Viral evolution during primary infection in immunocompromised hosts

The immune response to viral infection is a delicate balance . By perturbing this balance , immunodeficiencies are expected to influence within-host viral evolution. Indeed , the presence of immunocompromised hosts has been argued to be a source of novel viral variants in some infectious diseases, including SARS-CoV-2. However, these arguments rest upon between-host models and so the role of immunodeficiencies on within-host evolution in primary infections is poorly understood. Using a mechanistic immunological model, here we consider how diVerent immunodeficiencies shape the orchestration of the immune response during primary infection. We study how this alters the viral fitness landscape, thus speeding and slowing viral evolution . We show that during acute infections, while immunodeficiencies in neutrophils and interferon initially speed viral evolution, by the time the infection is cleared, mutations are at lower frequencies than in immunocompetent hosts. In persistent infections, we show that while T cell d eficiencies slow viral evolution , interleukin-6 and macrophage deficiencies speed viral evolution. Finally, we show that positive epistatic interactions arising due to the immunological response will accelerate the evolution of viral mutations aVecting the ability of virions to evade diVerent aspects of the immune response and to enter host cells.

immune response; mechanistic mathematical model; viral evolution; epistasis .CC-BY-NC 4.0 International licenseavailable under a (which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity. It is made The copyright holder for this preprintthis version posted September 4, 2025. ; https://doi.org/10.1101/2025.09.01.673077doi: bioRxiv preprint 2

The immune response to viral infection is tightly coordinated through a series of integrated, nonlinear networks 1. Immunodeficiencies perturb this delicate balance, altering immunological responses and prolonging infection . In turn, immunodeficiencies alter the viral fitness landscap e, influencing viral evolution and the speed of adaptation. Indeed, many viruses of importance for public health, including HIV 2, influenza 3,4, SARS -CoV-25,6, and HSV 7, show increased mutations and frequent within -host viral evolution in immunocompromised hosts8,9, depending on the type and severity of the immunodeficiency10. Because immunocompromised hosts may shed virus throughout the course of infection, within-host evolution creates the possibility of forward transmission of novel viral mutations or variants . For example , several SARS -CoV-2 variants have been hypothesized to have originated from accelerated evolution in immunocompromised hosts11-14, in part due to their genetic diVerences from other variants circulating in the population 15. Mathematical modelling of this hypothesis predicts that the longer duration of infection in immunocompromised hosts can speed viral evolution16,17 relative to transmission chains of acute infections, and can produce genetically diVerentiated variants in the presence of fitness valleys in the viral fitness landscape18,19. However, by focusing on the between-host dynamics, this previous work ignores within-host processes. Mutations are treated as neutral within -host, with the fitness valley occurring because “intermediate” mutations are selected against during between-host transmission, before a “jackpot” mutation is acquired 18,19. Recent within-host model ling, on the other hand, predicts the strength of immune pressure is critical to reproducing within-host evolutionary patterns20. Yet, as this work relies on statistical fitness landscapes that abstract the immunological dynamics, how the nature and severity of immunodeficiencies aVect the speed of within-host viral adaptation remains poorly understood. Here, we consider how immunodeficiencies aVecting the orchestration of the immune response determine the speed of within -host viral evolution in acute and persistent infections. We show that d uring acute infections, immunodeficiencies in neutrophils and interferon have the largest evolutionary role: while they can speed evolution early in the infection, by the time the infection has been cleared, mutations will be at a lower frequency in immunocompromised hosts. During persistent infections, T cell concent rations play a pivotal role: T cell deficiencies slow viral evolution, while interleukin-6 and macrophage deficiencies overstimulate T cells, speeding viral evolution. We then show how immunological interactions between mutations aVecting evasion of diVerent aspects of the .CC-BY-NC 4.0 International licenseavailable under a (which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity. It is made The copyright holder for this preprintthis version posted September 4, 2025. ; https://doi.org/10.1101/2025.09.01.673077doi: bioRxiv preprint 3 immune response can generate epistasis in fitness, further aVecting the speed of viral evolution during persistent infections. Together, this work underlines how dysregulated immune response s impact within-host viral evolution , with potential consequences for individual and population health. Model Immune response to primary infection We focus on an immunological model capturing the dynamics of a primary infection caused by a respiratory virus. In what follows, we highlight the key aspects of the model; more details are provided in the Supplementary Information 1, including information on parameter estimation. A model schematic is presented in Figure 1. A primary infection consists of virions, 𝑉(𝑡), and infected cells, 𝐼(𝑡). Virions infect susceptible cells, 𝑆(𝑡), via mass-action transmission with rate constant 𝛽. Infected cells undergo an eclipse phase lasting 𝜏! days during which cells are non -productively infected. Following the eclipse phase, lysis occurs with a burst size of 𝑝 virions. Virions naturally decay at a per-capita rate 𝑑", while the per-capita rate of infected cell death is 𝑑!. Primary infection induces an immune response that can be divided into the innate, adaptive, and humoral responses. The innate immune response consists of three parts. First, following contact with virions, infected cells secrete Type I interferon (IFN) making them refractory to infection while also blocking viral entry and replication in neighbouring cells. Second, neutrophils, 𝑁(𝑡), clear virions through neutrophil extracellular traps and kill infected cells. Neutrophils are recruited to the site of infection by signalling from the cytokines interleukin- 6 ( IL-6) and granulocyte colony -stimulator factor (G -CSF). Third, inflammatory macrophages, 𝑀#! (𝑡), destroy virions and infected cells through phagocytosis 21. Inflammatory macrophages are created from either monocytes, 𝑀(𝑡), by signalling from the cytokines IL-6 and granulocyte macrophage colony -stimulator factor (GM -CSF), or from tissue-resident macrophages, 𝑀#$ (𝑡), after contact with cells that are infected, 𝐼(𝑡), or dead, 𝐷(𝑡). The adaptive immune response consists of CD8+ eVector T cells, 𝑇(𝑡), and is activated 𝜏% days following viral infection. The activation delay accounts for the time necessary for naïve CD8+ T cells to be presented antige n, and to convert to the eVector phenotype following a primary exposure22. Following activation, CD8+ eVector T cells exhibit density-dependent saturable killing of infected cells 23. CD8+ T cells are recruited by infected cells and IFN signalling and suppressed by IL-6, as IL-6 serves as a proxy for anti-inflammatory signalling22. .CC-BY-NC 4.0 International licenseavailable under a (which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity. It is made The copyright holder for this preprintthis version posted September 4, 2025. ; https://doi.org/10.1101/2025.09.01.673077doi: bioRxiv preprint 4 The humoral immune response consists of antibodies, 𝐴(𝑡), which are generated by the presence of virions after 𝜏& days. The delay in antibody production accounts for the time required for antigen presentation to CD4+ T cells until the production of antibodies by B cells24. Following activation, antibodies reduce viral loads through saturable neutralization of virions. All antibodies are assumed to be neutralizing in our model. Under these assumptions the dynamics of the virions and infected cells can be written 𝑑𝑉 𝑑𝑡 = 𝑝𝐼(𝑡) − 4𝑑" + δ",(𝑁(𝑡) + δ",)𝑀*,!(𝑡) + δ",&𝐴(𝑡)+! 7ϵ",& +! + 𝐴(𝑡)+! 9 : ;<<<<<<<<<<<<<<<=<<<<<<<<<<<<<<<> ," 𝑉(𝑡), (1) 𝑑𝐼 𝑑𝑡 = 𝛽𝜖-,!𝑆(𝑡 − 𝜏!) 𝜖-,! + 𝐹.(𝑡)BCCCDCCCE /# 𝑉(𝑡 − 𝜏! ) − 4𝑑! + δ!,(𝑁(𝑡)+$ 𝐼𝐶01,( +$ + 𝑁(𝑡)+$ + δ!,)𝑀*,!(𝑡) + δ𝐼,𝑇𝑇(𝑡) ϵδ,𝑇 + 𝐼(𝑡)ℎ𝑇 : BCCCCCCCCCCCCCCCCDCCCCCCCCCCCCCCCCE ,# 𝐼(𝑡), (2) where 𝐹.(𝑡) is the concentration of bound IFN at time 𝑡. In equations (1) and (2) we use 𝑚" and 𝑚! to denote the per-capita loss of virions and infected cells, respectively, and 𝐵! to denote the per-capita production of infected cells by virions. Immunodeficiencies Immunocompromised hosts may experience protracted “acute”, or persistent, infection and so experience a loss of IFN eVects on refractory cells 25-28, as well as a deficient innate, adaptive, and/or humoral immune response29. To capture the loss of IFN eVects on refractory cells, we suppose they revert to susceptibility at a per capita rate 𝛿$ after 𝜏$ days. The delay in reversion accounts for the loss of increased Type I IFN signalling and interferon stimulated genes30,31 through e.g., receptor internalization and downregulation32. To capture the deficient immune response, we suppose th at th e rate of production, and hence concentration, of diVerent state variables of the immune response are reduced. For example, a monocyte production deficiency translates to a lower concentration of monocytes at homeostasis and during infection. The interconnectedness of the immune response means that production deficiencies in one variable may also impact other variables (e.g., the cytokine G-CSF recruits neutrophils, and so a G-CSF deficiency will also lead to a reduction in neutrophil concentrations). Therefore, for each immunodeficiency we first establish homeostasis at the specified level of production deficiency before introducing virions (see Sup. Info. Section 3 for more details). We will assume immunodeficiencies reduce the target variable by at least 50% as compared to healthy hosts. For all state variables, except T cells and antibodies, this translates to reducing the production rate of the variable in question by 35% (Sup. Info. Section 3). To mimic clinically observed T cell immunodeficiencies, such as advanced HIV disease defined .CC-BY-NC 4.0 International licenseavailable under a (which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity. It is made The copyright holder for this preprintthis version posted September 4, 2025. ; https://doi.org/10.1101/2025.09.01.673077doi: bioRxiv preprint 5 by CD4+ T cell counts <200 cells /𝜇𝑙33, a T cell production deficiency is assumed to reduce the production of T cells by 75%. To capture the near complete lack of antibodies in certain primary B cell immunodeficiencies 34, including B cell lymphomas 8, antibody immunodeficiencies are assumed to reduce antibody production by 90%. In all cases, varying the strength of the immunodeficiency quantitatively aVects the reduction in the target variable, as well as the magnitude of concomitant changes in the non-target variables, but does not qualitatively aVect our results (Sup. Info. Fig. S1). Viral evolution As our interest is how immunodeficiencies shape (and speed) selection on viral evasion of the immune response, we ignore de novo mutations and assume that the viral mutation(s) of interest are present at a low frequency of 0.01 in the initial viral inoculum . Each viral mutation we consider targets evasion of a single immunological variable (e.g., neutrophils) in a particular life history stage (i.e., virion or infected cell). Each viral mutation is beneficial, and for simplicity, cost-free for the virus (the addition of costs is a straightforward extension). We will refer to the immunological variable directly targeted by the viral mutation as the “target” and the other immunological variables as “nontargets” . The mutations considered can be divided into two groups , based on the life history stage (i.e., virion, infected cells) whose fitness they directly benefit. In the first group are mutations that directly increase a component of virion fitness (henceforth, virion mutations ). Virion mutations target virion evasion of neutrophils (decreased δ",(), macrophages (decreased δ",)), or antibodies (decreased δ",&), or target virion infection of susceptible cells, by increasing virion host cell entry35,36 (increased 𝛽), or by decreasing the ability of interferon to block viral entry37,38 (increased ϵ-,!). In the second group are mutations that directly increase a component of infected cell fitness (henceforth, infected cell mutations ). Infected cell mutations target infected cell evasion of neutrophils (decreased δ!,(), macrophage (decreased δ!,)), or T cells (decreased δ!,%). .CC-BY-NC 4.0 International licenseavailable under a (which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity. It is made The copyright holder for this preprintthis version posted September 4, 2025. ; https://doi.org/10.1101/2025.09.01.673077doi: bioRxiv preprint 6 Figure 1. Schematic of immune response to viral infection. Virions, 𝑉, infect susceptible cells, 𝑆, producing infected cells, 𝐼. Following the eclipse phase, i nfected cells lyse, producing virions . Cells refractory to infection, 𝑅, are created through upregulation of interferon (IFN) signalling by infected cells. Inflammatory macrophages, 𝑀*!, are converted from monocytes, 𝑀, or tissue-resident macrophages, 𝑀*$, and along with neutrophils, 𝑁, phagocytose virions and infected cells. Phagocytosis of infected cells produces dead and damaged cells, 𝐷. Antibodies, 𝐴, are mobilized by the presence of viral antigens, and neutralize virions. CD8+ eVector T cells, 𝑇, are stimulated by, and subsequently kill, infected cells. The action of the immune system is orchestrated through the antiviral cytokine IFN as well as the proinflammatory cytokines IL-6, G-CSF, and GM -CSF. Circles and boxes along arrows indicate the action of either a cell or cytokine, respectively. Dashed arrows: recruitment. Solid arrow: production or diVerentiation. Double arrows: induced death. Pink arrows: decreased eVect due to immunodeficiency. Yellow arrows: immune eVect targeted by mutations.

