibodies and different susceptibilities [ 15 ]. By integrating demographic data, vaccination coverage, and disease parameters, these models provide a robust framework for identifying high-risk age groups and optimizing location-specific vaccination strategies [ 2 ; 16 ]. Several age-structured measles models have been previously employed to study vaccination scenarios. Verguet et. al. [ 17 ] developed an age-stratified measles model called DynaMICE to compute the maximum allowable time between two consecutive SIAs for achieving measles control. They found that multiple SIAs at high coverage levels and regular intervals is a viable strategy to prevent measles outbreaks. Prada et. al. [ 18 ] found that a single high-coverage SIA can effectively maintain measles elimination, with follow-up campaigns potentially requiring smaller investments in 4 African populations. Trentini at. al. [ 16 ] calibrated a compartmental model for 9 countries using historical serological data to estimate disease burden reduction and age-specific susceptibility. Another compartmental modelling study [ 19 ] found that current vaccination policies are not be sufficient to achieve and maintain measles elimination countries under study. They concluded that strategies targeting unvaccinated children before they enter primary school can remarkably enhance the fulfillment of WHO targets. Fu et. al. [ 20 ] observed that high coverage of both RI doses and SIA is essential to achieve and maintain measles elimination goals. Winter et. al. [ 21 ] concluded that achieving and maintaining the elimination of measles and rubella worldwide will require innovative vaccination strategies, technologies to address the inequities in routine coverage, and continued investment in surveillance and outbreak response. Auzenbergs et. al. [ 2 ] found that to achieve high levels of vaccination coverage and meet targets for measles elimination in areas of high burden, SIAs must be strengthened and made more efficient, but routine two-dose coverage must also be improved over time. Recently, Goult et. al. [ 22 ] used an agestratified compartmental model to estimate the optimal age for infant measles vaccination. They found large heterogeneity in the optimal MCV1 ages, ranging 6–20 months of age. They showed that the optimal MCV1 age depends on the local epidemiology, with a lower optimal age predicted in populations having lower vaccination coverage or suffering from higher transmission. The requirement of controlling measles in infants under 9 months of age remains an important problem in measles endemic countries. In addition, very few of these models were calibrated according to the observed country-specific measles incidence data. The literature on examining the potential effect of vaccination of children in younger age groups ( < 9 months) through mathematical models is sparse. Therefore, there is scope to study model-based projections of measles elimination for various scenarios, such as early routine MCV1 vaccination, additional doses of vaccine before 9 months of age, and SIAs with revised lower age limits combined with RI. In this work, we propose a new age-stratified compartmental model of measles transmission to take into account measles infections in children who are too young to be vaccinated. The novelty of our model lies in the incorporation of maternal immunity loss and infection progression within the maternal immunity compartment, addressing a crucial dynamic that affects disease susceptibility over time in very young age groups. To achieve the elimination of measles by addressing the reduction in measles cases in the very young age group, we primarily explore the following vaccination strategies. Strategy 1: Routine MCV1, MCV2 with conventional SIAs at different frequencies. Strategy 2: Early MCV1 (with reduced efficacy), routine MCV2 with or without conventional SIAs at different frequencies. Strategy 3: Routine MCV1, MCV2 plus targeted SIAs with revised lower age limit (starting from 6 months). Strategy 4: Routine MCV1, MCV2 with conventional SIAs at different frequencies and an additional zero dose measles (MCV0) at 6-9 months. The zero dose of the measles vaccine, known as the MCV0 dose or supplementary dose, is specifically recommended for infants aged 6 to 9 months in selected geographies where the risk of measles infection is particularly high (see page 37 in https://iris.who.int/bitstream/\handle/10665/380106/9789240103399-eng.pdf?sequence=1 ). Although routine immunization schedules typically begin at 9 months of age with the MCV1 dose, the MCV0 dose acts as an early protective measure for vulnerable populations, especially during outbreaks or in settings with elevated exposure risk [ 23 ]. In strategy 4, we implement the zero-dose option at a single time point in a given year and not as part of RI activities. In addition, we calibrate our proposed model to country-specific measles data for six countries where measles is prevalent, namely Nigeria, Philippines, Indonesia, Democratic Republic of the Congo (DR), India, and Pakistan. We use MCV1 and MCV2 coverages, SIA coverages, and demographic variables (population, birth rate, death rate) as inputs to the model. The findings of this research may be instrumental in the shaping of future public health policies and vaccination strategies, potentially leading to a revision of current guidelines. Understanding the implications of early measles vaccination is crucial to achieve broader coverage of immunization and advance toward global goals for measles elimination. 