Mathematical model of the immune response to primary infection

Our model describes the immune dynamics following a primary infection by a respiratory virus. In what follows, we highlight the key aspects of the model; complete details are provided in S1 Text, including information on parameter estimation. A model schematic is presented in Fig 1.

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Fig 1. Schematic of immune response to viral infection.

Virions, , infect susceptible cells, , producing infected cells, . Following the eclipse phase, infected cells lyse, producing virions. Cells refractory to infection, , are created through upregulation of interferon (IFN) signalling by infected cells. Inflammatory macrophages, , are converted from monocytes, , or tissue-resident macrophages, , and along with neutrophils, , phagocytose virions and infected cells. Phagocytosis of infected cells produces dead and damaged cells, . Antibodies, , are mobilized by the presence of viral antigens, and neutralize virions. CD8 + effector T cells, , 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 differentiation. Double arrows: induced death. Pink arrows: decreased effect due to immunodeficiency. Yellow arrows: immune effect targeted by mutations.

https://doi.org/10.1371/journal.pcbi.1013967.g001

A primary infection occurs when virions, , infect susceptible cells, to create infected cells, , via mass-action at rate . The parameter thus captures the process of viral engulfment and cellular entry by endocytosis [22–25]. Infected cells then enter an eclipse phase lasting days during which they are non-productively infected. Following the eclipse phase, infected cells are productively infected and die at per-capita rate , releasing new virions (burst size) upon lysis. Virions naturally decay at a per-capita rate .

Primary infection stimulates the innate, adaptive, and humoral immune responses. The innate immune response consists of three parts. First, infected cells secrete Type I interferon (IFN) at a saturable rate with a maximal secretion rate of . Secretion of IFN makes infected cells refractory to infection while also blocking viral entry and replication in neighbouring cells. Second, neutrophils, , clear virions at per-capita rate and kill infected cells at per-capita rate . Neutrophils are recruited to the site of infection through interleukin-6 (IL-6) and granulocyte colony-stimulator factor (G-CSF) cytokine signalling. Third, inflammatory macrophages, , destroy virions and infected cells through phagocytosis [26] at per-capita rates and , respectively. Inflammatory macrophages are created from monocytes, , through IL-6 and granulocyte macrophage colony-stimulator factor (GM-CSF) signalling, or from tissue-resident macrophages, , after contact with infected or dead, , cells.

The adaptive immune response consists of CD8 + effector T cells, , and is activated days following primary infection. CD8 + T cells are recruited by infected cells and IFN signalling and suppressed by IL-6, which serves as a proxy for anti-inflammatory signalling [27]. CD8 + effector T cells kill infected cells through density-dependent saturable killing [28] at a maximal rate of .

The humoral immune response consists of antibodies, , which are generated by the presence of virions after days. Following activation, antibodies remove virions through saturable neutralization of virions. We assume all antibodies are neutralizing, with a maximal neutralization rate of , half-effect concentration , and Hill coefficient .

Under these assumptions, the dynamics of the virions and infected cells are given by:

(1)

(2)

where is the concentration of bound IFN at time . In equations (1) and (2), and denote the per-capita loss of virions and infected cells, respectively, and denotes the per-capita production of infected cells by virions.

Modelling immunodeficiencies

Immunocompromised hosts may experience protracted “acute” (i.e., persistent) infection leading to a loss of IFN effects on refractory cells [29–32], as well as a deficient innate, adaptive, and/or humoral immune response [33]. To capture the loss of IFN effects on refractory cells, we suppose they revert to susceptibility at per capita rate after days (see S1 Text Section 1). To capture immunodeficiencies, we reduce the rate of production, and hence concentration, of different state variables of the immune response. For example, a monocyte production deficiency translates to a lower concentration of monocytes at homeostasis and throughout the course of the 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 for all model variables at the specified level of production deficiency prior to infection (see S1 Text Section 3 for more details).

Previous studies have shown immunodeficiencies lead to aberrant production and/or generation rates of different aspects of the immune response [34], with cascading effects to other cell types [35]. We therefore assume immunodeficiencies reduce the target variable by at least 50% compared to healthy hosts. To mimic clinically observed T cell immunodeficiencies, such as advanced HIV disease defined by CD4 + T cell counts <200 cells/ [36], 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 [37], including B cell lymphomas [8], antibody immunodeficiencies are assumed to reduce antibody production by 90%. For all other state variables, to achieve at least a 50% reduction in state variable concentration, we reduce production rates by 35% (S1 Text Section 3). In all cases, varying the strength of the immunodeficiency quantitatively affects the reduction in the target variable and the magnitude of concomitant changes in the non-target variables, but does not qualitatively affect our results (Fig A in S1 Text).