The speed of within-host evolution of individual mutations First, we ask how the immunological environment determines the speed of evolution of a single segregating mutation . To do so , we note that i f 𝑝2 (𝑡) denotes the frequency of mutation 𝑥 within the viral population, the frequency of mutation 𝑥 changes according to 𝑑𝑝2 𝑑𝑡 = 𝑠2(𝑡)𝑝2 (1 − 𝑝2 ), (3) where 𝑠2(𝑡) is the time -varying selection coeVicient. Thus, t he strength of selection on mutation 𝑥 at time 𝑡 is determined by the magnitude of 𝑠2(𝑡); this dictates the instantaneous V S N D I R M MɸR MɸI T A Induced death Recruitment Production or differentiation Effect of immunodeficiency Mutation Virions Susceptible cells Infected cells Dead cells Resistant cells CD8+ T cells Neutrophils Monocytes Tissue-resident macrophages Inflammatory macrophages Antibodies IFN G-CSF IL-6 GM-CSF Virus Tissue Innate Response Adaptive Response Humoral Response Cytokines .CC-BY-NC 4.0 International licenseavailable under a (which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity. It is made The copyright holder for this preprintthis version posted September 4, 2025. ; https://doi.org/10.1101/2025.09.01.673077doi: bioRxiv preprint 7 rate of increase of the mutation due to selection. In turn, 𝑠2(𝑡) can be used to calculate the time-averaged strength of selection on mutation 𝑥 from time 𝑡1 to time 𝑡: ⟨𝑠2(𝑡)⟩ = 1 𝑡 − 𝑡1 S 𝑠2(𝑡)𝑑𝑥. 3 3% (4) Equation (4) measures how strong constant selection would have to be to yield the observed change in mutation frequency by time 𝑡. Thus, 𝑠2(𝑡) and ⟨𝑠2(𝑡)⟩ capture how the immune response determines the short- and long-term strength of selection, respectively, dictating the speed of adaptation , and therefore will be our focus here . Because the strength of selection will depend on the size of the mutational eVect, in our figures we will normalize the selection coeVicients, and the time -averaged strength of selection, by their maximum values over the duration of infection (e.g., 𝑠2 (𝑡)/ max 𝑠2 (𝑡)). Calculating selection coeVicients in populations with diVerent life history stages (e.g., virions and infected cells) is not trivial, particularly for populations with temporally varying per-capita growth rates39-42. Therefore, we will primarily rely on two approximations for 𝑠2(𝑡). First, we approximate 𝑠2(𝑡) from simulation data as 𝑠2 (𝑡) ≈ − 1 𝑡4 − 𝑡456 ln ]𝑝2 ! (𝑡456)71 − 𝑝2 ! (𝑡4 )9 𝑝2! (𝑡4 )71 − 𝑝2! (𝑡456)9^ (5) where 𝑝2 ! (𝑡) is the frequency of mutation 𝑥 at time 𝑡 in infected cells, and 𝑡4 − 𝑡456 is some (short) increment of time (e.g., hours, days). ⟨𝑠2(𝑡)⟩ can be approximated using equation (5) by setting 𝑡456 = 𝑡1. Although equation (5) only considers the frequency of the mutation in infected cells, simulation results indicate that it yields similar predictions if it is calculated using the frequency of the mutation in virions (Sup. Info. Fig. S 2). The principal area of divergence between the diVerent approximations of selection occurs at the outset of the infection. This is because each infection starts with only virions present, and so the delay in production of infected cells due to the eclipse phase means equat ion (5) cannot be accurately calculated during th e initial few hours of the infection ( Sup. Info. Fig. S2). Therefore, in all simulation results we plot equation (5) from hour 12 onwards. Second, if we suppose mutation 𝑥 is of weak phenotypic eVect, and that the immunological variables, and density of susceptible cells, change slowly relative to the change in infected cells and virions (e.g., see Day et al.39), we can analytically approximate 𝑠2(𝑡) as 𝑠2(𝑡) ≈ `1 2 `1 − 𝑚a 𝜃 c 𝜕𝑚" 𝜕𝑥 + 1 2 `1 + 𝑚a 𝜃 c 𝜕𝑚! 𝜕𝑥 + 𝑝𝐵! 𝜃 𝜕𝑓 𝜕𝑥c |𝛥𝑥| + 𝒪(Δ𝑥7), (6) where |Δ𝑥| is the (small) phenotypic diVerence between mutant and wildtype, 𝑚a ≡ 𝑚" − 𝑚!, θ ≡ m4𝑝𝐵! + 𝑚a 7, 𝐵! is evaluated when 𝜏! = 0, and ∂𝑓/ ∂𝑥 = 1 if 𝑥 = 𝛽 and ∂𝑓/ ∂𝑥 = 𝐹./7ϵ-,! + 𝐹. 9 if 𝑥 = ϵ-,! (Sup. Info. Section 4). .CC-BY-NC 4.0 International licenseavailable under a (which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity. It is made The copyright holder for this preprintthis version posted September 4, 2025. ; https://doi.org/10.1101/2025.09.01.673077doi: bioRxiv preprint 8 Equation (6) reveals that selection on each mutation is the product of two components. The first is the direct eVect of mutation 𝑥 on the viral fitness component 𝑧 ∈ {𝑚", 𝑚!, 𝑓}, captured by 89 82 |𝛥𝑥|. Since each mutation is beneficial and has a single eVect on viral life history, only one of the direct eVects in equation (6) will be nonzero and positive. The second component of selection in equation (6) is how each of the direct eVects are weighted. These weights are based on the frequency of the mutation in the diVerent life history stages (virions and infected cells) as well as the relative value of each life history stage for their contribution to future generations (reproductive value39-42). As the per-capita rate of virion destruction is expected to be higher than that of infected cells, 𝑚a > 0, a decrease in 𝐵! and/or 𝑚! decreases the value of virion mutations and increases the value of infected cell mutations, whereas a decrease in 𝑚" has the opposite eVect. This has important consequences for the strength of selection. Availability of susceptible cells, and reversion to susceptibility of refractory cells, determines dynamical phases of primary infections The availability of susceptible cells, and the reversion to susceptibility of refractory cells , partitions the infection into three phases exhibiting qualitatively distinct dynamics (Fig. 2). In the first phase of infection, susceptible cells are abundant and so the infection grows rapidly while the innate response (neutrophils, IFN, and macrophage) mobilizes (Fig. 2A -2B). As susceptible cells are abundant, the ratios −𝑚a/θ and 𝑝𝐵!/θ are maximized. Thus, the value of virion mutations is at its maximum while the value of infected cell mutations is at its minimum (equation (6); Fig. 2G and 2H). In the second phase of infection, susceptible cells are largely depleted, and the innate immune response is fully mobilized. Because susceptible cells are depleted, 𝐵! → 0, and so equation (6) reduces to 𝑠2(𝑡) ≈ 𝜕𝑚! 𝜕𝑥 |Δ𝑥|. (7) Thus, the value of virion mutations is zero and so virion mutations are selectively neutral, while the value of infected cell mutations is maximized (Fig. 2G and 2H). In immunocompetent hosts (i.e., 𝛿$ = 0), the depletion of susceptible cells combined with the mobilized immune response leads to clearance of the infection (Fig. 2A-2C). However, in immunocompromised hosts (Fig. 2D -2F), the loss of IFN signalling and reversion to susceptibility of refractory cells creates a third phase of infection shortly after 𝑡 = 𝜏$. In the third phase of infection, there is a temporary abundance of susceptible cells as the first wave of refractory cells revert to susceptibility . T he dynamics then settle down to a quasi - equilibrium where the infection is limited by a combination of susceptible cell availability and the mobilized immune response (Fig. 2D-2F; dark grey shaded region). The rebound in .CC-BY-NC 4.0 International licenseavailable under a (which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity. It is made The copyright holder for this preprintthis version posted September 4, 2025. ; https://doi.org/10.1101/2025.09.01.673077doi: bioRxiv preprint 9 susceptible cells in immunocompromised hosts increases the value of virion mutations and tends to decrease the value of infected cell mutations, as in the long term, 𝑝𝐵! → 𝑚"𝑚! (Fig. 2G,H). Moreover, the mobilization of the adaptive (T cells) and humoral (antibodies) response (Fig. 2 C and 2 F) generates a direct eVect on selection for antibody and T cell evasion mutations, increasing the strength of selection on these mutations from the second to third phase of infection (Fig. 2G and 2H). The time of onset of each phase of infection is aVected by the size of the viral inoculum, 𝑉(0), in a predictable fashion. As the size of the inoculum decreases, the amount of time required for the viral population to exhaust susceptible cells increases, prolonging the first phase of infection, and so delaying the onset of the second phase (Sup. Info. Fig. S4). Although changes to the size of the viral inoculum aVect the quantitative time of onset of the diVerent phases, it does not qualitatively change the key predictions ( Sup. Info. Fig. S4; see also Jenner et al. 22). Therefore, in what follows we fix the initial inoculum at 𝑉(0) = 4.5 log10(copies/mL). For this inoculum size, the second and third phase of infection begin at approximately 4.5 and 9 days, respectively (Fig. 2D and 2F). 0 1 2Cells (cells/mL) #108 0 3.75 .5 Virus (log10(copies/mL)) Susceptible Infected 7 Refractory Virions 0 1 2Cells (cells/mL) #108 0 3.75 7.5 Virus (log10(copies/mL)) 0 5 10 Cells (cells/mL) #106 0 175 350 Interferon (pg/mL) IFN Macrophages Neutrophils 0 5Cells (cells/mL) #106 0 175 350 Interferon (pg/mL) 100 =T =R 102 Time (log10(days)) 0 7.5 15 Cells (cells/mL) #105 0 1.5 3 Antibodies (AU/mL) #104 T cells Antibodies 100 =T =R 102 Time (log10(days)) 0 5 10 15 Cells (cells/mL) #105 0 1.5 3 Antibodies (AU/mL) #104 A C D F G H B E 10 - 0FI /VN /VA /VM /IT /IM /IN Susceptible Infected Refractory Virions IFN Macrophages Neutrophils T cells Antibodies - 0FI /VN /VA /VM /IT /IM /IN 0 0 .5 1 0 0 .5 1 0 0.5 1 Time-averaged selection coefficient (normalized) 100 =T =R 102 Time (log10(days)) 0 0.5 1 Selection coefficient (normalized) .CC-BY-NC 4.0 International licenseavailable under a (which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity. It is made The copyright holder for this preprintthis version posted September 4, 2025. ; https://doi.org/10.1101/2025.09.01.673077doi: bioRxiv preprint 10 Figure 2. Immune dynamics distinguish the phases of infection. In immunocompetent hosts, the infection can be divided into two phases (panels A-C). In the first phase, indicated by the white area on each panel , susceptible cells are abundant, while the immune response is mobilizin g. The abundance of susceptible cells means the value of virion mutations (i.e., mutations aVecting cell entry 𝛽, as well as virion evasion of IFN, ϵ-,!, neutrophils, δ",(, macrophage, δ",), and antibodies, δ",&) is maximized, while the value of infected cell mutations (i.e., mutations aVecting infected cell evasion of neutrophils, δ!,(, macrophage, δ!,), and T cells, δ!,%) is minimized (panels G,H). At the same time, the mobilizing immune response (panels B,E) increases the direct eVect of selection for immune evasion. In the second phase, indicated by light grey shading, susceptible cells are depleted, and the immune response eliminates the infection. During the second phase, the absence of susceptible cells means virion mutations are selectively neutral, while the value of infected cell mutations increases (panels G,H). In immunocompromised hosts (panels D- F), the reversion to susceptibility of refractory cells (δ$ > 0) leads to a third infection phase, indicated by dark grey shading on panels D -H. This causes a rebound in the value of virion mutations, and a decline in the value of infected cell mutations (panels G,H), while the mobilizing adaptive and humoral response (panels C,F) selects for T cell and antibody evasion. Note that in panels D -F we do not consider any additional immunodeficiencies other than δ$ > 0. In all panels, the dashed vertical grey line indicates 12 hours (see Sup. Info. Fig. S2). Immunodeficiencies speed and slow the evolution of individual mutations Next, we ask how diVerent immunodeficiencies aVect the speed of viral evolution. The most straightforward prediction occurs for target immunodeficiencies , that is, a production deficiency in the immunological variable directly targeted by the viral mutation. As the direct eVect of mutation 𝑥 on viral fitness component 𝑧 ∈ {𝑚", 𝑚!, 𝑓} is an increasing function of the concentration of the target variable , target variable immunodeficiencies reduce the strength of selection on mutation 𝑥 and so will slow the evolution of mutation 𝑥. This is intuitive: if, for example, mutation 𝑥 targets antibody evasion and antibody concentrations are reduced, there are fewer antibodies to “evade” and so mutation 𝑥 will be less beneficial, weakening selection and hence slowing evolution of mutation 𝑥. The interconnectedness of the immune response, however, mean s that non-target immunodeficiencies may not only feedback on the target variable, aVecting the direct eVect of unrelated mutations, but trigger changes in mutational value by, for example, aVecting the availability of susceptible cells. In what follows, we consider these eVects across the diVerent phases of infection. Because mutations targeting infected cell evasion of neutrophils (decreased 𝛿!,() and evasion of macrophage (decreased 𝛿",) and 𝛿!,)) are under very weak selection, regardless of the size of the mutational eVect (Sup. Info. Fig. S2), in what follows we ignore these three .CC-BY-NC 4.0 International licenseavailable under a (which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity. It is made The copyright holder for this preprintthis version posted September 4, 2025. ; https://doi.org/10.1101/2025.09.01.673077doi: bioRxiv preprint 11 mutations. W e will therefore restrict our attention to mutation s aVecting cell entry (increased 𝛽), virion evasion of interferon, neutrophils, and antibodies (increased ϵ-,! or decreased δ",( and δ",&, respectively), and infected cell evasion of T cells (decreased δ!,%). Neutrophil and IFN deficiencies tend to speed viral evolution in the first phase of infection, and slow viral evolution in the second phase During the first two phases of infection, the immune response targeting infected cells and virions consists primarily of neutrophils, IFN, and macrophages (Fig. 2B,E). Of the three, neutrophils are the most important for controlling the initial infection (Fig. 3 A). Neutrophils are recruited by the cytokine G-CSF; G-CSF is produced by monocytes; and monocytes are generated from bone marrow precursors by the cytokine GM-CSF (Fig. 1). A deficiency in any component of this pathway reduces neutrophil concentrati ons, either directly (i.e., a neutrophil deficiency) or indirectly (i.e., a G -CSF , GM-CSF , and/or monocyte deficiency), sharply increasing infection size (Fig. 3 B and 3C). IFN deficiencies have a similar, but less strong eVect. Since reduced neutrophil concentrations decrease the destruction of virions (decrease 𝑚"), while reduced IFN concentrations aid the infection of susceptible cells (increase 𝐵!), reductions in neutrophils and/or IFN increase the value of virion mutations and so tend to speed the evolution of non-target virion mutations targeting cell entry, neutrophil evasion, and IFN evasion during the first phase of infection (Fig. 3 D and 3E, see also Sup. Info. Fig. S3A and S3H). However, reduced concentrations of neutrophils and/or IFN and the increase in infection size during the first phase of infection lead to a more rapid depletion of susceptible cells . This triggers an earlier onset of the second phase of infection. Since virion mutations are neutral during this phase (see equation (6)), by the end of the second phase of infection the normalized time-averaged selection coeVicient is larger in immunocompetent hosts (Fig. 3D and 3E). Thus, if refractory cells do not revert to susceptibility (i.e., there is no third phase of infection), by the end of the infection , mutations are at a higher frequency in immunocompetent hosts. .CC-BY-NC 4.0 International licenseavailable under a (which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity. It is made The copyright holder for this preprintthis version posted September 4, 2025. ; https://doi.org/10.1101/2025.09.01.673077doi: bioRxiv preprint 12 Figure 3. Viral evolution during acute infections is most strongly a=ected by neutrophil and IFN deficiencies . During acute infections (phase one and two; 𝛿$ = 0), IFN and neutrophil deficiencies tend to speed evolution of non -target mutations during the first phase of infection, before slowing evolution during the second phase of infection (see also Sup. Fig. S3A and S3H ). For acute infections, o nly mutations aVecting cell entry, 𝛽, virion evasion of IFN, ϵ-,!, and virion evasion of neutrophils, δ",(, experience significant selection. In panel A, t he percent change in each immunological variable is calculated as (𝑋 − 𝑌)/𝑋 × 100, where 𝑋 and 𝑌 are the integrals of the indicated variable over the length of the phase in immunodeficient and healthy hosts, respectively , while the percent change in susceptible cells is calculated using the integral of 𝑆,:2 − 𝑆(𝑡). In panels D and E , the diVerence in the time -averaged selection coeVicient between immunodeficient and immunocompetent hosts is normalized by its maximum (absolute) value. The dashed vertical grey line in panels B-E indicates 12 hours. T cell concentrations play a pivotal role in speeding or slowing viral evolution during phase three If refractory cells do not revert to susceptibility, the infection is cleared during phase two. Therefore, to provide a point of comparison for the immunodeficiency of interest, in this section we allow for the loss of refractory cells (i.e., δ$ > 0) in both immunodeficient and immunocompetent hosts. During the third phase of infection, T cells play an important role in killing infected cells, reducing the size of infection and increasing the availability of susceptible cells . This -100 -50 -25 -15 -5 5 15 25 50 100 300 NaN Percent change A B C selection coefficient (normalized) Difference in time-averaged D E 0 9 18 Cells (cells/mL) #107 Susceptible Infected 0 10 20 Virions (log10(copies/mL)) Healthy IFN def Neut def Healthy IFN def Neut def selection coefficient (normalized) Difference in time-averaged Monocyte Neutrophil G-CSF GM-CSF Macrophage IL-6 IFN Antibody T cell Deficiency Phase 1 Virus Infected cells Susceptible cells Refractory cells Dead cells Macrophages Monocytes Neutrophils Antibodies T cells Bound G-CSF Bound GM-CSF Bound IFN Bound IL-6 Variable Monocyte Neutrophil G-CSF GM-CSF Macrophage IL-6 IFN Antibody T cell Deficiency Phase 2 100 =T =R Time (log10(days)) -1 0 1 IFN deficiency - 0FI /VN 100 =T =R Time (log10(days)) -1 0 1 Neutrophil