2. Model description In this section, we propose an age-stratified model for studying the effects of early vaccination on measles transmission. Since measles predominantly affects children under 5 years [ 24 ], we use age stratifications within this range (0-3, 3-6, 6-9, 9-12 months, and 1-year intervals from 1 to 5 years) to capture key developmental and immunological changes, particularly waning of maternal immunity. For the remaining age groups, for simplicity, broader intervals (5-15 years and 15-80 years) are considered, since this work focuses mostly on the impact of various vaccination strategies targeting children under 5 years of age. The population is divided into age groups and divided into eight epidemiological compartments: maternally immune ( M ( t )), susceptible ( S ( t )), infected ( I ( t )), recovered ( R ( t )), vaccine failures ( F ( t )), successfully vaccinated with MCV1 ( V 1 ( t )), successfully vaccinated with MCV2 ( V 2 ( t )), and successfully vaccinated through SIA ( V SIA ( t )), where t represents chronological time. Successfully vaccinated children are those who have received any dose of the measles vaccine and have developed immunity to measles. We assume that children in class M are partially immune and that their immunity decays as they age. The compartments in a particular age group i are denoted by subscript i in the respective compartments. The youngest age group, 0 to 3 months, is represented by i = 1, and the index increases with age groups. The oldest age group, which includes individuals aged 15 to 80 years, is indicated by i = 10. Maternally immune children can become fully susceptible due to loss of maternal immunity. Susceptible individuals can either acquire infection or receive vaccination. Vaccination can lead to successful immunization or failure, in which case individuals remain susceptible and may require additional doses [ 16 ]. If infected, individuals move to the I compartment, where they contribute to disease transmission before moving to the R class, gaining long-term immunity. The flow diagram of the proposed model is shown in Fig. 2 . We denote this model as the primary model. Later, we introduce an alternative model to implement strategies 3 and 4. Download figure Open in new tab Figure 2: Flow diagram of the proposed model for a single age group without births and deaths. To reiterate, the proposed model incorporates loss of maternal immunity and progression of infection in the maternal immunity compartment, allowing us to focus on the problem of susceptibility to measles over time in very young age groups. This addition is significant because it captures an important subgroup that could contribute to disease transmission and is usually ignored in the literature on measles modeling. A similar model was explored by [ 16 ], but unlike their approach, our model explicitly incorporates the interaction between infection risk in maternally immune individuals and the reduction in vaccine effectiveness due to early measles vaccination in children 6-9 months of age, offering a more comprehensive understanding of how these factors shape the dynamics of the measles epidemic. The dynamics of disease transmission are described by the following system of differential equations: Where Here n represents the number of age groups. The birth and natural death rates in age group i are denoted by b i and d i , respectively. The disease dynamics are governed by the recovery rate γ and the transfer rate µ i from maternally immune individuals ( M i ) to susceptible individuals ( S i ) within each age group. The transmission dynamics are further characterized by β i ( t ), the age-specific transmission rate, which includes seasonal variations controlled by , and , the age-specific rate of transmission. For simplicity, the transmission rates are assumed to be the same within the four age groups under 1 year , the same within the four age groups between 1 and 5 years , and the same within the two age groups between 5 and 15 years , and the same for the 15+ years age group ( ). These rates are assumed to be different due to the variation in contact patterns among different age classes [ 25 ]. Additionally, κα 2 ( i ) represents an adjustment factor or transition probability that modifies the force of infection (Λ i ( t )) for maternally immune individuals in age group i , reflecting the interaction between disease exposure and waning maternal immunity, lastly δ denotes the Kronecker delta function. Vaccination strategies are defined by the efficacy of the first dose ( ϵ 1 ), second dose ( ϵ 2 ), and SIAs ( ϵ s ), along with their respective coverage levels c 1 ( t ), c 2 ( t ), and c s ( t ). V 1 S and V 2 S account for MCV1 coverage with SIA and MCV2 coverage with SIA respectively. The term ω i accounts for changes in the age-group distribution due to natural population growth in age group i . The targeted age groups for vaccination are defined by the parameters , and . Here, i 1 denotes the age group in which the first dose (MCV1) is administered, while i 2 represents the age group receiving the second dose (MCV2). The SIAs are targeted at age groups ranging from to . The flexibility of these parameters allows for the representation of various vaccination strategies. Mathematically, all parameters , and range from 1 to n . The model parameters and some descriptions are reported in Table 2 . 3. Mathematical properties 3.1 Positivity and boundedness Demonstrating the well-posedness of the model requires proving that it is both positive and bounded. Positivity guarantees that solutions remain within meaningful and realistic ranges, while boundedness ensures the solutions exhibit stable behavior. These attributes are crucial for establishing the model’s reliability and practical utility, instilling confidence in its analysis and predictive capabilities within the given context. 3.1.1 Positivity To prove the positivity of the system 1, we use the approach described in [ 29 ]. We set, for j = 1, 2, · · ·, 10. Now for fixed i ∈ {1, 2, · · ·, 10}, we have the following: Hence, is invariant under the system (1). This completes the proof of positivity of the system (1). 