Modelling viral evolution

As our interest is how immunodeficiencies affect the strength of selection on viral evasion of the immune response, we ignore de novo mutations and assume that any 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). We refer to the immunological variable directly targeted by the viral mutation as the “target” and the other immunological variables as “nontargets”. Each viral mutation is beneficial, and for simplicity, cost-free for the virus (the addition of costs is a straightforward extension).

We considered mutations to be divided into two groups based on the life history stage 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 ), macrophages (decreased ), or antibodies (decreased ), or target virion infection of susceptible cells, by increasing virion host cell entry [38,39] (increased ), or by decreasing the ability of interferon to block viral entry [40,41] (increased ). 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 ), macrophage (decreased ), or T cells (decreased ).

The speed of within-host evolution of individual mutations

To study how the immunological environment determines the speed of evolution of a single segregating mutation, note that the frequency of mutation within the viral population, , changes according to the equation

(3)

where is the time-varying selection coefficient. Thus, the strength of selection on mutation at time is determined by the magnitude of , which dictates the instantaneous rate of increase of the frequency of a mutation due to selection. The time-averaged strength of selection on mutation from time to time is given by

(4)

Equation (4) measures how strong constant selection would have to be to yield the observed change in mutation frequency by time . Thus, and capture how the immune response determines the short- and long-term strength of selection, respectively, dictating the speed of adaptation. Because the strength of selection depends on the size of the mutational effect, in our figures we will normalize the selection coefficients and the time-averaged strength of selection by their maximum values over the duration of infection (e.g., ).

Calculating selection coefficients in populations with different life history stages (e.g., virions and infected cells) is not trivial, particularly for populations with temporally varying per-capita growth rates [42–45]. Therefore, we first approximate from simulation data as

(5)

where is the frequency of mutation at time in infected cells and is some (short) increment of time (e.g., hours, days); can be approximated using equation (5) by setting . Although equation (5) only considers the frequency of the mutation in infected cells, simulation results yield similar predictions if equation (5) is calculated using the frequency of the mutation in virions (Fig B in S1 Text). The principal area of divergence between the different approximations of selection occurs at the outset of the infection: because each infection starts with only virions present, the delay in production of infected cells due to the eclipse phase means equation (5) cannot be accurately calculated during the initial few hours of the infection (Fig B in S1 Text). Therefore, in all simulation results we plot equation (5) from hour 12 onwards.

Second, if we suppose mutation is of weak phenotypic effect 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. [42]), we can analytically approximate as

(6)

where is the (small) phenotypic difference between mutant and wildtype, , , is evaluated when , and if and if (Sup. Info. Section 4). Equation (6) reveals that selection on each mutation is the direct effect of mutation on the viral fitness component , captured by and weighted by the frequency of the mutation in the different 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 value [42]). Since each mutation is beneficial and has a single effect on virion life history, only one of the direct effects in equation (6) will be nonzero and positive.

As the per-capita rate of virion destruction is expected to be higher than that of infected cells (i.e., ), a decrease in and/or decreases the value of virion mutations and increases the value of infected cell mutations, whereas a decrease in has the opposite effect. 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 the infection grows rapidly while the innate response (neutrophils, IFN, and macrophage) mobilizes (Fig 2A and 2B). As susceptible cells are abundant, the ratios and 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).

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Fig 2. Immune dynamics distinguish the phases of infection.

A-C) In immunocompetent hosts, the infection can be divided into two phases. In the first phase (white shaded backgrounds), susceptible cells are abundant while the immune response is mobilizing. In the second phase (light grey shaded backgrounds), susceptible cells are depleted, and the immune response eliminates the infection. D-F) The reversion to susceptibility of refractory cells () leads to a third infection phase (dark grey shaded backgrounds) in immunocompromised hosts. This timescale is consistent with observations of persistent infections lasting months [8,9,47]. G-H) In both immunocompetent and immunocompromised hosts, the abundance of susceptible cells during the first phase of infection means the adaptive value of virion mutations (i.e., mutations affecting cell entry , as well as virion evasion of IFN, , neutrophils, , macrophage, , and antibodies, ) is maximized, while the value of infected cell mutations (i.e., mutations affecting infected cell evasion of neutrophils, , macrophage, , and T cells, ) is minimized. At the same time, the mobilizing immune response (B and E) increases the direct effect of selection for immune evasion. During the second phase, the absence of susceptible cells means virion mutations are selectively neutral, while the adaptive value of infected cell mutations increases. This causes a rebound in the value of virion mutations, and a decrease in the value of infected cell mutations. The mobilizing adaptive and humoral response (C and F) selects for T cell and antibody evasion. D-F) No additional immunodeficiencies other than are considered. A-F) White background: first phase of infection. Light grey background: second phase of infection. Dark grey background: third phase of infection. Dashed vertical grey line: 12 hours post-infection (see Fig B in S1 Text).