deficiency - 0FI /VN .CC-BY-NC 4.0 International licenseavailable under a (which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity. It is made The copyright holder for this preprintthis version posted September 4, 2025. ; https://doi.org/10.1101/2025.09.01.673077doi: bioRxiv preprint 13 increases the value of virion mutations and decreases the value of infected cell mutations. T cell production is suppressed by bound IL -6 concentrations43,44; bound IL -6 is generated from stimulation of unbound IL -6 by neutrophils and monocytes, while unbound IL -6 is generated from infected cells, monocytes, and macrophages (Fig. 1; see also Sup. Info. 1). Consequently, while T cell deficiencies decrease T cell concentrations, deficiencies in neutrophils (arising either directly, or indirectly through deficiencies in G -CSF , GM-CSF , and/or monocytes), monocytes, macrophages and /or IL-6 will decrease bound IL -6 concentrations, increasing T cell concentrations (Fig. 4A and 4C), decreasing infection size and increasing the availability of susceptible cells (Fig. 4B). IFN and antibody deficiencies have limited eVect on bound IL-6 or T cell concentrations (Fig. 4A). Because a T cell deficiency both reduces the direct eVect of T cell evasion a s well as decreases the value of virion mutations by increasing susceptible cells , a T cell deficiency will slow the rate of increase of each of the mutations considered (Fig. 4 F). On the other hand, since IL-6 and macrophage deficiencies increase T cell concentrations, they have the opposite eVect, speeding the rate of increase of each of the five mutations considered (Fig. 4D and 4E ). T cell counts are also increased by lower n eutrophil concentrations arising through deficiencies in monocytes, G-CSF , GM-CSF , and/or neutrophils, as lower neutrophil concentrations will increase infection size. However, these deficiencies also reduce IFN (Fig. 4A). Consequently, deficiencies in monocytes, G -CSF , GM-CSF , and/or neutrophils speed the evolution of mutations targeting cell entry, antibody evasion, and T cell evasion, but slow the evolution of mutations targeting IFN and neutrophil evasion by reducing the concentration of IFN and neutrophils (Fig. 4D). .CC-BY-NC 4.0 International licenseavailable under a (which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity. It is made The copyright holder for this preprintthis version posted September 4, 2025. ; https://doi.org/10.1101/2025.09.01.673077doi: bioRxiv preprint 14 Figure 4 . Immunodeficiencies a=ecting T cell concentrations strongly a=ect viral evolution during persistent infections. T cell killing of infected cells is a key determinant of the size of the infection and availability of susceptible cells (panels A and B) during persistent infections ( i.e., 𝛿$ > 0). A deficiency in T cells will increase infection size and decrease the availability of susceptible cells, decreasing the value of virion mutations and so slowing the evolution of all mutations considered (panel D). A deficiency in IL -6 (either directly, or due to a macrophage deficiency; Sup. Info. Fig . S3F) will increase T cell concentrations and so speed the evolution of all mutations considered (panel E). A deficiency in neutrophils (either directly, or due to a deficiency in the cytokines G-CSF and GM-CSF or monocytes, as shown here; see also Sup. Info. Fig. S3B-S3D), will increase T cell concentrations, but will also reduce neutrophils and IFN (panel F). In panel A, the percent change in each immunological variable is calculated as (𝑋 − 𝑌)/𝑋 × 100, where 𝑋 and 𝑌 are the integrals of the indicated variable over the length of the phase in immunodeficient and healthy hosts, respectively, while the percent change in susceptible cells is calculated using the integral of 𝑆,:2 − 𝑆(𝑡). In panels D-F, the diVerence in the time -averaged selection coeVicient between immunodeficient and immunocompetent hosts is normalized by its maximum (absolute) value. In panels B-F, the dashed vertical grey line indicates 12 hours. Speed of evolution of multiple mutations Finally, we ask how the speed at which individual mutations increase in frequency is aVected by the presence of other segregating mutations. The possibility of genetic variation for multiple viral traits can yield fitness interactions between mutations (epistasis in fitness) . Epistasis aVects the speed of evolution by altering the fitness of a mutation depending on its genetic background, a s well as producing linkage disequilibrium (LD) between mutations, - /IT /VA /VN 0FI Difference in time-averaged selection coefficient (normalized) 0 1.25 2.5 Cells (cells/mL) #108 Susceptible Infected 0 3.875 7.75 Cells (cells/mL) #106 0 175 350 Interferon (pg/mL) Neutrophils IFN T cells -100 -50 -25 -15 -5 5 15 25 50 100 300 NaN Percent change Healthy IL-6 def Monocyte def T cell def A B C D E F - /IT /VA /VN 0FI Difference in time-averaged selection coefficient (normalized) - /IT /VA /VN 0FI Difference in time-averaged selection coefficient (normalized) Virus Infected cells Susceptible cells Refractory cells Dead cells Macrophages Monocytes Neutrophils Antibodies T cells Bound G-CSF Bound GM-CSF Bound IFN Bound IL-6 Variable Monocyte Neutrophil G-CSF GM-CSF Macrophage IL-6 IFN Antibody T cell Deficiency Phase 3 E FVariable 100 =T =R Time (log10(days)) -1 0 1 100 =T =R Time (log10(days)) 1 0 1 100 =T =R Time (log10(days)) -1 0 1 .CC-BY-NC 4.0 International licenseavailable under a (which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity. It is made The copyright holder for this preprintthis version posted September 4, 2025. ; https://doi.org/10.1101/2025.09.01.673077doi: bioRxiv preprint 15 that is, non-random assortment between mutations45; and the amount of LD in deterministic models will tend to scale with the strength of epistasis . For beneficial mutations, such as those studied here, these combined eVects mean positive epistasis speeds evolution, whereas negative epistasis slows evolution45-47 (Sup. Info. 4.5). Under the same assumptions that allowed us to derive equation (6), we can calculate a weak selection approximation of epistasis between pairs of mutations (Sup. Info. 4.5). From this approximation, two predictions emerge. First, if the two mutations both directly aVect the fitness of virions (so virion evasion of neutrophils, antibodies, IFN, or increased cell entry) or both directly aVect the fitness of infected cells , epistasis is positive, speeding their joint evolution (Sup. Info. 4.5). If instead, one mutation directly aVects virion fitness, whereas the other mutation directly aVects the fitness of infected cells (e.g., evasion of T cells), epistasis is negative, slowing their joint evolution (Sup. Info. 4.5). Simulation results, however, suggest that while positive epistasis between virion mutations can dramatically speed evolution by enhancing fitness and by the generation of large amounts of LD, the ability of negative epistasis between virion and infected cell mutations to slow evolution is weak (Fig. 5). This occurs due to the source of epistasis, which depends upon the mutations under consideration. For example, mutations targeting cell entry and IFN evasion multiplicatively interact to aVect the rate of infection of new cells, which will directly produce epistasis48. On the other hand, the source of negative epistasis in our model is strictly due to the division of the viral life cycle into virions and infected cells. This can be seen by supposing virion turnover is much more rapid than infected cell turnover, that is, 𝑚" ≫ 𝑚!. As this diVerence grows , negative epistasis between virion and infected cell mutations becomes increasingly weak (Sup. Info. 4.5). Consequently, as our model parameterization assumes virion turnover is relatively rapid, negative epistasis is comparatively weak. .CC-BY-NC 4.0 International licenseavailable under a (which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity. It is made The copyright holder for this preprintthis version posted September 4, 2025. ; https://doi.org/10.1101/2025.09.01.673077doi: bioRxiv preprint 16 Figure 5 . The sign of epistasis and the speed of viral evolution var y based upon the combination of mutations considered. When diVerent mutations target immune evasion by the same life history stage , i.e., infected cells or virions , the speed of evolution can be dramatically increased through positive epistasis and the production of significant amounts of linkage disequilibrium (panels A and B, solid lines) . Conversely, when one mutation targets T cell evasion, while the other targets virion evasion of the immune response, negative epistasis occurs (panels A and B, dashed lines) . However, in contrast to positive epistasis, negative epistasis in our model is weak , and so is incapable of generating much linkage disequilibrium (panel B) . That negative epistasis is weak arises due to the rapid turnover of virions relative to infected cells. In panel A, the diVerence in the time -averaged selection coeVicients is normalized by their respective maximum (absolute) value.

Acute infections can become persistent in immunodeficient hosts due to weakened immune defences. Such persistent infections can provide an ideal environment for the generation of, and selection for, viral mutations33,49 increasing immune evasion15,50,51 and viral replication10,52,53. Indeed, between-host models have shown that persistent infections can speed viral evolution across the host population16-19,54. However, this previous work ignores how immunodeficiencies alter the within -host environment and the implications for viral evolution. Here, we work towards filling this gap by studying viral evolution during primary infection in immunocompromised hosts using a mechanistic model of the immune response. The viral mutations considered in our model can be classified as “virion” or “infected cell” mutations, depending on the viral life stage they directly aVect. The strength of selection for Difference in time-averaged selection coefficient (normalized) /VN /VA /IT /VN + /VA /VN + /IT 100 =T=R Time (log10(days)) -0.14 0 0.14 Linkage disequilbrium /VN + /VA /VN + /IT 100 =T =R -1.5 0 #10-4 A B 100 =T=R Time (log10(days)) 1 0 1 .CC-BY-NC 4.0 International licenseavailable under a (which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity. It is made The copyright holder for this preprintthis version posted September 4, 2025. ; https://doi.org/10.1101/2025.09.01.673077doi: bioRxiv preprint 17 virion mutations tends to be positively correlated with the availability of susceptible cells and so follows a U-shaped dynamic through the phases of infection (Fig. 2G and 2H): virion mutations tend to be under strong selection at the outset of the infection when susceptible cells are abundant ( phase one ), they are selectively neutral once susceptible cells have been exhausted ( phase two ), and they rebound in value as refractory cells revert to susceptibility in persistent infections (phase three). Conversely, the strength of selection on infected cell mutations tends to be negatively correlated with the availability of susceptible cells and so follows an inverted U-shaped dynamic through the phases of infection. The strength of selection on virion and infected cell mutations also depends on the type and severity of immunodeficiency. During an acute infection (phase one and two), immunodeficiencies in the orchestration of the innate immune response, specifically, deficiencies in IFN, neutrophils, monocytes and/or the cytokines G -CSF and GM -CSF play the most significant evolutionary role. Although such deficiencies strengthen selection on non-target virion mutations during the first phase of infection, selection is weakened during the second phase and by the end of an acute infection the frequency of virion mutations tends to be lower in hosts with these immunodeficiencies (Fig. 3D and 3 E). During a persistent infection (phase three), T cell deficiencies increase infection size and so decrease the availability of susceptible cells . This slows the evolution of virion mutations, while also reducing selection on T cell evasion (Fig. 4D) . Conversely, a n IL -6 and/or macrophage deficiency overstimulates T cells, thus speeding the evolution of virion mutations and T cell evasion (Fig. 4E). While reduced neutrophil concentrations increase T cell counts, they also reduce IFN concentrations and so speed the evolution of mutations targeting cell entry, and antibody and T cell evasion, but slow the evolution of evasion of neutrophil and IFN (Fig. 4F). Previous work using between -host models indicates that fitness valleys (i.e., negative epistasis), are necessary for immunocompromised hosts to promote the emergence of novel viral strains18,19. While fitness valleys are important at the between -host level, at the within-host level, it is probable positive epistasis is more likely to produce genetically distinct strains by accelerating evolution during persistent infections where viral diversity is more common . Indeed, o ur model indicates positive epistasis occurs between combinations of virion mutations and can significantly accelerate within-host evolution (Fig. 5). Conversely, negative epistasis, which occurs between virion mutations and mutations aVecting T cell evasion, has a negligible eVect (Fig. 5). Importantly, epistasis emerges in our model due to the immunological dynamics (i.e., from the phenotype -fitness mapping ); previous work generated epistasis by building it into the genotype-phenotype map18-20. Macrophages are an important component of the innate immune response, helping to bridge the innate and adaptive immune response 55 and clear cellular debris 56. As macrophage is .CC-BY-NC 4.0 International licenseavailable under a (which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity. It is made The copyright holder for this preprintthis version posted September 4, 2025. ; https://doi.org/10.1101/2025.09.01.673077doi: bioRxiv preprint 18 primarily an orchestrator of the immune response, our model indicates evasion of macrophage is under weak selection (Sup. Info. Fig. S2) and macrophage deficiencies have limited evolutionary impact during acute infection . However, during persistent infections, macrophage deficiencies overstimulate T cells, speeding viral evolution. Mathematical modelling indicates macrophage deficiencies through dysregulated monocyte -to- macrophage diVerentiation, are predictive of disease severity 22, as they cause T cell lymphopenia and hyperinflammation . Clinical studies have similarly reported shifts in the proportions of monocyte and macrophage subsets in the blood and lungs of severe COVID- 19 patients, causing reductions to T cell recruitment of T cell and counts57. In combination with our analysis, this would suggest severe disease can also provide an optimal environment for within-host evolution. To isolate how immunodeficiencies aVect the orchestration of the immune response and the condition of the immunocompromised host at homeostasis , we intentionally constrained each immunodeficiency to reducing the production of a single immunological state variable. Real immunodeficiencies, however, involve multiple aspects of the immune response. For example, hosts with uncontrolled HIV show decreased production of naïve T cells, and CD4+ T cell lymphopenia 58, resulting in a failure to produce antibodies 59. Importantly, most clinically observed immunodeficiencies exhibit decreased T cells (see Sup. Info. 2.1), which our analysis reveals will slow viral evolution (Fig. 4D). A notable exception are hosts with B cell deficiencies and/or lymphomas 60, who have weakened antibody response s and neutrophil deficiencies34,61; indeed, the latter speeds the evolution of T cell evasion (Fig. 4F). In a secondary infection, the adaptive and humoral response will more rapidly activate leading to an earlier onset of selection for antibody and T cell evasion. Although beyond the scope of the current work, this raises the question of whether these mutations should be more strongly selected for in a primary infection in an immunocompromised host, or in a secondary infection in an immunocompetent host? Our analysis suggests that, owing to the heightened value of virion mutations at the beginning of the infection when susceptible cells are abundant, antibody evasion is more likely to be selected for in a secondary infection in an immunocompetent host. Conversely, as the value of infected cell mutations tends to be inversely correlated with the availability of susceptible cells, T cell mutations are more likely to experience stronger selection in a primary infection in an immunocompromised host. To isolate the role of selection, we ignored the generation of de novo mutations, instead assuming mutations were present at low frequencies in the viral inoculum. It is well understood that, owing to the exponential within -host growth of viral populations during acute infections, most de novo mutations in acute infections will appear just before the susceptible cells are exhausted 62,63, greatly restricting the amount of time for selection to .CC-BY-NC 4.0 International licenseavailable under a (which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity. It is made The copyright holder for this preprintthis version posted September 4, 2025. ; https://doi.org/10.1101/2025.09.01.673077doi: bioRxiv preprint 19 operate. This constraint is alleviated in persistent infections, providing another reason for more rapid adaptation in immunocompromised hosts , as compared to chains of acute infections63. Furthermore, we assumed each mutation had a single beneficial eVect on viral fitness and so ignored potential costs or trade-oVs. Fitness trade-oVs, however, are likely to aVect the forward transmission process as within -host evolution is unlikely to optimize transmission between -hosts. Indeed, there is some data in SARS -CoV-2 evolution in immunocompromised hosts of a trade -oV between antibody evasion and between -host transmissibility53. Thus, integrating our within -host evolutionary analysis into a between - host framework represents an important next step for future work to quantify the population- level risks posed by within-host evolution during persistent infections.