3.1.2 Boundedness Adding all the equations in the system (1) we obtain: Suppose , and b ( t ), where T is the maximum time of observation. Then we can write Suppose , and ω 0 = ω n = 0. Summing i from 1 to n and assuming we obtain: This completes the proof of positivity and boundedness of solution of the system 1. 3.2 Effective reproduction number The effective reproduction number, R e ( t ), is a fundamental metric for evaluating the transmissibility of an infectious disease and assessing the potential impact of control measures. It is defined so that if R e ( t ) > 1, the infection will spread, whereas if R e ( t ) < 1, the infection will decline at time t . To derive its expression at a specific time t = t 0 , we assume that, for all i ∈ {1, 2, · · ·, n }, around t 0 , the infected population I i ( t ) follows the solution , where λ is the rate of exponential growth or decline of the infection whose sign characterizes the effective reproduction number. By substituting into equation (1d), evaluating at t = t 0 , and summing over i , we obtain Since ω 0 = ω n = 0, we have . Consequently, Defining the proportion of infections in each age group i , by, , we obtain Thus, we can define the effective reproduction number at time t 0 as follows Here, the numerator reflects the transmission potential normalized by the effective population at risk, while the denominator quantifies the overall rate at which infections are removed from the population, either by recovery or death. This characterization of ℛ e ( t 0 ) will be utilized in the results section to analyze epidemic growth over the years across different countries. Additional mathematical analysis on local and global stability was conducted for a simplified age-independent version of the model (1), see Appendix Appendix A for details. In the following sections, using our proposed age stratified model, we study the incidence and various strategies for the elimination of measles in six selected countries. A detailed description of the country-specific data that are needed as inputs to the model is provided. Estimation of age-specific transmission rates by calibrating the model with yearly measles incidence country specfic data is discussed. Using these calibration results, we analyze the sensitivity of model parameters on cumulative cases based on the partial rank correlation coefficients (PRCCs) method. The variation of the effective reproduction number estimated by the model over time is examined for each country. The effects of various vaccination strategies for the treatment of measles in very young age groups are studied and the relative number of averted cases for each country are estimated. We also explore the country-specific expected year for measles elimination under different strategies. 4. Data description We consider data from six measles prevalent countries in Africa and South East Asian regions: Nigeria, Philippines, Indonesia, Democratic Republic of Congo (DR Congo), India and Pakistan [ 2 ]. The reported cases of measles from 2000 to 2023, the coverages of MCV1 and MCV2 for the six countries have been obtained from the WHO immunization dashboard https://immunizationdata.who.int/ , while the coverages of SIA for the period 2000 to 2020 have been taken from Auzenbergs et al. [ 2 ]. SIA administration ages for the countries Nigeria, Philippines, Indonesia, DR Congo and Pakistan are taken as 9-59 months, while for India it is a age group of 9 months - 15 years. The initial population and initial age distribution of the countries are extracted from https://www.populationpyramid.net/ . Immunization coverage of MCV1, MCV2 and SIA along with WHO reported cases for the six countries is shown in Fig. 3 . Among the six countries, India reports the lowest measles cases per million with at least 85% coverages for both MCV1 and MCV2. Download figure Open in new tab Figure 3: Immunization coverage for MCV1, MCV2 and SIA are depicted for - Nigeria, Philippines, Indonesia, DR Congo, India and Pakistan. Reported measles cases per million are also shown. 5. Model calibration We calibrate the model (1) to yearly reported measles cases from the six countries of interest. Corresponding coverage data of MCV1, MCV2 and SIA for the six countries are given as inputs in the calibration process. The built-in optimization function lsqcurvefit in MATLAB ( https://in.mathworks.com/products/matlab.html ) was used to perform a non-linear least-squares fitting [ 30 ; 31 ]. In addition, we utilize the parametric bootstrapping technique to quantify the uncertainty of the parameters and obtain the 95% confidence intervals (CIs) of the estimated parameters [ 32 ; 33 ]. We generate 1000 bootstrap samples from the best-fitted model assuming a normal error structure [ 32 ]. For the 1000 reconstructed time series data, the model was fitted using nonlinear least squares. The 1000 sets of parameters obtained are used to characterize the empirical distribution and the corresponding 95% CIs of the parameters. Consider the model (1) to be of the form where denotes the rate of change in the state variables ξ i ( t ) where i = 1, …, 90 and Θ is the set of model parameters. The steps for performing the model calibration are given below for a specific country: Derive the parameter estimates through least-square fitting using time series data Y = ( y 1 , y 2 , y 3 , …, y n ) to obtain the best fitted model. Generate bootstrap replications of the obtained best fitted model assuming a normal error structure. Re-estimate the parameters by fitting the model to the 1000 simulated time series data. Construct empirical distribution and confidence intervals of the estimated parameters. A schematic flow diagram of the model