The authors wish to thank the institutions who funded this study . MC is the Canada Research Chair in Computational Immunology and t his research was undertaken, in part, thanks to funding from the Canada Research Chairs Program. MC was also funded by NSERC Discovery Grant s RGPIN-2018-04546 and RGPIN-2025-04412. XD was funded by a Bourse de fin d’études from the Université de Montréal. DVM was funded by NSERC Discovery Grant RGPIN-2024-04608. MC and DVM were also funded by the NOVA – FRQNT-NSERC PROGRAM for junior researchers (https://doi.org/10.69777/342478). Funders played no role in study design, data collection, analysis and interpretation of data, or the writing of this manuscript. Competing interests: The authors declare no competing interests. .CC-BY-NC 4.0 International licenseavailable under a (which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity. It is made The copyright holder for this preprintthis version posted September 4, 2025. ; https://doi.org/10.1101/2025.09.01.673077doi: bioRxiv preprint 20 Supplementary Information 1. Immunological model of primary infection Virions, 𝑉(𝑡), infect susceptible cells, 𝑆(𝑡), producing infected cells, 𝐼(𝑡), via mass action transmission with rate constant 𝛽. Infected cells undergo an eclipse phase lasting 𝜏! days during which cells are non -productively infected. Following the eclipse phase, lysis occurs with a burst size of 𝑝 virions. Virions naturally decay at a per -capita rate 𝑑" and are destroyed by neutrophils, 𝑁(𝑡), inflammatory macrophages, 𝑀#! (𝑡), and antibodies, 𝐴(𝑡). Infected cells die at a per-capita rate 𝑑!, and are killed by neutrophils, inflammatory macrophages, and T cells, 𝑇(𝑡). Infected cells can become refractory to infection, 𝑅(𝑡), through Type -I interferon (IFN) signalling, 𝐹.(𝑡). After 𝜏$ time units, refractory cells revert to susceptibility at a per-capita rate 𝛿$. The delay30 in reversion accounts for the loss of increased Type I IFN signalling and IFN stimulated genes31 through e.g., receptor internalization and downregulation32. Susceptible and refractory cells are killed by neutrophil toxicity through bystander damage. These cells also undergo logistic growth , which depends on the concentration of susceptible, refractory, and infected cells, as well as the concentration of d ead (and damaged) cells, 𝐷(𝑡). Dead cells are generated by death or damage of infected, susceptible, and refractory cells. Dead cells naturally decay and are destroyed by macrophages. Let 𝑚𝑉 ≡ 𝑑& + 𝛿&,'𝑁 + 𝛿&,(𝑀)* + δ&,+ 𝐴," ϵ&,+ ," + 𝐴," (S1) 𝑚𝐼 ≡  𝑑* + δ*,'𝑁,# 𝑁,# + 𝐼𝐶-.,' ,# + δ*,(𝑀ΦI+ δ*,1𝑇 ϵ2,1 + 𝐼,$ (S2) denote the per -capita decay rates of virions and infected cells, respectively. T hen the dynamics of virions, and infected, susceptible, refractory, and dead cells are given by the following equations: 𝑑𝑉 𝑑𝑡 = 𝑝𝐼 − 𝑚&𝑉, (S3) 𝑑𝐼 𝑑𝑡 = 𝛽𝜖3,* 𝜖3,* + 𝐹4 𝑆(𝑡 − 𝜏* )𝑉(𝑡 − 𝜏* ) − 𝑚*𝐼, (S4) 𝑑𝑆 𝑑𝑡 = κ5 A1 − 𝐶 𝑆678 B S − C𝛽𝑉 + ρδ'𝑁,# 𝑁,# + 𝐼𝐶-.,' ,# E 𝑆 + 𝛿9𝑅(𝑡 − 𝜏9), (S5) .CC-BY-NC 4.0 International licenseavailable under a (which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity. It is made The copyright holder for this preprintthis version posted September 4, 2025. ; https://doi.org/10.1101/2025.09.01.673077doi: bioRxiv preprint 21 𝑑𝑅 𝑑𝑡 = 𝜅5 A1 − 𝐶 𝑆678 B 𝑅 + 𝛽𝐹 4 𝐹4 + 𝜖3,* 𝑆(𝑡 − 𝜏*)𝑉(𝑡 − 𝜏*) − C 𝜌𝛿'𝑁,# 𝑁,# + 𝐼𝐶-.,' ,# E 𝑅 − 𝛿9𝑅(𝑡 − 𝜏9 ), (S6) 𝑑𝐷 𝑑𝑡 = 𝑚*𝐼 + 𝜌𝛿'(𝑆 + 𝑅)𝑁,# 𝑁,# + 𝐼𝐶-.,' ,# − (𝑑; + (𝛿;,(% − 𝛿(%,;)(𝑀<9 + 𝑀<*))𝐷, (S7) where 𝐶 ≡ 𝑆 + 𝐼 + 𝑅 + 𝐷 is the total density of cells, regardless of state. Infection first induces an innate response by neutrophils, 𝑁(𝑡), and inflammatory macrophages, 𝑀*!(𝑡). Neutrophils are generated by the bound cytokines interleukin-6 (IL-6; 𝐿.(𝑡)) and granulocyte colony-stimulating factor (G-CSF; 𝐶.(𝑡)) at a rate proportional to the concentration of neutrophils in the bone marrow, 𝑁$ ∗, taken to be a constant for simplicity (here and in what follows we use a subscript 𝑏 and 𝑢 to denote bound and unbound cytokines). Neutrophils have a natural per-capita death rate of 𝑑(. Inflammatory macrophages are diVerentiated from monocytes, 𝑀(𝑡), at a per-capita rate that depends on the cytokines IL-6 and granulocyte macrophage-colony stimulating factor (GM-CSF; 𝐺.(𝑡)) given by 𝜃) ≡ 𝑝𝑀Φ𝐼,𝐺 𝐺𝑏 ℎ𝑀,𝑀Φ 𝐺𝑏 ℎ𝑀,𝑀Φ + ϵ𝐺,𝑀Φ𝐼 ℎ𝑀,𝑀Φ + 𝑝𝑀Φ𝐼,𝐿 𝐿𝑏 𝐿𝑏 + ϵ𝐿,𝑀Φ . Inflammatory macrophages are also converted from tissue-resident macrophages, 𝑀*$(𝑡), at a per-capita rate depending on infected and dead cells: 𝜃𝑀Φ𝑅,𝐼 ≡ 𝑎𝐼,𝑀Φ(𝐼 + 𝐷). In turn, inflammatory macrophages convert to tissue-resident macrophages at a per-capita rate depending on tissue-resident macrophages and virions: 𝜃𝑀Φ𝐼,𝑅 ≡ !1 − '%& '%'() $ (*% )*++,*% . Macrophages are destroyed through phagocytosis and natural decay. Monocytes are generated by the cytokine GM-CSF at a rate proportional to the concentration of monocyte precursors in the bone marrow, 𝑀$ ∗ (taken to be a constant for simplicity). Monocytes are also generated through stimulation by infected cells. The dynamics of neutrophils, inflammatory and tissue-resident macrophages, and monocytes are given by 𝑑𝑁 𝑑𝑡 = A𝑁ABCD ∗ + Q𝜓' 678 − 𝑁ABCD ∗ S 𝐶4/(ζF 𝑁) − 𝐶43 ∗ 𝐶4/(ζF 𝑁) − 𝐶43 ∗ + 𝜖F,' B 𝑁9 ∗ + 𝑝',G𝐿4 𝐿4 + 𝜖G,' − 𝑑'𝑁, (S8) 𝑑𝑀)* 𝑑𝑡 = 𝜃𝑀Φ𝑅,𝐼 𝑀)9 + 𝜃𝑀𝑀 − Q𝑑(,- + δ(,,;𝐷 + 𝜃𝑀Φ𝐼,𝑅 S𝑀)*, (S9) 𝑑𝑀)9 𝑑𝑡 = 𝜃𝑀Φ𝐼,𝑅 𝑀)* − Q𝑑(,. + δ(,,;𝐷 + 𝜃𝑀Φ𝑅,𝐼 S𝑀)9, (S10) 𝑑𝑀 𝑑𝑡 = C𝑀ABCD ∗ + Q𝜓( 678 − 𝑀ABCD ∗ S 𝐺4 ,/ 𝐺4 ,/ + 𝜖H,( ,/ E 𝑀9 ∗ + 𝑝(,*𝐼 𝐼 + 𝜖*,( 𝑀 − (𝑑( + 𝜃𝑀)𝑀. (S11) .CC-BY-NC 4.0 International licenseavailable under a (which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity. It is made The copyright holder for this preprintthis version posted September 4, 2025. ; https://doi.org/10.1101/2025.09.01.673077doi: bioRxiv preprint 22 While the innate response is being mobilized, the humoral response mediated by antibodies, 𝐴(𝑡), is generated by the presence of virions. The production of antibody by virions starts after 𝜏& days. The delay in antibody production accounts for the time required for the uptake of antigen by antigen presenting cells and their presentation to CD4+ T cells, the initial production of antibodies by short -lived plasma cells after CD4+ T cell priming, and th e production of long -lived plasma cells and memory B cells from germinal centre B cells undergoing somatic hypermutation 24. A ntibodies neutralize virions and are lost due to natural decay. Their dynamics are given by 𝑑𝐴 𝑑𝑡 = 𝑝+,&𝑉(𝑡 − 𝜏+) − 𝑑+𝐴 − 𝛿&,+ 𝐴," 𝑉 𝜖&,+ ," + 𝐴," . (S12) After 𝜏% days, the adaptive immune response mediated by CD8+ eVector T cells, 𝑇(𝑡), is activated. The activation delay accounts for the time necessary for naïve CD8+ T cells to be presented antigen, and to convert to the eVector phenotype following a primary exposure22,23,64,65. CD8+ T cells are recruited by infected cells through a process mediated by IL-666, are stimulated by IFN, and are lost due to natural decay. Their dynamics are given by 𝑑𝑇 𝑑𝑡 = 𝑝%,!𝜖?,% 𝐿. + 𝜖?,% 𝐼(𝑡 − 𝜏%) + 𝑝%,- 𝐹. 𝐹. + 𝜖-,% 𝑇 − 𝑑%𝑇. (S13) The action of the immune system is mediated by cytokines. We model unbound and bound cytokines, with bound cytokines responsible for immune eVects. Let 𝑌@(𝑡) and 𝑌.(𝑡) denote the concentration of unbound and bound cytokine 𝑌 at time 𝑡, and 𝑌ABCD (𝑡) denote the rate of endogenous cytokine production. Then the general pharmacokinetic model of cytokine unbinding/binding is given by 𝑑𝑌@ 𝑑𝑡 = 𝑌ABCD − 𝑘E4F𝑌@ − 𝑘. (𝑋𝜁 − 𝑌.)(𝑌@)GHI + 𝑘@𝑌., 𝑑𝑌. 𝑑𝑡 =   − 𝑘4F𝑌. + 𝑘. (𝑋ζ − 𝑌.)(𝑌@)GHI − 𝑘@𝑌.. Here, 𝑘. and 𝑘@ are the binding and unbinding rates, respectively, 𝑘4F is the rate of bound cytokine internalization, 𝑘E4F is the rate of elimination, 𝑃𝑂𝑊 is a stoichiometric constant, 𝑋 is the sum of all cells modulated by the cytokine, and ζ is a scaling factor satisfying Xζ = 𝑝̂𝑌)I K10FX, in concentration units of pg/mL. We used the equation 𝑌)I = 𝑀𝑀/𝑁& to calculate the molecular weight of each cytokine, where 𝑀𝑀 is the molar mass and 𝑁& = 6.02214 × 107J is Avogadro’s number. In the equation above, 𝑝̂ is a constant relating the stoichiometry between cytokine molecules and their receptors, 𝐾 is the number of receptors specific to each cytokine on a cell’s surface and 10F is a factor correcting for cellular units, giving: ζ3 = 𝑀𝑀3 𝑁+ Q𝐾3,1  + 𝐾3,*S ∙ A10IJ 5000B , .CC-BY-NC 4.0 International licenseavailable under a (which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity. It is made The copyright holder for this preprintthis version posted September 4, 2025. ; https://doi.org/10.1101/2025.09.01.673077doi: bioRxiv preprint 23 ζG = 𝑀𝑀G 𝑁+ Q𝐾G,' + 𝐾G,1 + 𝐾G,(S ∙ A10IJ 5000B , ζH = 𝑀𝑀H 𝑁+ 𝐾H,( ∙ A10IJ 5000B , ζF = 𝑝̂ 𝑀𝑀F 𝑁+ 𝐾F,' ∙ A 10K 5000B . In our immunological model of primary infection, there are four key cytokines 22: interferon (IFN), interleukin -6 (IL -6), granulocyte colony -stimulating factor (G -CSF), and granulocyte macrophage-colony stimulating factor (GM -CSF). Unbound IFN, 𝐹@(𝑡), is produced by infected cells, monocytes, and inflammatory macrophages. IFN binds to infected cells and CD8+ T cells, producing bound IFN, 𝐹.(𝑡), that leads to refractory cells. The dynamics of IFN are captured by 𝑑𝐹L 𝑑𝑡 = 𝑝3,*𝐼 𝐼 + 𝜂3,* + 𝑝3,(𝑀 𝑀 + 𝜂3,( + 𝑝3,(,- 𝑀)* 𝑀)* + 𝜂3,(,- − 𝑘MNO0 𝐹L − 𝑘40 Q(𝐼 + 𝑇)ζ3 − 𝐹4S𝐹L + 𝑘L0 𝐹4, (S14) 𝑑𝐹4 𝑑𝑡 = −(𝑘NO3+𝑘L0 )𝐹4 + 𝑘40 Q(𝐼 + 𝑇)ζ3 − 𝐹4S𝐹L. (S15) Unbound IL -6, 𝐿@(𝑡), is produced by infected cells, monocytes, and inflammatory macrophages. IL-6 binds to neutrophils, monocytes, and CD8+ T cells, forming bound IL -6, 𝐿.(𝑡), which stimulates neutrophil production and the diVerentiation of monocytes into inflammatory macrophage, and inhibits the production of CD8+ T cells. The dynamics of IL- 6 are given by 𝑑𝐿L 𝑑𝑡 = 𝑝G,*𝐼 𝐼 + 𝜂G,* + 𝑝G,(𝑀 𝑀 + 𝜂G,( + 𝑝G,(,- 𝑀)* 𝑀)* + 𝜂G,(,1 − 𝑘MNO2 𝐿L − 𝑘42 Q(𝑁 + 𝑀 + 𝑇)ζG − 𝐿4 S𝐿L + 𝑘L2 𝐿4, (S16) 𝑑𝐿4 𝑑𝑡 = −Q𝑘NO2 + 𝑘L2 S𝐿4 + 𝑘42 Q(𝑁 + 𝑀 + 𝑇)ζG − 𝐿4S𝐿L. (S17) Unbound G -CSF , 𝐶@(𝑡), is produced by monocytes. G -CSF binds to neutrophils, forming bound G-CSF , 𝐶.(𝑡), which stimulates neutrophil production. The dynamics of G -CSF are given by 𝑑𝐶L 𝑑𝑡 = 𝑝F,(𝑀 𝑀 + 𝜂F,( − 𝑘MNO3 𝐶L − 𝑘43 (𝑁ζF − 𝐶4)(𝐶L)QRS + 𝑘L3 𝐶4, (S18) 𝑑𝐶4 𝑑𝑡 = −𝑘NO3 𝐶4 + 𝑘43 (𝑁ζF − 𝐶4)(𝐶L)QRS − 𝑘L3 𝐶4. (S19) Finally, unbound GM-CSF , 𝐺@(𝑡), is produced by monocytes and inflammatory macrophages and binds to monocytes, forming bound GM -CSF, 𝐺.(𝑡). Bound GM -CSF produces monocytes and induces diVerentiation of monocytes into inflammatory macrophages. The dynamics of GM-CSF are given by .CC-BY-NC 4.0 International licenseavailable under a (which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity. It is made The copyright holder for this preprintthis version posted September 4, 2025. ; https://doi.org/10.1101/2025.09.01.673077doi: bioRxiv preprint 24 𝑑𝐺L 𝑑𝑡 = 𝑝H,(𝑀 𝑀 + 𝜂H,( + 𝑝H,(,- 𝑀)* 𝑀)* + 𝜂H,(, − 𝑘MNO4 𝐺L − 𝑘44 (𝑀ζH − 𝐺4)𝐺L + 𝑘L4 𝐺4, (S20) 𝑑𝐺4 𝑑𝑡 = −𝑘NO4 𝐺4 + 𝑘44 (𝑀ζH − 𝐺4)𝐺L − 𝑘L𝐺4. (S21) 2. Immunological dynamics in immunocompromised hosts Next, we modify the immunological model to allow for immunodeficiencies . We assume immunodeficiencies have two eVects. First, they lead to prolonged infection due to the loss of IFN eVects on refractory cells. To capture this, after 𝜏$ time units, refractory cells revert to susceptibility at a per-capita rate 𝛿$ in immunocompromised hosts (in immunocompetent hosts, 𝛿$ = 0). Second, immunodeficiencies reduce the rate of production, and hence availability, of diVerent aspects of the immune response. For example, a monocyte production deficiency translates to a lower concentration of monocytes at homeostasis and during infection. Let 𝜆4 denote a production deficiency in immune variable 𝑖 ∈ {𝑁, 𝑀K,!, 𝐴, 𝑇, 𝐹@, 𝑀, 𝐿@, 𝐶@, 𝐺@}. Then production deficiencies modify the model as follows: 𝑑𝑁 𝑑𝑡 = 𝜆' _A𝑁ABCD ∗ + Q𝜓' 678 − 𝑁ABCD ∗ S 𝐶4/(𝐴F 𝑁) − 𝐶43 ∗ 𝐶4/(𝐴F 𝑁) − 𝐶43 ∗ + 𝜖F,' B 𝑁9 ∗ + 𝑝',G𝐿4 𝐿4 + 𝜖G,' ` − 𝑑'𝑁, (S24) 𝑑𝑀)* 𝑑𝑡 = 𝜃(,.,- 𝑀)9 + 𝜆(,. 𝜃(𝑀 − Q𝑑(,- + δ(,,;𝐷 + 𝜃(,-,. S𝑀)*, (S25) 𝑑𝐴 𝑑𝑡 = 𝜆+𝑝+,&𝑉 − 𝑑+𝐴 − 𝛿&,+ 𝐴," 𝑉 𝜖&,+ ," + 𝐴," , (S26) 𝑑𝑇 𝑑𝑡 = 𝜆1 _ 𝑝1,*𝜖G,1 𝐿4 + 𝜖G,1 𝐼(𝑡 − 𝜏1) + 𝑝1,3𝐹4 𝐹4 + 𝜖3,1 𝑇` − 𝑑1𝑇, (S27) 𝑑𝐹L 𝑑𝑡 = 𝜆3 _ 𝑝3,*𝐼 𝐼 + 𝜂3,* + 𝑝3,(𝑀 𝑀 + 𝜂3,( + 𝑝3,(,- 𝑀)* 𝑀)* + 𝜂3,(,- ` − 𝑘MNO0 𝐹L − 𝑘40 Q(𝑇 + 𝐼)𝐴3 − 𝐹4 S𝐹L + 𝑘L0 𝐹4, (S28) 𝑑𝑀 𝑑𝑡 = 𝜆( C𝑀ABCD ∗ + Q𝜓( 678 − 𝑀ABCD ∗ S 𝐺4 ,/ 𝐺4 ,/ + 𝜖H,( ,/ E 𝑀9 ∗ + 𝜆( 𝑝(,*𝐼 𝐼 + 𝜖*,( 𝑀 − (𝑑( + 𝜃()𝑀, (S29) 𝑑𝐿L 𝑑𝑡 = 𝜆G5 A 𝑝G,*𝐼 𝐼 + 𝜂G,* + 𝑝G,(𝑀 𝑀 + 𝜂G,( + 𝑝G,(,- 𝑀)* 𝑀)* + 𝜂G,(,1 B − 𝑘MNO2 𝐿L − 𝑘42 Q(𝑁 + 𝑀 + 𝑇)𝐴G − 𝐿4 S𝐿L + 𝑘L2 𝐿4, (S30) 𝑑𝐶L 𝑑𝑡 = 𝜆F5 𝑝F,(𝑀 𝑀 + 𝜂F,( − 𝑘MNO3 𝐶L − 𝑘43 (𝑁𝐴F − 𝐶4)(𝐶L)QRS + 𝑘L3 𝐶4, (S31) 𝑑𝐺L 𝑑𝑡 = λH5 A 𝑝H,(𝑀 𝑀 + ηH,( + 𝑝H,(,1 𝑀)T 𝑀)* + ηH,(, B − 𝑘MNO4 𝐺L − 𝑘44 (𝑀𝐴H − 𝐺4)𝐺L + 𝑘L4 𝐺4 (𝑆32) Production deficiencies also modify the initial conditions, as they can aVect homeostasis in the absence of infection. Therefore, for all simulations we first allow the immune system to reach homeostasis before challenging the host with the viral inoculum. .CC-BY-NC 4.0 International licenseavailable under a (which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity. It is made The copyright holder for this preprintthis version posted September 4, 2025. ; https://doi.org/10.1101/2025.09.01.673077doi: bioRxiv preprint 25 2.1. More realistic immunodeficiencies While we treat immunodeficiencies as a reduction in the production rate of a single variable, clinically observed immunodeficiencies tend to be multifaceted, simultaneously aVecting diVerent aspects of the immune response. Four common immunodeficiencies , and their most prominent immunological features, are: 1. B cell deficiencies and/or lymphomas : These individuals have a weakened antibody response and may also experience neutrophil deficiencies 34,61 (reduced 𝐴(𝑡), 𝑁(𝑡)), depending on the type of deficiency or lymphoma60. 2. Uncontrolled HIV and AIDS : Individuals with uncontrolled HIV show decreased production of naïve T cells, and CD4+ T cell lymphopenia 58, resulting in a failure to produce antibodies 59. Further, natural interferon -α producing cells are greatly reduced in individuals with AIDS67 (reduced 𝐴(𝑡), 𝑇(𝑡), 𝐹. (𝑡)) 3. Non-B cell cancers : Individuals with non -B cell malignancies are treated with myelosuppressive chemotherapies that suppress aspects of the innate immune system (e.g., deficiencies in neutrophils and macrophages68) as well as CD8+ T cells69 (reduced 𝑁(𝑡), 𝑇(𝑡), 𝑀*,! (𝑡)). 4. Solid organ transplant recipients : These individuals are treated with lifelong immunosuppressive drugs 70 and so are deficient in all aspects of the immune response (reduced 𝐴(𝑡), 𝑁(𝑡), 𝑇(𝑡), 𝑀*,! (𝑡)). We expect that more realistic immunodeficiencies recapitulate the strongest eVects we observed when considering single-variable immunodeficiencies, and so we do not consider them further here. 3. Parameter definitions and parameter values used in simulations Our immunological model is based oV a previous published model22; we have made several modifications to this model which we detail here . First, we allow for the reconversion of refractory to susceptible cells after 𝜏$ days in immunocompromised hosts. The rate of this reconversion, δ$, was taken to be 0.05 days-1, and 𝜏$ was set to 8 days to fall within the range of previous estimates27. Second, consistent with studies from the lungs of influenza infected mice 23, we modelled infected cell killing by T cells to be density dependent and saturable using the function δ!,% 𝑇 ϵL,% + 𝐼+, . By comparing the total number of infected cells in our model versus that of Myers et al.64, we set the maximal rate of T cell killing to be δ!,% = 7 days-1, the half-maximal cell concentration ϵL,% to be 0.01 x 109 cells/mL, and the Hill coeVicient ℎ% to be 0.5. .CC-BY-NC 4.0 International licenseavailable under a (which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity. It is made The copyright holder for this preprintthis version posted September 4, 2025. ; https://doi.org/10.1101/2025.09.01.673077doi: bioRxiv preprint 26 Third, given that primary infections can become protracted in immunocompromised hosts, we included neutralizing antibodies in our model. We assume antibodies are generated by the presence of virions 𝜏& days post-infection. The delay in antibody production accounts for the time required for the uptake of antigen by antigen presenting cells and their presentation to CD4+ T cells, the initial production of antibodies by short-lived plasma cells after CD4+ T cell priming, and t he production of long -lived plasma cells and memory B cells from germinal centres24. We assumed antibodies are produced at a constant rate of 𝑝&," = 500 days-1 and are cleared at rate 𝑑& = 0.033 days-1. We model the neutralization of virions by antibodies using the Hill function given by δ",& 𝐴+! ϵ",& +! + 𝐴+! . Based on our previous work where we fit this Hill function to data of SARS -CoV-2 neutralization by antibodies24,71, we set δ",& = 5 days-1, ℎ& = 1.19, and ϵ",& = 1000 AU/mL. Fourth, we model immunodeficiencies as a reduction in the production of the target variable (see Sup. Info. 2) . In our model, production of cells (i.e., neutrophils, monocytes, macrophages, and T cells) and cytokines (i.e., IFN, G -CSF , GM-CSF , and IL -6) is often generated through multiple pathways ( see equations S24-S32). By assuming that immunodeficient hosts had at least a 50% reduction in the target variable compared to healthy hosts, we reduce the rate of production of neutrophils, monocytes, macrophages, IFN, G-CSF , GM-CSF , and IL-6 by 35%. We further reduce T cell and antibody production to more closely resemble clinical observations of lymphopenia33 and the absence of antibodies in certain patient groups 8,34. Hence, T cell production is reduced by 75% and antibody production by 90%. All other parameters in the immunological model were set to their previously estimate d values. Full details are provided in the Supplementary Information to Jenner et al. 22. Briefly, Jenner et al.22 applied a hierarchical estimation procedure in which certain parameter values were fixed directly from the literature, while others were fit using nonlinear least squares or nonlinear mixed eVects methods to dose response data or time-series measurements. In simulations incorporating immunodeficiencies, we calculated any remaining parameter values at physiological homeostasis to ensure that the immunological components return to basal concentrations . In particular, we set the diVerential equations for the diVerent components of the immune system to 0 (in the absence of the virus), and then we calculated the values of 𝐿/,1, 𝐶/,1, 𝐺/,1, 𝐹/,1, 𝑀-.,0, ϵ?,)1# , ϵM,), 𝑝%,-, 𝑀ABCD ∗ , 𝑁ABCD ∗ , and 𝜂3,(,- . The complete list of parameter values is provided in Table S1. .CC-BY-NC 4.0 International licenseavailable under a (which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity. It is made The copyright holder for this preprintthis version posted September 4, 2025. ; https://doi.org/10.1101/2025.09.01.673077doi: bioRxiv preprint 27 Figure S1. Changes in the immunological response in hosts with varying degrees of immunodeficiencies. Increasing the severity of the immunodeficiencies results in a quantitative, but not qualitative, change in the concentrations of the diVerent immunological variables. From panel A to C, we increase the severity of the immunodeficiencies (panel B shows the severity of immunodeficiencies used in the main text). In panel A, the severity of immunodeficiencies was reduced by 25% from its baseline value shown in panel B (see

in the Main Text). In panel C, the severity of immunodeficiencies was increased by 25% from its baseline value (panel B) . In all panels, the percent change in each immunological variable is calculated as (𝑋 − 𝑌)/𝑋 × 100, where 𝑋 and 𝑌 are the integrals of the indicated variable over the length of the phase in immunodeficient and healthy hosts, respectively, while the percent change in susceptible cells is calculated using the integral of 𝑆,:2 − 𝑆(𝑡). Virus Infected cells Susceptible cells Refractory cells Dead cells Macrophages Monocytes Neutrophils Antibodies T cells Unbound G-CSF Bound G-CSF Unbound GM-CSF Bound GM-CSF Unbound IFN Bound IFN Unbound IL-6 Bound IL-6 Variable Monocyte Neutrophil G-CSF GM-CSF Macrophage IL-6 IFN Antibody T cell Deficiency Virus Infected cells Susceptible cells Refractory cells Dead cells Macrophages Monocytes Neutrophils Antibodies T cells Unbound G-CSF Bound G-CSF Unbound GM-CSF Bound GM-CSF Unbound IFN Bound IFN Unbound IL-6 Bound IL-6 Variable Virus Infected cells Susceptible cells Refractory cells Dead cells Macrophages Monocytes Neutrophils Antibodies T cells Unbound G-CSF Bound G-CSF Unbound GM-CSF Bound GM-CSF Unbound IFN Bound IFN Unbound IL-6 Bound IL-6 Variable Monocyte Neutrophil G-CSF GM-CSF Macrophage IL-6 IFN Antibody T cell Deficiency Monocyte Neutrophil G-CSF GM-CSF Macrophage IL-6 IFN Antibody T cell Deficiency Phase 1 Phase 2 Phase 3A B C -100 -50 -25 -15 -5 5 15 25 50 100 300 NaN Percent change -100 -50 -25 -15 -5 5 15 25 50 100 300 NaN Percent change -100 -50 -25 -15 -5 5 15 25 50 100 300 NaN Percent change .CC-BY-NC 4.0 International licenseavailable under a (which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity. It is made The copyright holder for this preprintthis version posted September 4, 2025. ; https://doi.org/10.1101/2025.09.01.673077doi: bioRxiv preprint 28 4. Viral evolution 4.1 Evolutionary model Next, we extend the immunological model to account for the possibility of viral evolution. To do so, we allow for strains carrying diVerent mutations to simultaneously circulate in the population. We focus throughout on the action of selection and so ignore de novo mutations. We therefore assume that the mutant strains under consideration (and so the mutations of interest) are present at low frequencies (specifically, a frequency of 0.01) in the initial viral inoculum. We consider eight possible mutations. Each of these mutations is beneficial and has a single eVect on viral life history. The eight mutations considered are: 1. Virion evasion of antibodies, 𝐴(𝑡), through decreased δ",&. 2. Virion evasion of neutrophils, 𝑁(𝑡), through decreased δ",(. 3. Virion evasion of macrophages, 𝑀*,! (𝑡), through decreased δ",). 