calibration process in displayed in Fig. 4 . We assume that only ρ proportion of cases are reported for the countries. Therefore, we fit the model generated infections multiplied by the reporting ρ with the actual reported cases to WHO. Using this method, Download figure Open in new tab Figure 4: Flow diagram of the model calibration and forecast scenarios. we estimate six parameters and ρ , for each country. The model fit is displayed in Fig. 5 for the six countries. These estimated parameter values presented in Table 2 are used in the numerical simulations presented in the following section. Download figure Open in new tab Figure 5: Fitting the model (1) to WHO reported measles cases in the countries Nigeria, Philippines, Indonesia, DR Congo, India and Pakistan respectively. The blue circles represent observed data, solid red line indicate mean model fit and the dotted red lines represent 95% CI. 6. Numerical results In this section we perform numerical simulations based on the proposed age stratified model (1). 6.1 Sensitivity analysis Global sensitivity analysis allows us to identify influential parameters while studying disease transmission. Here we use the PRCC method as discussed in Marino et al. [ 34 ]. The input parameters for the sensitivity analysis are: transmission rates for the different age groups ; modification parameter for transmission rates of the maternal immune class, κ ; respective coverage and efficacies of MCV1 and MCV2, c 1 , c 2 , ϵ 1 , ϵ 2 . The model (1) is simulated for a general hypothetical framework considering the effects of only routine doses of MCV1 and MCV2 vaccinations. The initial population size is taken to be 1 million. All parameters are varied in the range (0,1) except ϵ 1 and ϵ 2 , these are, respectively, varied between (0,0.95) and (0,0.98). The response variable is the aggregate incidence of measles over 30 years . This global sensitivity analysis quantifies the impact of the parameters on the response variable. Latin Hypercube Sampling (LHS) is used in generating 5000 samples of each parameter from the corresponding ranges. The PRCCs are then calculated using the MATLAB functions provided in [ 34 ]. The results are shown in Fig. 6 . Download figure Open in new tab Figure 6: Sensitivity of model parameters with respect to aggregated measles incidence over 30 years. Significant PRCCs are indicated as * (p-value < 0.05). Fixed parameters are taken as in Table 1 . It is observed that , that is, the transmission rate for age groups 1-5 years is the most positively correlated parameter with the response variable , followed in decreasing order by and . On the other hand, ϵ 1 is the parameter most negatively correlated with , followed by c 1 . These results imply that while an increase in age group-wise transmission rates significantly (at level 5%) increases total measles cases in the next 30 years, an increase in coverage and efficacy of MCV1 vaccination significantly decreases measles cases in the next 30 years. View this table: View inline View popup Download powerpoint Table 1: Description of the model parameters with corresponding values or ranges. View this table: View inline View popup Download powerpoint Table 2: Table of estimated parameters for the six countries with mean and corresponding confidence intervals. 6.2 Estimates of effective reproduction number The effective reproduction number ( R e ( t )) is a key metric in assessing the transmissibility of an infectious disease, representing the average number of secondary infections generated by an infected individual in a population made up of both susceptible and non-susceptible hosts. Using parameter estimates from Table 2 and fixed parameter values from Table 1 in Equation (5), we plot the estimated R e ( t ) for the six countries shown in Fig. 7 . During the period 2020 to 2023, we observe that the R e ( t ) values remain within the range (0.85, 1.15) with variations between countries, reflecting differences in vaccination coverage, demographic variables, and measles surveillance systems. A similar trend was observed in R e ( t ) over the years in [ 35 ], with oscillations around the endemic situation. Download figure Open in new tab Figure 7: Plot of the estimated effective reproduction number over time from 2000 to 2023. The solid blue line represents the mean estimates, while the dashed blue lines indicate the 95% confidence interval. The horizontal red dashed line corresponds to R e ( t ) = 1. The slope of R e ( t ) is positive during 2022 − 2023 in Philippines and Indonesia, indicating the possibility of future outbreaks unless remedial measures are taken. The mean values of R e ( t ) are > 1 in multiple countries reflecting the need for more intense vaccination campaigns. For Nigeria and DR Congo, the 95% CIs of R e ( t ) are wider than other countries, indicating the possibility of higher values of R e . During 2021-2022, the values of R e ( t ) for DR Congo, Pakistan, Nigeria, and India are seen to be ( > 1), possibly indicating the effect of the pandemic on vaccination campaigns. Differences in estimates of R e over time and space reveal the varying effect of interventions in each country in any given year, while highlighting the need to tailor vaccination strategies to achieve measles elimination and mitigate measles transmission according to the epidemiological setting. 6.3 Scenario analysis To assess the impact of vaccination interventions on measles control and elimination, we consider 4 strategies. Further, under each broad strategy, we study different scenarios (see Table 3 ). We start with a baseline scenario that consists only of RI activities. RI with the standard timing of vaccination plus conventional SIAs (varying frequencies over years) form scenarios 1-3. Pre-ponement of MCV1 to 6-9 months and MCV2 at standard timing with and without conventional SIAs at different frequencies form scenarios 4-7. In scenarios 4-7, it is also assumed that MCV1 preponing induces 25% a reduction in vaccine effectiveness [ 36 ]. RI plus targeted SIAs with revised lower age limit starting from 6 months constitute scenarios 8 and 9. In scenarios 10 and 11 which are under strategy 4, we have RI combined with measles zero-dose (MCV0) at 6-9 months and conventional SIAs at different frequencies. To recap, Scenarios 1-3 represent Strategy 1 , Scenarios 4-7 represent Strategy 2 , Scenarios 8-9 represent Strategy 3 and lastly, Scenarios 10-11 represent Strategy 4 . For children over 9 months, the conventional SIA coverages vary by country: 35% in Nigeria, the Philippines, Indonesia, and DR Congo; 16.5% in India; and 80% in Pakistan, which are based on the range of the last SIA coverages observed during the training period for each country. In scenarios 8 and 9, the targeted SIAs have mixed vaccine coverages: 28% for children 6-9 months and country-specific coverages for children over 9 months. Specifically, for India in scenario 9, 6-9 months, children receive SIA doses with 28% coverage and over 9 months, children receive SIA doses with 16. 5% coverage. In Scenarios 10 and 11 with MCV0, we have considered the MCV0 coverage to be 28% for children aged 6 to 9 months in all countries, and as in Scenarios 8 and 9, the conventional SIA coverages for children over 9 months are country-specific. View this table: View inline View popup Download powerpoint Table 3: Summary of vaccination scenarios considered for analysis We use parameter values from Tables 1 and 2 to generate the infections from the model (1) for the period 2024 to 2050. The routine MCV1 and MCV2 coverages in the years 2024 to 2050 are assumed to remain the same as of 2023 in a given country. Thus, we do not consider intensified vaccine coverages in the forecasting period. Furthermore, SIA administration ages for the countries Nigeria, Philippines, Indonesia, DR Congo, and Pakistan are taken as 9-59 months, whereas for India it is 9 months - 15 years age group [ 2 ]. The birth and death rates in the forecasting period are assumed to follow a linear trend similar to that in the calibration period. For implementing the zero dose measles (MCV0) in strategies 3 and 4, we propose an alternative model (1) with added equations to incorporate the additional dose. Individuals successfully vaccinated with MCV0 are assumed to be in the ( V 0 ) compartment. The additional differential equation for V 0 and the modified equations for compartments, M and V 1 , are given below. The efficacy of zero dose is indicated by ϵ 0 with the corresponding coverage level c 0 ( t ) = 0.28. For zero doses, we use efficacy ϵ 0 = 0.70 [ 36 ] same as ϵ s . Note that infants who receive the MCV0 dose will subsequently receive all routine doses, including MCV1 and MCV2. For simplicity, it is assumed that the individuals for whom MCV0 fails remain in the M compartments so that they receive routine vaccines according to the schedule. To accommodate the additional compartment of V 0 , in the flow diagram in Fig. 2 , we should add the compartment V 0 that receives the inflow of individuals from the M compartment and is connected by an outward arrow to V 1 . The country-specific forecasted cases in the period 2024-2050 are shown in Figs. B.10 – B.15 . For Nigeria, we note that the forecasted measles counts show a steady increase in case counts if Strategy 1 is followed. Preponement of MCV1 with and without SIAs forming Strategy 2 fails to stop the increase in the forecasted case counts. However, the use of additional doses of MCV0 under Strategies 3 and 4 is more effective in curbing the increase in the case counts though there are still 0.25 × 10 5 reported cases in 2050. In Philippines, Strategy 1 is not helpful; however, Strategy 2 is able to decrease the case counts compared to strategy 1. As before, Strategies 3 and 4 bring down the reported measles cases. For Indonesia, we see a steep decrease in case counts with conventional Strategy 1. The cases are further reduced with the second Strategy. The addition of MCV0 to vaccination campaigns proves to be very successful, showing the possibility of elimination by 2030 with no resurgence. The partial success of the second strategy in both the Philippines and Indonesia may indicate higher infections in children younger than 9 months in these two countries. From the literature, we noted a recent outbreak in Indonesia where 10% of the cases were found in children less than 9 months [ 37 ]. Also, Domai et al. [ 38 ] reported the interquartile age range of measles infection in the Philippines to be between 7-28 months. Strategy 4 is able to bring down the cases of measles to zero without further resurgence in Indonesia by 2030, while for the Philippines, it takes much longer. The forecasted incidence of measles in DR Congo shows a steep upward trend under RI activities. Adding SIAs to RI under Strategy 1, is not very effective in reducing reported cases unless the frequency of SIAs is high. Strategy 2 performs better as compared to Strategy 1, particularly with more frequent SIAs. The additional MCV0 combined with frequent SIAs shows that the cases drop to zero by 2030 with no further resurgence in DR Congo. The high coverages of MCV1 and MCV2 reported in India show continued RI activities at these coverages, and combining them with conventinal SIAs is effective in reducing the burden of measles. Strategy 2 shows a small improvement over Strategy 1 in the forecasted measles cases. The addition of MCV0 to the vaccination schedules and increasing the frequency of SIAs shows a drop in the measles cases before 2030 with no further resurgence till 2050. In-sample measles cases in Pakistan are on average