4. Increased virion ability to enter the host cell, 𝑆(𝑡), through increased 𝛽 (e.g., by inhibiting PAMPs35,36). 5. Virion evasion of interferon, by reducing the ability of interferon to block viral entry through increased ϵ-,!. 6. Infected cell evasion of T cells, 𝑇(𝑡), through decreased 𝛿!,%. 7. Infected cell evasion of neutrophils, 𝑁(𝑡), through decreased δ!,(. 8. Infected cell evasion of macrophages, 𝑀*,! (𝑡), through decreased δ!,). The mutations can be grouped as either “virion mutations” , which are mutations that target components of virion fitness (mutations 1-5, that is, decreased δ",(, δ",), δ",& or increased 𝛽, ϵ-,!), or “infected cell mutations” , which are the mutations that target components of infected cell fitness (mutations 6 -8, that is, decreased δ!,(, δ!,), δ!,%). Note that ϵ-,! is classified here as a “virion mutation” mainly because the eclipse phase is short, and so it is more closely linked to virion fitness than infected cell fitness. For each mutation, we divide the immunological variables into the variable directly targeted by the mutation (henceforth the “target”) and all other immunological variables not targeted by the focal mutation but targeted by a diVerent mutation (henceforth , “nontargets”). For example, if the focal mutation targets virion evasion of neutrophils (decreased δ",(), then the target variable is neutrophils, while the nontarget variables are macrophage, interferon, antibodies, T cells, and susceptible cells. .CC-BY-NC 4.0 International licenseavailable under a (which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity. It is made The copyright holder for this preprintthis version posted September 4, 2025. ; https://doi.org/10.1101/2025.09.01.673077doi: bioRxiv preprint 29 Next, let 𝐼4 (𝑡) and 𝑉4(𝑡) denote the density of cells infected with strain 𝑖 and strain 𝑖 virions, respectively. Strain 𝑖 has phenotype {𝛿",( 4 , 𝛿",) 4 , 𝛿",& 4 , 𝛽4, 𝜖-,! 4 , 𝛿!,( 4 , 𝛿!,) 4 , 𝛿!,% 4 }. Then we can write the dynamics of strain 𝑖 as 𝑑𝑉N 𝑑𝑡 = 𝑝𝐼N − C𝑑& + 𝛿&,' N 𝑁 + 𝛿&,( N 𝑀<,* + 𝛿&,+ N 𝐴," Q𝜖&,+ ," + 𝐴," S E deeeeeeeeeeeefeeeeeeeeeeeeg 66 7 𝑉N, (S32) 𝑑𝐼N 𝑑𝑡 = 𝛽N𝜖3,* N 𝜖3,* N + 𝐹4 𝑆(𝑡 − 𝜏*) hiiiiijiiiiik U- 7 𝑉N(𝑡 − 𝜏*) − C𝑑* + 𝛿*,' N 𝑁,# 𝑁,# + 𝐼𝐶-.,' ,# + 𝛿*,( N 𝑀<,* + δ*,1𝑇 ϵ2,1 + 𝐼,$ E hiiiiiiiiiiiiijiiiiiiiiiiiiik 6- 7 𝐼N. (S33) where 𝐵! 4 is the per -capita rate at which strain 𝑖 virions produce cells infected by strain 𝑖, while 𝑚" 4 and 𝑚! 4 are the per -capita decay rates of strain 𝑖 virions and infected cells, respectively. 4.2 Evolution of individual mutations First consider the evolution of each individual mutation. To do so, we note that if 𝑝2 (𝑡) denotes the frequency of mutation 𝑥 within the viral population, the frequency of mutation 𝑥 changes according to the equation 𝑑𝑝2 𝑑𝑡 = 𝑠2(𝑡)𝑝2 (1 − 𝑝2 ), (𝑆34) where 𝑠2(𝑡) is the time-varying selection coeVicient. T he strength of selection on mutation 𝑥 at time 𝑡 is determined by the magnitude of 𝑠2(𝑡); this dictates the instantaneous rate of increase of the mutation, given an observed mutation frequency. In turn, 𝑠2(𝑡) can be used to calculate the time-averaged strength of selection on mutation 𝑥 from time 𝑡1 to time 𝑡: ⟨𝑠2(𝑡)⟩ = 1 𝑡 − 𝑡1 S 𝑠2(𝑡)𝑑𝑥. 3 3% (S35) Equation ( S35) measures how strong (constant) selection would have to be to yield the observed change in frequency by time 𝑡. Thus, 𝑠2(𝑡) and ⟨𝑠2 (𝑡)⟩ capture how the immune response determines the short - and long- term strength of selection, respectively. This in turn dictates the speed of adaptation and so will be our focus here. Specifically, we will use 𝑠2(𝑡) and ⟨𝑠2(𝑡)⟩ to understand how immunodeficiencies aVect the speed of evolution of diVerent mutations targeting viral evasion of the immune response. 4.2.1 Approximating per-capita growth rate To calculate the selection coeVicient, we need to compute the per-capita growth rates, 𝑟4, of the diVerent strains. This is because if there are two strains, 𝑖 ∈ {𝑥N, 𝑥,}, diVering based on a single mutation, then the selection coeVicient acting on mutation 𝑥 is .CC-BY-NC 4.0 International licenseavailable under a (which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity. It is made The copyright holder for this preprintthis version posted September 4, 2025. ; https://doi.org/10.1101/2025.09.01.673077doi: bioRxiv preprint 30 𝑠2(𝑡) = 𝑟22 (𝑡) − 𝑟23 (𝑡). (S36) There are two challenges to computing per -capita growth rate in our model. The first challenge is the presence of the delay due to eclipse time, 𝜏!. To remove the delay, define 𝐸4(𝑡) to be the density of cells that have been infected by strain 𝑖, but are not yet productively infected. Then we can apply the approximation 𝑑𝑉4 𝑑𝑡 = 𝑝𝐼4 − 𝑚" 4 𝑉4, (S37) 𝑑𝐸4 𝑑𝑡 = 𝛽4𝑆𝑉4 − 1 τ! 𝐸4, (S38) 𝑑𝐼4 𝑑𝑡 = 1 τ! 𝜖-,! 4 𝜖-,! 4 + 𝐹. 𝐸4 − 𝑚! 4 𝐼4. (S39) The second challenge to calculating per-capita growth rate arises due to the fact individuals belong to diVerent classes (i.e., virions , infected cells , and productively infected cells) . Calculating per -capita growth rates in populations where individuals belong to diVerent classes (i.e., virions and infected cells) is not trivial, particularly for populations with temporally varying per-capita growth rates39-42. Therefore, to gain some analytic insight into how immunodeficiencies aVect the strength of selection, suppose the immunological variables, and density of susceptible cells, change slowly relative to the change in infected cells and virions (e.g., see Day et al. 202239). Then the per-capita growth rate of strain 𝑖 is the dominant eigenvalue, 𝑟4, of the matrix 𝑅4 = ⎝ ⎜⎜ ⎛ −𝑚" 4 0 𝑝 𝛽4𝑆 − 1 τ! 0 0 1 τ! 𝜖-,! 4 𝜖-,! 4 + 𝐹. −𝑚! 4 ⎠ ⎟⎟ ⎞ . (S40) Because 𝑅4 is a 3 × 3 matrix, finding the dominant eigenvalue means finding the roots of the cubic polynomial: λJ + `𝑚! 4 + 𝑚" 4 + 1 τ! c λ7 + ]𝑚! 4 𝑚" 4 + 𝑚" 4 + 𝑚! 4 τ! ^ λ + 𝑚! 4 𝑚" 4 τ! − 𝑝 τ! 𝛽4𝑆ϵ-,! 4 ϵ-,! 4 + 𝐹. . This is numerically straightforward but poses a challenge analytically. However, as the eclipse phase is short ( τ! is small), it is not unreasonable to suppose that the dynamics of non-productively infected cells occurs on a fast time scale, relative to the dynamics of virions and productively infected cells, that is 𝑑𝐸4/𝑑𝑡 ≈ 0, and so on the slow scale 𝐸4(𝑡) ≈ 𝛽4𝑆𝑉4 τ! .CC-BY-NC 4.0 International licenseavailable under a (which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity. It is made The copyright holder for this preprintthis version posted September 4, 2025. ; https://doi.org/10.1101/2025.09.01.673077doi: bioRxiv preprint 31 This approximation yields the reduced system on the slow time scale, 𝑑𝑉4 𝑑𝑡 = 𝑝𝐼4 − 𝑚"𝑉4 (S41) 𝑑𝐼4 𝑑𝑡 = 𝛽4𝑆 𝜖-,! 4 𝜖-,! 4 + 𝐹/ 𝑉4 − 𝑚! 4 𝐼4 = 𝐵! 4𝑉4 − 𝑚! 4 𝐼4, (S42) where 𝐵! 4 = 𝛽4𝑆 𝜖-,! 4 𝜖-,! 4 + 𝐹/ . For the reduced system, the per-capita growth rate of strain 𝑖 is the dominant eigenvalue, 𝑟4, of the matrix 𝑅4 = ]−𝑚" 4 𝑝 𝐵! 4 −𝑚! 4 ^ . (S43) Thus 𝑟4 = 𝜃4 − 𝑚" 4 − 𝑚! 4 2 (S44) where θ4 = š4𝐵! 4𝑝 + (𝑚a4 )7 and 𝑚a4 ≡ 𝑚" 4 − 𝑚! 4 . Numerical results indicate that approximating 𝑟4 using equation (S44) and using this to calculate the selection coeVicient, 𝑠2, is very good, with the only deviation occurring at the beginning of the infection (Fig. S2). For the numerical results presented in the main text, 𝑠2 can be approximated using simulation data as 𝑠2 (𝑡) ≈ − 1 𝑡4 − 𝑡456 ln ]𝑝2! (𝑡456)71 − 𝑝2! (𝑡4 )9 𝑝2! (𝑡4 )71 − 𝑝2! (𝑡456)9^ (S45) where 𝑝2 ! (𝑡) is the frequency of mutation 𝑥 at time 𝑡 in infected cells, and 𝑡4 − 𝑡456 is some (short) increment of time (e.g., hours, days). ⟨𝑠2(𝑡)⟩ can be approximated using equation (S45) by setting 𝑡456 = 𝑡1. Although equation (S4 5) only considers the frequency of the mutation in infected cells, simulation results indicate that it largely yields similar predictions if it is calculated using the frequency of the mutation in virions ( Fig. S2), even though the “true” selection coeVicient involves a weighted average of the frequencies of virions and infected cells. The principal area of divergence between the diVerent measures occurs at the onset of the infection. This is because each infection starts with only virions present, and so the delay in production of infected cells due to the eclipse phase means equation ( S45) cannot be accurately calculated during the initial few hours of infection (Fig. S2). Moreover, the calculations of the selection coeVicient using eigenvalues capture the strength of selection after the initial transient dynamics owing to the initial conditions are finished. Therefore, in all simulation

in the main text we plot equation (S45) from hour 12 onwards. .CC-BY-NC 4.0 International licenseavailable under a (which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity. It is made The copyright holder for this preprintthis version posted September 4, 2025. ; https://doi.org/10.1101/2025.09.01.673077doi: bioRxiv preprint 32 The other key point from Figure S2 is that mutations aVecting virion and infected cell evasion of macrophage, and mutations targeting infected cell evasion of neutrophils are under very weak selection (Figure S2F -S2H). Specifically, the time for a mutation with a constant selection coeVicient, 𝑠, to increase in frequency from 𝑝1 to 𝑝6 is 𝑡 = − 1 𝑠 ln ]𝑝1(1 − 𝑝6) 𝑝6(1 − 𝑝1)^. The time-averaged selection coeVicient for mutations aVecting evasion of macrophage and infected cell evasion of neutrophils are on the order of 105J or less, and so using the above equation it would take approximately 9,190 days for a mutation to increase from a frequency of 𝑝1 = 0.01 to 𝑝6 = 0.99. This very weak selection remains true regardless of the size of the mutational eVect; indeed, in Figure S2F-S2H, the mutational eVect s for virion and infected cell evasion of macrophage, and infected cell evasion of neutrophils, are maximal. The immunological reasoning for the negligible selection on mutations aVecting δ",), δ!,), and δ!,( are as follows. The primary role of neutrophils during viral infections is to neutralize virions72 and not infected cells. Consequently, neutrophils have a saturated killing rate of infected cells22, which translates to weak selection on infected cell evasion of neutrophils. Macrophages, on the other hand, have two key eVects. First, they balance inflammation and tissue repair, helping bridge the innate and adaptive immune responses55. Second, they clear pathogenic and cellular debris 56 through phagocytosis once cells have become damaged or die. While the first eVect is important for immune system functioning, it will not select for macrophage evasion, and the second eVect occurs near the end of the infected cells life and so has limited selective consequences. 