much lower than other countries except India, this may be due to low reporting rates as noted previously by [ 39 ; 32 ]. This large under-reporting of cases is suspected to produce lower forecasted measles case values. For Pakistan, almost all strategies seem to be effective in lowering future cases, showing an early decrease in measles before 2030 with no further resurgence. Also, RI combined with high-frequency SIAs (scenario 7) gives slightly better results than those with MCV0. From these results, it is evident that SIA, pre-exposure of MCV1 to 6-9 months and adding a measles zero-dose have different rates of beneficial effects on reducing the measles cases in the six countries. Overall, scenarios 10 and 11 perform the best for all countries. The relative averted cases (RACs) are computed as the relative difference in the total number of cases (over 2024-2050) under the base scenario and the total number of cases for a certain scenario. These RACs and their interval estimates are depicted in Fig. 8 for the six countries. Download figure Open in new tab Figure 8: Different color bars represent percent of relative averted cases from 2024 to 2050, for different Scenarios, with respect to the base scenario, as discussed in the text. The averted cases are shown for the countries Nigeria, Philippines, Indonesia, DR Congo, India and Pakistan respectively. As mentioned above, Nigeria performs the worst among all countries, reporting less than 80% RAC even for scenario 11. Philippines and Indonesia show at least 80% RACs for Scenarios 8-11. DR Congo measles case counts are greatly affected by combining RI with high-frequency SIAs, reaching at least 80% RAC for scenarios 3, 6-11. India shows a reasonable trend of decreasing cases of measles even under the base scenario of RI activities, thus, we note an above 80% RAC value for enhanced scenarios 7-11. Except for scenarios 1 and 4, in Pakistan, all others show more than 80% RAC value. To better understand the interaction between RACs and year-to-elimination (i.e., achieving less than 5 cases per million per year with no further resurgence), we refer to Fig. 9 . We observe that scenarios with zero dose vaccination for measles (scenarios 8-11) perform better than other scenarios (RI + SIA, routine MCV1 preponed + MCV2 + SIA) in terms of RACs except in Nigeria. It is also observed that these same scenarios lead to the elimination of measles in all countries except Download figure Open in new tab Figure 9: Relative averted cases and year to elimination under different scenarios for the six countries. The size of the bubbles indicates the RACs and the colour of the bubbles represents year to elimination for the corresponding scenarios. Nigeria. Among the zero-dose scenarios, scenario-11 outperforms other settings. However, scenario 9 (Strategy 3) is much more economically viable as we are implementing an RI with targeted SIAs every three years with lower age limit reduced to 6 months of age. In terms of measles elimination, Strategy 1 and Strategy 2 appear to be less effective, leading to delays in the achievement of elimination in all countries. In contrast, Strategies 3 and 4 perform better in most countries, with the potential to eliminate measles everywhere except in Nigeria. Based on the available resources in a given country, policymakers may adopt any one of the suitable scenarios to eliminate measles. 7. Discussion and conclusion Efforts towards early elimination of measles through various vaccination campaigns are ongoing in all six WHO regions. However, measles remains a major problem for healthcare organizations in many countries with frequent outbreaks. In this study, we formulate an age-stratified measles transmission model to examine the benefits of vaccination to infants younger than 9 months of age. This study explores the dynamics of transmission, immunity gaps, and the effectiveness of interventions in six countries where measles is still endemic. Another distinctive feature of this study is the proposal to include a zero-dose measles vaccination compartment to tackle the problem of measles in children too young to be vaccinated. We establish the positivity and boundedness of the model solutions and analytically obtain the expression of the effective reproduction number for the proposed model. From the sensitivity analysis, we observed that an increase in transmission rates significantly increases the total number of measles cases, while an increase in the efficacy of MCV1 coverage significantly decreases total measles cases ( Fig. 6 ). Using reported measles cases, coverages of MCV1, MCV2, SIA and demographic variables from the six countries, we calibrated the proposed model ( Fig. 4 ). The effective reproduction number ( R e ( t )) is crucial for assessing measles transmissibility and evaluating the impact of vaccination strategies [ 40 ]. For the six countries, between 2020 and 2023, R e ( t ) values generally ranged from 0.85 to 1.15. The estimated trends of R e ( t ) provide a valuable means to predict potential outbreaks, enabling timely planning of targeted interventions, such as Outbreak Response Immunization (ORI) campaigns, to mitigate epidemic risks. Furthermore, the modeling framework utilized in this study allows for forecasting R e ( t ) trends under various scenarios, offering critical insights into the anticipated impact of different vaccination strategies and guiding evidence-based policymaking to enhance measles control efforts. Scenario analysis for the forecast period (2024-2050) highlights the effectiveness of integrated vaccination strategies in reducing measles cases. Among the scenarios evaluated, Strategies 3 and 4, combining SIAs with routine MCV and zero-dose vaccine administration for children aged 6 to 9 months, emerged as the most impactful ( Fig. 8 ). However, if strategy 4 is not viable in all countries due to logistical and economical constraints, we can rely on the targeted SIAs in scenario 9. If zero-dose administration is not pursued due to concerns about its cost-effectiveness, other approaches still remain possible. These findings emphasize the importance of tailored vaccination strategies that consider local epidemiological contexts, resource availability, and infrastructure capacity to achieve long-term measles elimination goals. In this work, we also analyse the local and global stability of a simplified age-independent version of the original model. Investigating the stability properties of the full-age stratified model remains an immediate goal for our future research. Another important future direction of research is to evaluate the cost-effectiveness of four proposed strategies using country-specific data [ 41 ]. Such an analysis could provide critical insights into the trade-offs between the additional logistical and financial investments required and the potential public health benefits. Moreover, in this work, we kept the coverage of SIAs and zero-dose administration fixed for simplicity. However, an important future goal is to investigate the impact of intensifying these coverages in different scenarios. Understanding how changes in SIA and zero-dose coverage levels influence the dynamics of measles transmission could provide valuable information to optimize the intervals between special vaccination campaigns such as SIA and zero-dose administration. Furthermore, it is important to mention that, in addition to the mathematical modeling perspective, it is crucial to study the operational feasibility and potential challenges of implementing these vaccination strategies in different countries. Disclaimer The work/opinion is based solely on the research findings of the authors and not the opinion of any government. Data and code availability The sources of all data used during this study are cited in the article. The codes are available at the following github link: https://github.com/nsamiran/Age-stratified-model.git . Acknowledgments Samiran Ghosh was supported by the Indian Institute of Technology Bombay, India, through the Institute Post-Doctoral Fellowship. Indrajit Ghosh and Siuli Mukhopadhyay were supported by the Gates Foundation (INV-044445). Appendix A. Age-independent model with constant parameters The age independent version of the model with κ = 1 is given by: where, all variables and parameters in the age-independent population retain the same meanings as in the age-dependent model (1), where they were previously denoted with the subscript i to indicate age group specificity. The positivity and boundedness of the system A.1 follow similarly from the proof of positivity and boundedness of the main system (1). Appendix A.1. Effective reproduction number Consequently, the effective reproduction number can be defined as: Appendix A.2. Disease-free equilibrium If we assume that b = d then N ( t ) = constant for all t ≥ 0. Under this assumption, the disease-free equilibrium (DFE) point is given by where The reproduction number around the DFE is given by: The plot of in the parametric space of c 1 and β/N is shown in Figure B.16 . Appendix A.3. Local stability of the DFE Theorem 1 . The DFE is locally asymptotically stable if and unstable if . Proof . The Jacobian matrix of the system A.1 around the DFE is given by: The eigen values of A are: − d , − d , −( c 2 ϵ 2 + d ), − d and the rest of the eigen values are determined by the eigen values of the following matrix: Now the eigen values of ℬ are given by: Consequently, all the eigen values of A are negative except the last one which can be written as . Hence, if the DFE ℰ * is locally asymptotically stable and if the DFE ℰ * is unstable. □ Appendix A.4. Global stability of the DFE In this subsection we study the global stability of the disease-free equilibrium. We have the following theorem: Theorem 2 . The DFE E * is globally asymptotically stable if . Proof . To prove this theorem we follow the approach described in [ 42 ]. We rewrite the system (A.1) in the following form: where, (corresponding to the uninfected compartments), I is the infected compartment as defined in the model (A.1) and Then the sufficient conditions for the global asymptotic stability of the DFE ℰ * = ( H * , 0) are as follows [ 42 ]: is globally asymptotically stable for the system . lim t →∞ I ( t ) = 0. Now, can be written in the form where, and Observe that all the eigen values of K are negative, and consequently, H * is globally asymptotically stable for the system . Now, From the first equation of the system (A.1) we get, Consider the comparison equation Note that is a globally asymptotically stable equilibrium of the above comparison equation. Hence, for any given ϵ > 0 there exists t 1 > 0 such that, Now from the second equation of the system (A.1), for t ≥ t 1 , we get, Consider the corresponding comparison equation It is easy to observe that Hence, there exists t 2 > t 1 > 0 such that where, . Now for t ≥ t 2 , from the third equation of the system (A.1), we get, Note that for the corresponding comparison equation we obtain Thus there exists t 3 > t 2 > 0 such that where . Now from the equation of I in the system (A.1), for t ≥ t 3 we get: Since, and . Moreover, note that initial choice of ϵ > 0 was arbitrary. Hence, we can choose ϵ > 0 in such a way that . Thus we can conclude that lim t →∞ I ( t ) = 0. This completes the proof. Appendix A.5. Existence of endemic equilibrium Suppose the endemic equilibrium point is denoted by . Then we have, and the solution of I e is obtained by the following equation: where Note that if the equation (A.7) has positive solution then there exists positive endemic equilibrium . Due to the