0 0.08 0.16 Selection coefficient -- Cubic Quadratic Inf cells Virions 0 0.25 0.5 00Fi Cubic Quadratic Inf cells Virions 0 0.71 1.42 //VN Cubic Quadratic Inf cells Virions 0 0.16 0.32 //VA Cubic Quadratic Inf cells Virions 100 =T =R 102 Time (log10(days)) 0 0.05 0.1 Selection coefficient //IT Cubic Quadratic Inf cells Virions 100 =T =R 102 Time (log10(days)) 0 2.5 5 #10-4 //VM Cubic Quadratic Inf cells Virions 100 =T =R 102 Time (log10(days)) 0 0.75 1.5 #10-3 //IM Cubic Quadratic Inf cells Virions 100 =T =R 102 Time (log10(days)) 0 4 8 #10-3 //IN Cubic Quadratic Inf cells Virions A B C D E F G H .CC-BY-NC 4.0 International licenseavailable under a (which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity. It is made The copyright holder for this preprintthis version posted September 4, 2025. ; https://doi.org/10.1101/2025.09.01.673077doi: bioRxiv preprint 33 Figure S 2. Comparison of di=erent approximations of the selection coe=icient . The selection coeVicient can be numerically calculated in four ways: (1) by using formula (S36), with 𝑟22 and 𝑟23 calculated using the matrix in equation (S40) (dashed-dot lines); (2) by using formula (S36), with 𝑟22 and 𝑟23 given by equation (S44) (dotted lines) ; (3) by applying equation (S45) to simulation data (solid lines); or (4) by applying equation (S45) to simulation data, except by measuring frequency amongst virions (dashed lines). Notice that virion and infected cell evasion of macrophage (panels F and G), and infected cell evasion of neutrophils (panel H) are under very weak selection, even though the mutational eVect size is maximal. In all panels, the dashed vertical grey line indicates 12 hours. 4.2.2 Weak selection approximation We can gain further insight into the selection coeVicient by supposing the mutational diVerence between the wildtype and mutant phenotypes, 𝛥𝑥 ≡ 𝑥, − 𝑥N, is small, that is, selection is weak. Then we can use 𝑟4 to approximate 𝑠2 as 𝑠2 ≈ ]− 1 2 `1 − 𝑚a N θN c 𝜕𝑚" N 𝜕𝑥N − 1 2 `1 + 𝑚a N 𝜃N c 𝜕𝑚! N 𝜕𝑥N + 𝑝𝐵! N 𝜃N 𝜕𝑓2N 𝜕𝑥N ^ Δ𝑥 + 𝒪(Δ𝑥7), (S46) where 𝜕𝑓N/𝜕𝑥N = 1 if 𝑥 = 𝛽 and 𝜕𝑓N/𝜕𝑥N = 𝐹./7ϵ-,! + 𝐹.9 if 𝑥 = ϵ-,!. Note that each of the ∂𝑧/ ∂𝑥N are non -negative for 𝑧 ∈ {𝑚" N, 𝑚! N, 𝑓N}, while Δ𝑥 0 if mutation 𝑥 targets the infection of susceptible cells ( 𝐵!). Consequently, each term in equation (S46) is positive, as observed in the main text. From our approximation of 𝑠2, the strength of selection (magnitude of 𝑠2), depends on two factors. The first factor is the direct impact of mutation 𝑥 on the relevant viral life -history quantity (i.e., the magnitude of O9 O23 Δ𝑥, for 𝑧 ∈ {𝑚", 𝑚!, 𝑓}. Since each mutation has a single eVect on viral life history, only one direct eVect will be non -zero. The second factor is how the direct eVect is weighted due to class -structure (e.g., the magnitude of (1/2)(1 − 𝑚a N/𝜃N)). These weights capture the distribution of mutations between the diVerent classes, as well as the reproductive value of each class. In the first phase of the infection, 𝑠2 is as given in equation (S4 5). In the second phase of infection, susceptible cells have been largely depleted, that is, 𝑆(𝑡) ≈ 0, and so 𝐵! N ≈ 0. Consequently, 𝜃N ≈ 𝑚" N − 𝑚! N (since 𝑚" N > 𝑚! N), and so equation (S46) reduces to 𝑠2 ≈ − 𝜕𝑚! 𝜕𝑥 Δ𝑥. (𝑆47) Thus, the value of virion mutations is zero, while the value of infected cell mutations is at its maximum. In the third phase of infection, eventually the concentration of virions and infected cells will be in a quasi-equilibrium state. Therefore, we have .CC-BY-NC 4.0 International licenseavailable under a (which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity. It is made The copyright holder for this preprintthis version posted September 4, 2025. ; https://doi.org/10.1101/2025.09.01.673077doi: bioRxiv preprint 34 0 ≈ 𝑝𝐼 − 𝑚"𝑉 and 0 ≈ 𝐵!𝑉 − 𝑚!𝐼 which implies 𝑉 𝐼 ≈ 𝑝 𝑚" ≈ 𝑚! 𝐵! or 𝑝𝐵! ≈ 𝑚"𝑚!. Consequently, θ ≈ 𝑚" + 𝑚!, and so equation (S46) reduces to 𝑠2 ≈ ]− 𝑚! N 𝑚! N + 𝑚" N 𝜕𝑚" N 𝜕𝑥N − 𝑚" N 𝑚! + 𝑚" N 𝜕𝑚! N 𝜕𝑥N + 𝑚" N𝑚! N 𝑚! N + 𝑚" N 𝜕𝑓2 N 𝜕𝑥N ^ Δ𝑥 + 𝒪(Δ𝑥7). (S48) Because 𝑚" N > 𝑚! N, the value of mutations targeting infected cell evasion of the immune response is higher than the value of mutations targeting virion evasion of the immune response. 4.3 Similar evolutionary dynamics are observed for di>erent immunodeficiencies In the Main Text, we observed that certain immunodeficiencies have qualitatively similar eVects on the strength of selection ; this is shown in Figure S3. Immunodeficiencies in neutrophils, monocytes, and/or the cytokines G -CSF and GM -CSF induce qualitatively similar evolutionary dynamic s (Fig. S3A -S3D). This is because each of these deficiencies trigger a reduction in neutrophil concentrations (Fig. S1). Similar ly, immunodeficiencies in IL-6 and macrophage overstimulate T cell concentrations and so induce qualitatively similar evolutionary dynamics during persistent infections (Fig. S3E-S3F). T cell deficiencies and IFN deficiencies, on the other hand, yield distinct evolutionary dynamics (Fig. S3G-S3H). -0.45 -0.15 0.15 Difference in time-averaged selection coefficient Neut def -0.45 -0.15 0.15 Mono def -0.45 -0.15 0.15 G-CSF def -0.45 -0.15 0.15 GM-CSF def 100 =T =R -0.05 -0.0125 0.025 Difference in time-averaged selection coefficient Mac def 100 =T =R -0.05 -0.0125 0.025 IL-6 def 100 =T =R -0.02 -0.01 0 T cell def 100 =T =R -0.15 -0.025 0.1 IFN def - 0FI /VN /VA /IT Inf cells Cubic A B C D E F HG .CC-BY-NC 4.0 International licenseavailable under a (which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity. It is made The copyright holder for this preprintthis version posted September 4, 2025. ; https://doi.org/10.1101/2025.09.01.673077doi: bioRxiv preprint 35 Figure S3. Qualitative similarities in evolutionary dynamics are determined by the type of immunodeficiency. Qualitatively similar viral evolutionary dynamics emerge for neutrophil, monocyte, G -CSF , and GM-CSF deficiencies (panels A -D, respectively; orange background), macrophage and IL -6 deficiencies (panels E and F , respectively; red background) while T cell (panel G; purple background) and IFN (panel H; blue background) deficiencies show distinct dynamics. In all panels, the time-averaged selection coeVicient was calculated in two ways: using formula (S36), with 𝑟22 and 𝑟23 calculated as the dominant eigenvalue of matrix (S40) (dashed lines), or by applying equation (S45) to simulation data (solid line). The divergence between these measures at the beginning of the infection arises due to two sources: (1) estimating the selection coeVicient from the simulation data only takes into account the frequency of mutations in infected cells, whereas the true selection coeVicient is a weighted average of infected cells and virions, and (2) at the beginning of the infection, there are only virions present and there is a delay owing to the eclipse phase before infected cells appear. The dashed vertical grey line in each panel indicates 12 hours. .CC-BY-NC 4.0 International licenseavailable under a (which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity. It is made The copyright holder for this preprintthis version posted September 4, 2025. ; https://doi.org/10.1101/2025.09.01.673077doi: bioRxiv preprint 36 4.4 E>ect of inoculum size on infection dynamics All infections initially consist of some concentration of virions, 𝑉(0) > 0, and no infected cells, 𝐼(0) = 0. However, we would expect some heterogeneity in the size of the initial inoculum across hosts. Heterogeneity in initial inoculum size aVects the duration of phases of infection as well as the strength of the immune response. A smaller inoculum size slows the growth of the initial infection, slowing the depletion of susceptible cells and so extending the first phase of infection ( Fig. S4). This will also tend to weaken the immune response across the first two phases of infection. The inoculum size has negligible consequences during the third phase of infection. Figure S4. Inoculum size influences the onset of infection phases. Dynamics of susceptible and refractory cells (panel A), infected cells and virions (panel B), macrophages and neutrophils (panel C), interferon (panel D), T cells (panel E), and antibodies (panel F) for initial viral inoculum size of 1, 4.5, and 9 log10(copies/mL). Shaded regions indicate the three infection phases for an inoculum of 4.5 log10(copies/mL), as in the results shown in the main text. Larger inoculum sizes shorten ( shift left) the first phase of infection due to higher viral loads and faster depletion of susceptible cells, while the reverse is true for smaller inoculum sizes. 0 0.5 1 1.5 2 Cells (cells/mL) #108 0 0.5 1 1.5 2 Cells (cells/mL) #108 Susceptible Refractory 0 0.5 1 1.5 2 Cells (cells/mL) #108 0 1 2 3 4 5 6 7 Virions (log10(copies/mL)) Infected Virions 0 2 4 6 8Cells (cells/mL) #105 0 2 4 6 8 Cells (cells/mL) #106 Macrophages Neutrophils 100 =T =R 102 Time (log10(days)) 0 50 100 150 200 250 Interferon (pg/mL) 5 V0 = 1 V0 = 4. V0 = 9 100 =T =R 102 Time (log10(days)) 0 2 4 6 8 10 12 14 T cells (cells/mL) #105 100 =T =R 102 Time (log10(days)) 0 0.5 1 1.5 2 2.5 3 Antibodies (AU/mL) #104 A D B E C F .CC-BY-NC 4.0 International licenseavailable under a (which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity. It is made The copyright holder for this preprintthis version posted September 4, 2025. ; https://doi.org/10.1101/2025.09.01.673077doi: bioRxiv preprint 37 4.5 Multiple mutations Next, we consider the evolution of multiple mutations. To do so, we suppose there are two mutations, 𝑥 and 𝑦, at diVerent loci aVecting diVerent aspects of the viral life history. This means there are four pathogen strains to consider, 𝑖 ∈ {𝑥N𝑦N, 𝑥,𝑦N, 𝑥N𝑦,, 𝑥,𝑦,}. When mutations at diVerent loci are simultaneously segregating in the population, in addition to selection on each individual mutation, we also need to account for epistasis in fitness between mutations. In continuous time models, if 𝑟4 is the per-capita growth rate of strain 𝑖, then epistasis in fitness is defined as 𝑠2P ≡ 𝑟22P2 + 𝑟23P3 − 𝑟23P2 − 𝑟22P3 . (S49) Thus, epistasis in fitness captures whether the fitness of a mutation depends on the genetic

or not. If epistasis in fitness is positive, strains carrying both mutations are fitter than would be expected, based upon the fitness contribution of each individual mutation, speeding their evolution. If epistasis in fitness is negative, strains carrying both mu tations are less fit than expected, slowing their evolution (in extreme cases, this can cause individually beneficial mutations to decrease in frequency). In addition, epistasis in fitness can produce linkage disequilibrium (LD) between mutations. The pres ence of LD, or the non -random associations between mutations, means that selection on mutation 𝑦, will cause changes in the frequency of mutation 𝑥, (so-called indirect selection). Because both mutations are individually beneficial, if epistasis is negative, then indirect selection will slow the evolution of both mutations, whereas if epistasis is positive, then indirect selection will speed the evolution of both mutations. Thus, in our model, indirect selection, and the direct consequences of epistasis work together to speed (positive epistasis) or slow (negative epistasis) evolution. 4.5.1 Approximating epistasis in fitness As before, per-capita growth rates are diVicult to calculate in class-structured populations. Therefore, to make analytic progress we apply the same approximation as before. We assume that immunological variables, and density of susceptible cells, change slowly relative to the change in infected cells and virions and that the eclipse phase is short. Then under weak selection (i.e., the diVerences 𝑥, − 𝑥N and 𝑦, − 𝑦N are small), a Taylor series expansion of 𝑠2P to leading order can be written 𝑠2P ≈ 𝑝𝐵! N (θN )J 4]𝜕𝑚" N 𝜕𝑥 − 𝜕𝑚! N 𝜕𝑥 ^ ]2 𝜕𝑚" N 𝜕𝑦 + 𝑚a N 𝜕𝑓P 𝜕𝑦 ^ + ∂𝑚! N ∂𝑥 ∂𝑚! N ∂𝑦 : |𝛥𝑥||𝛥𝑦| + 𝑝𝐵! N (θN)J (2𝑝𝐵! N + (𝑚aN)7) 𝜕𝑓2 𝜕𝑥 𝜕𝑓P 𝜕𝑦 |𝛥𝑥||𝛥𝑦| , (S50) .CC-BY-NC 4.0 International licenseavailable under a (which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity. It is made The copyright holder for this preprintthis version posted September 4, 2025. ; https://doi.org/10.1101/2025.09.01.673077doi: bioRxiv preprint 38 where 𝜕𝑓9/𝜕𝑧 = 1 if the mutation 𝑧 aVects transmissibility, 𝛽, and 𝜕𝑓9/𝜕𝑧 = 𝐹./7𝜖-,! + 𝐹. 9 if the mutation 𝑧 aVects evasion of IFN. As each mutation has a single direct eVect, only one of the 𝜕𝑞/𝜕𝑥 and one of the 𝜕𝑞/𝜕𝑦 are non-zero for 𝑞 ∈ {𝑚", 𝑚!, 𝑓}. 4.5.2 The sign of epistasis in fitness In our model, epistasis in fitness comes from two sources. First, if the two mutations target infection of susceptible cells (increased 𝛽) and interferon evasion (increased ϵ-,!), these two terms multiplicatively aVect the production of infected cells. This produces a positive epistatic interaction aVecting 𝐵! that will ultimately translate to positive epistasis in fitness. Second, class structure means that even in the absence of epistatic interactions aVecting one component of viral life history, epistasis in fitness tends to be generated48. Recall that we previously identified two groups of mutations based on their direct fitness eVects. In the first group are virion mutations. These mutations target virion infection and entry into cells (i.e., increased 𝛽, ϵ-,!) or target virion evasion of the immune response (i.e., decreased δ",&, δ",(, δ",)). In the second group are infected cell mutations. These mutations target infected cell evasion of the immune response (i.e., decreased δ!,%, δ!,(, δ!,)). Our approximation of 𝑠2P makes two predictions concerning these groups. If both mutations belong to the same group, epistasis in fitness is positive. If one mutation belongs to one group, and the other mutation belongs to the other group, epistasis in fitness is negative. Negative epistasis is weak in our model. This occurs because the per-capita decay rate of virions is higher than the per-capita decay rate of infected cells. Consider the case in which 𝑚" 4 ≫ 𝑚! 4 . Then we can approximate the density of virions of strain 𝑖 as 𝑉4 ≈ 𝑝 𝑚" 4 𝐼4. (S51) Using this relationship, we have 𝑑𝐼4 𝑑𝑡 ≈ ]𝐵! 4𝑝 𝑚" 4 − 𝑚! 4 ^ 𝐼4, (S52) and so 𝑟4 ≈ 𝐵! 4𝑝 𝑚" 4 − 𝑚! 4 . (S53) If we use this per-capita growth rate in the formula for epistasis given in equation (S4 9), we can see that negative epistasis is not possible between pairs of beneficial mutations. .CC-BY-NC 4.0 International licenseavailable under a (which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity. It is made The copyright holder for this preprintthis version posted September 4, 2025. ; https://doi.org/10.1101/2025.09.01.673077doi: bioRxiv preprint 39 Parameter Units Description Value Viral kinetic parameters 𝑝 1/day ×log(cop/ml) /109cells Lytic viral production 2.59 κ4 1/day Proliferation of epithelial cells 0.7397 𝑆567 109cells Epithelial cells carrying capacity 𝑆0 k'- log(cop/ml)/day Production of alveolar macrophages 5943 𝑀-567 109cells/ml Alveolar macrophage carrying capacity 𝑀-8,0 𝛽 1/day× 1/ log(cop/ml) SARS-CoV-2 virus infection rate 0.30 𝜏. days Eclipse time 0.17 𝜏9 days Delay in CD8+ T cell arrival 4.5 τ8 days Delay in refractory cell re-entry 8 τ: days Delay in antibody production 5 Cell production, recruitment, and activation rates 𝑝',-,; 1/day Monocyte to macrophage diPerentiation by GM-CSF 1.68 𝑝'%.,< 1/day Monocyte to macrophage diPerentiation by IL-6 1.68 𝑎.,'- ml/(109cells)× (1/day) Activation of macs by infected and dead cells 1.1 × 10= 𝑝',. 1/day Monocyte recruitment rate by infected cells 0.22 𝑝9,> 1/day CD8+ T cell production rate by IFN 4 𝑝?,< 1/day Neutrophils recruitment rate by IL-6 0.3 𝑝9,. 1/day CD8+ T cell proliferation rate 0.02 𝑝:,) 1/day Antibody production by virus 500 𝑀@ABC ∗ 1/day Homeostasis reservoir release rate 0.13 𝜓' 567 1/day Maximal reservoir release rate 11.55 𝑁@ABC ∗ 1/day Homeostasis reservoir release rate 0.21 𝜓? 567 1/day Maximal reservoir release rate 4.13 𝐶E> ∗ Dimensionless Homeostasis neutrophil receptor bound fraction 1.6 × 10FG Cell-related half-e8ect (𝜖), IC50 (𝐼𝐶G0), and Hill coe8icient (ℎ) parameters 𝜖>,. pg/ml IFN inhibition of viral production 2 × 10FH 𝜖,9 pg/ml IFN production of CD8+ T cells 399.3 𝜖I,? unitless G-CSF recruitment of neutrophils 1.89× 10FH 𝜖<,? pg/ml IL-6 recruitment of neutrophils 57.2 𝜖.,' 109cells/ml Infected cell monocyte recruitment 0.05 𝜖<,9 pg/ml IL-6 production of CD8+ T cells 1.5× 10FG 𝜖),'- log(cop/ml) Viral load for mac replenishing 905.22 ϵJ,9 109 cells/ml Killing of infected cells by CD8+ T cells 0.01 ϵ),: AU/ml Antibody neutralization 1000 ℎ' Dimensionless GM-CSF monocyte recruitment 1.67 ℎ','- Dimensionless GM-CSF monocyte to macrophages 2.03 ℎ? Dimensionless Neutrophil induced damage 3.02 ℎ: Dimensionless Antibody neutralization 1.19 ℎ9 Dimensionless Killing of infected cells by CD8+ T cells 0.5 𝐼𝐶G0,? 109 cells/ml Neutrophil induced damage 4.7 x 10-5 .CC-BY-NC 4.0 International licenseavailable under a (which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity. It is made The copyright holder for this preprintthis version posted September 4, 2025. ; https://doi.org/10.1101/2025.09.01.673077doi: bioRxiv preprint 40 Parameter Units Description Value Cell/virus-induced death rates 𝛿),'- ml/(109cells)× 1/day Rate of viral clearance by macrophages 768 𝛿),? ml/(109cells)× 1/day Rate of viral clearance by neutrophils 1152 𝛿? 1/day Rate of neutrophil inflicted damage 1.68 𝜌 Dimensionless Bystander death modulation constant 0.5 𝛿.,'- ml/(109cells)× (1/day) Rate macrophages phagocytose infected cells 121.20 𝛿.,9 ml/(109cells)× (1/day) Rate CD8+ T cells induce apoptosis in infected cells 7 𝛿'K,L ml/(109cells)× (1/day) Rate macrophages die from phagocytosis 6.06 𝛿L,'- ml/(109cells)× (1/day) Rate macrophages phagocytose dead cells 8.03 δ8 1/day Rate of refractory cell reconversion 0 (Immunocompetent) 0.05 (Immunodeficiency) δ),: log(copies/ml) x (1/day) Rate of antibody neutralization 5 Cell death and decay rates 𝑑) 1/day Viral decay rate 1.81 𝑑. 1/day Infected cell death rate 0.1 𝑑L 1/day Degradation rate of apoptosed cells 8 𝑑'%& 1/day Alveolar macrophage death rate 0.01 𝑑'%- 1/day Inflammatory macrophage death rate 0.3 𝑑' 1/day Monocyte death rate 0.76 𝑑? 1/day Neutrophil death rate 0.4 𝑑9 1/day CD8+ T cell death rate 0.4 𝑑: 1/day Antibody clearance rate 0.033 Cytokine production rates 𝑝<,. pg/ml/day IL-6 production by infected cells 11.89 𝑝<,'%- pg/ml/day IL-6 production by inflammatory macrophages 1872 𝑝<,' pg/ml/day IL-6 production by monocytes 72.56 𝑝;,'%- pg/ml/day GM-CSF production by inflammatory macrophages 2626 𝑝I,' ng/ml/day G-CSF production by monocytes 26.26 𝑝;,' pg/ml/day GM-CSF production by monocytes cells 123.4 𝑝>,. pg/ml/day IFN production by infected cells 2.82 𝑝>,',- pg/ml/day IFN production by inflammatory macrophages 1.3 𝑝>,' pg/ml/day IFN production by monocytes 3.56 Half-e8ect parameters 𝜂<,. 109 cells/ml IL-6 production by infected cells 0.7 𝜂<,' 109 cells/ml IL-6 production by monocytes 0.0045 𝜂<,'%. 109 cells/ml IL-6 production by inflammatory macrophages 3.6× 10FG 𝜂;,'- 109 cells/ml GM-CSF production by macrophages 3.6× 10FG 𝜂;,' 109 cells/ml GM-CSF production by monocytes 0.15 𝜂I,' 109 cells/ml G-CSF production by monocytes 3.05 𝜂>,. 109 cells/ml IFN production by infected cells 0.011 .CC-BY-NC 4.0 International licenseavailable under a (which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity. It is made The copyright holder for this preprintthis version posted September 4, 2025. ; https://doi.org/10.1101/2025.09.01.673077doi: bioRxiv preprint 41 Parameter Units Description Value 𝜂>,',- 109 cells/ml IFN production by inflammatory macrophages 1× 10FG 𝜂>,' 109cells/ml IFN production by monocytes 0.54 Cytokine clearance and internalization rates 𝑘MNO/ 1/day Rate of IL-6 renal clearance 16.6 𝑘MNO0 1/day Rate of GM-CSF renal clearance 11.7 𝑘MNO1 1/day Rate of G-CSF renal clearance 0.16 𝑘MNO2 1/day Rate of IFN renal clearance 18 𝑘NO/ 1/day Internalization rate of IL-6 61.8 𝑘NO0 1/day Internalization rate of GM-CSF 73.4 𝑘NO1 1/day Internalization rate of G-CSF 462 𝑘NO2 1/day Internalization rate of IFN 17 Cytokine binding/unbinding rates and stoichiometric constant 𝑘E/ ml/pg/day IL-6 binding rate 0.0018 𝑘E0 ml/pg/day GM-CSF binding rate 0.0021 𝑘E1 ml/ng/day G-CSF binding rate 2.24 𝑘E2 ml/pg/day IFN binding rate 0.011 𝑘P/ 1/day IL-6 unbinding rate 22.3 𝑘P0 1/day GM-CSF unbinding rate 522 𝑘P1 1/day G-CSF unbinding rate 184 𝑘P2 1/day IFN unbinding rate 6.07 𝑃𝑂𝑊 Dimensionless Stoichiometric constant (G-CSF) 1.4608 Stoichiometric constant (IL-6, GM-CSF, IFN) 1 𝑝̂ Dimensionless Stoichiometry relating constant (G-CSF) 1.46 Stoichiometry relating constant (IL-6, GM-CSF, IFN) 1 Number of cellular receptors and cytokine molecular weights 𝐾<,? sites/cell No. IL-6 receptors on neutrophils 720 𝐾<,9 sites/cell No. IL-6 receptors on T cells 300 𝐾,9 sites/cell No. of IFN receptors on T cells 1000 𝐾>,. sites/cell No. of IFN receptors on infected cells 1300 𝑀𝑀 g/mol Molecular weight of IFN-𝛽 19000 Initial conditions 𝑉0 log(copies/ml) Initial total viral load 4.5 𝑆0 109 cells/ml Initial susceptible cells 0.16 𝐼0 109 cells/ml Initial infected cells 0 𝑅0 109 cells/ml Initial refractory cells 0 𝑀-8,0 109 cells/ml Initial resident macrophages 2.73 × 10FG 𝑀-.,0 109 cells/ml Initial inflammatory macrophages 2.9 × 10FQ 𝑀0 109 cells/ml Initial monocytes 0.0004 𝑀8 ∗ 109 cells/ml Concentration of reservoir monocytes 0.0023 𝑁0 109 cells/ml Initial neutrophils 0.0053 𝑁8 ∗ 109 cells/ml Concentration of reservoir neutrophils 0.0316 .CC-BY-NC 4.0 International licenseavailable under a (which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity. It is made The copyright holder for this preprintthis version posted September 4, 2025. ; https://doi.org/10.1101/2025.09.01.673077doi: bioRxiv preprint 42 Parameter Units Description Value 𝑇0 109 cells/ml Initial CD8+ T cells 1.1 × 10FH 𝐿R,0 pg/ml Initial unbound IL-6 1.1 𝐿S,0 pg/ml Initial bound IL-6 0 𝐺R,0 pg/ml Initial unbound GM-CSF 2.43 𝐺S,0 pg/ml Initial bound GM-CSF 1.6× 10FT 𝐶R,0 ng/ml Initial unbound G-CSF 0.025 𝐶S,0 ng/ml Initial bound G-CSF 6.5× 10FU0 𝐹R,0 pg/ml Initial unbound IFN 0.015 𝐹S,0 pg/ml Initial bound IFN 1.1× 10FT 𝐴0 AU/ml Initial antibodies 0 Table S1. Parameter values for the immunological model. .CC-BY-NC 4.0 International licenseavailable under a (which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity. It is made The copyright holder for this preprintthis version posted September 4, 2025. ; https://doi.org/10.1101/2025.09.01.673077doi: bioRxiv preprint 43

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