complexity of the equation (A.7), we are unable to perform further analytical study. However, using equivalent parameter setup as in Table 1 , numerically we studied the existence of forward bifurcation as shown in Figure (B.17) . Appendix B. Supplementary figures Download figure Open in new tab Figure B.10: Nigeria. Model fitting for the time period 2000 to 2023 and forecasting for different scenarios from 2024 to 2050 are shown with 95% confidence intervals. Panels (a), (d), (g), and (j) correspond to the base scenario and Scenarios 1, 2, and 3, respectively; (b), (e), (h), and (k) correspond to Scenarios 4, 5, 6, and 7, respectively; and (c), (f), (i), and (l) correspond to Scenarios 8, 9, 10, and 11, respectively. Download figure Open in new tab Figure B.11: Philippines. Model fitting for the time period 2000 to 2023 and forecasting for different scenarios from 2024 to 2050 are shown with 95% confidence intervals. Panels (a), (d), (g), and (j) correspond to the base scenario and Scenarios 1, 2, and 3, respectively; (b), (e), (h), and (k) correspond to Scenarios 4, 5, 6, and 7, respectively; and (c), (f), (i), and (l) correspond to Scenarios 8, 9, 10, and 11, respectively. Download figure Open in new tab Figure B.12: Indonesia. Model fitting for the time period 2000 to 2023 and forecasting for different scenarios from 2024 to 2050 are shown with 95% confidence intervals. Panels (a), (d), (g), and (j) correspond to the base scenario and Scenarios 1, 2, and 3, respectively; (b), (e), (h), and (k) correspond to Scenarios 4, 5, 6, and 7, respectively; and (c), (f), (i), and (l) correspond to Scenarios 8, 9, 10, and 11, respectively. Download figure Open in new tab Figure B.13: DR Congo. Model fitting for the time period 2000 to 2023 and forecasting for different scenarios from 2024 to 2050 are shown with 95% confidence intervals. Panels (a), (d), (g), and (j) correspond to the base scenario and Scenarios 1, 2, and 3, respectively; (b), (e), (h), and (k) correspond to Scenarios 4, 5, 6, and 7, respectively; and (c), (f), (i), and (l) correspond to Scenarios 8, 9, 10, and 11, respectively. Download figure Open in new tab Figure B.14: India. Model fitting for the time period 2000 to 2023 and forecasting for different scenarios from 2024 to 2050 are shown with 95% confidence intervals. Panels (a), (d), (g), and (j) correspond to the base scenario and Scenarios 1, 2, and 3, respectively; (b), (e), (h), and (k) correspond to Scenarios 4, 5, 6, and 7, respectively; and (c), (f), (i), and (l) correspond to Scenarios 8, 9, 10, and 11, respectively. Download figure Open in new tab Figure B.15: Pakistan. Model fitting for the time period 2000 to 2023 and forecasting for different scenarios from 2024 to 2050 are shown with 95% confidence intervals. 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Share An age-stratified mathematical model to inform optimal measles vaccination strategies Samiran Ghosh , Indrajit Ghosh , Siuli Mukhopadhyay medRxiv 2025.04.01.25325066; doi: https://doi.org/10.1101/2025.04.01.25325066 Share This Article: Copy Citation Tools An age-stratified mathematical model to inform optimal measles vaccination strategies Samiran Ghosh , Indrajit Ghosh , Siuli Mukhopadhyay medRxiv 2025.04.01.25325066; doi: https://doi.org/10.1101/2025.04.01.25325066 Citation Manager Formats BibTeX Bookends EasyBib EndNote (tagged) EndNote 8 (xml) Medlars Mendeley Papers RefWorks Tagged Ref Manager RIS Zotero Tweet Widget Facebook Like Google Plus One Subject Area Infectious Diseases (except HIV/AIDS) Subject Areas All Articles Addiction Medicine (572) Allergy and Immunology (864) Anesthesia (302) Cardiovascular Medicine (4448) Dentistry and Oral Medicine (444) Dermatology (383) Emergency Medicine (609) Endocrinology (including Diabetes Mellitus and Metabolic Disease) (1515) Epidemiology (15240) Forensic Medicine (30) Gastroenterology (1130) Genetic and Genomic Medicine (6613) Geriatric Medicine (669) Health Economics (1000) Health Informatics (4550) Health Policy (1372) Health Systems and Quality Improvement (1613) Hematology (543) HIV/AIDS (1267) Infectious Diseases (except HIV/AIDS) (15927) Intensive Care and Critical Care Medicine (1105) Medical Education (624) Medical Ethics (147) Nephrology (668) Neurology (6620) Nursing (346) Nutrition (999) Obstetrics and Gynecology (1147) Occupational and Environmental Health (957) Oncology (3341) Ophthalmology (977) Orthopedics (369) Otolaryngology (421) Pain Medicine (436) Palliative Medicine (130) Pathology (665) Pediatrics (1695) Pharmacology and Therapeutics (693) Primary Care Research (714) Psychiatry and Clinical Psychology (5459) Public and Global Health (9247) Radiology and Imaging (2205) Rehabilitation Medicine and Physical Therapy (1371) Respiratory Medicine (1197) Rheumatology (597) Sexual and Reproductive Health (715) Sports Medicine (530) Surgery (714) Toxicology (99) Transplantation (289) Urology (265) (function(){function c(){var b=a.contentDocument||a.contentWindow.document;if(b){var d=b.createElement('script');d.innerHTML="window.__CF$cv$params={r:'a027c2ad2c7273de',t:'MTc3OTkxMzY0OA=='};var a=document.createElement('script');a.src='/cdn-cgi/challenge-platform/scripts/jsd/main.js';document.getElementsByTagName('head')[0].appendChild(a);";b.getElementsByTagName('head')[0].appendChild(d)}}if(document.body){var a=document.createElement('iframe');a.height=1;a.width=1;a.style.position='absolute';a.style.top=0;a.style.left=0;a.style.border='none';a.style.visibility='hidden';document.body.appendChild(a);if('loading'!==document.readyState)c();else if(window.addEventListener)document.addEventListener('DOMContentLoaded',c);else{var e=document.onreadystatechange||function(){};document.onreadystatechange=function(b){e(b);'loading'!==document.readyState&&(document.onreadystatechange=e,c())}}}})();