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In the second phase of infection, susceptible cells are largely depleted, and the innate immune response is fully mobilized. Because susceptible cells are depleted, , and equation (6) reduces to

(7)

Thus, the value of virion mutations is zero and are selectively neutral, while the value of infected cell mutations is maximized (Fig 2G and 2H).

In immunocompetent hosts, and thus the depletion of susceptible cells combined with the mobilized immune response clears the infection (Fig 2A-2C). However, in immunocompromised hosts,  > 0, and the loss of IFN signalling and reversion to susceptibility of refractory cells creates a third phase of infection shortly after . In this phase, there is a temporary abundance of susceptible cells as the first wave of refractory cells revert to susceptibility (Fig 2D-2F). The dynamics then settle down to a quasi-equilibrium (akin to the viral setpoint [46]) where infection size is limited by susceptible cell availability and the mobilized immune response (Fig 2D-2F; dark grey shaded region). Since in the long term, the rebound in susceptible cells in immunocompromised hosts increases the value of virion mutations and tends to decrease the value of infected cell mutations (Fig 2G-2H). Moreover, the mobilization of the adaptive and humoral responses (Fig 2C and 2F) generates a direct effect on selection and increases the strength of selection for T cell and antibody evasion mutations from the second to third phase of infection (Fig 2G and 2H).

The time of onset of each phase is affected by the size of the viral inoculum, . Smaller inoculums increase the amount of time required for the viral population to exhaust susceptible cells, thus prolonging the first phase of infection and delaying the onset of the second phase (Fig D in S1 Text). Although changes to the size of the viral inoculum affect the time of onset of each infection phase, they do not qualitatively change our key predictions, provided they are large enough to cause an infection (i.e., log(copies/mL; see Fig D in S1 Text and Jenner et al. [27]). Therefore, in what follows, we fix the initial inoculum at log₁₀(copies/mL) as in Jenner et al. [27]. For this inoculum size, the second and third phases of infection begin at approximately 4.5 and 9 days, respectively (Fig 2D and 2F).

Immunodeficiencies speed and slow the evolution of individual mutations

Next, we ask how different immunodeficiencies affect the speed of viral evolution. The most straightforward prediction occurs for target immunodeficiencies, i.e., a production deficiency in the immunological variable directly targeted by the viral mutation. The direct effect of mutation on viral fitness component is an increasing function of the concentration of the target variable. Thus, target variable immunodeficiencies reduce the strength of selection on mutation , slowing its evolution. This is intuitive; for example, if mutation targets antibody evasion and antibody concentrations are reduced, there are fewer antibodies to “evade” and so mutation will be less beneficial, weakening selection and, hence slowing the increase in frequency of mutation . However, the interconnectedness of the immune response means that non-target immunodeficiencies may not only feedback on the target variable. Off-target effects directly impact unrelated mutations and trigger changes in mutational value by, e.g., affecting the availability of susceptible cells. In what follows, we consider these effects across the different phases of infection.

Our model predicts that mutations targeting infected cell evasion of neutrophils (decreased ) and evasion of macrophage (decreased and ) are under very weak selection, regardless of the size of the mutational effect (Fig B in S1 Text). Thus, we will ignore these three mutations and restrict our attention to mutations affecting cell entry (increased ), virion evasion of interferon, neutrophils, and antibodies (increased , or decreased and , respectively), and infected cell evasion of T cells (decreased ).

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 innate response, consisting mainly of neutrophils, IFN, and macrophages, is largely responsible for infected cell and virion clearance (Fig 2B and 2E). Of the three, neutrophils are the most important for controlling the initial infection (Fig 3A). Neutrophils are recruited by G-CSF which is produced by monocytes generated through GM-CSF signalling (Fig 1). A deficiency in any component of this pathway either directly (i.e., a neutrophil deficiency) or indirectly (i.e., a G-CSF, GM-CSF, and/or monocyte deficiency) reduces neutrophil concentrations, sharply increasing infection size (Fig 3B and 3C). IFN deficiencies have a similar, but weaker effect. Since reduced neutrophil concentrations decrease the destruction of virions (decrease ), while reduced IFN concentrations lead to larger infection sizes (increase ), reductions in neutrophils and/or IFN increase the value of virion mutations. This increase tends to speed the evolution of non-target virion mutations targeting cell entry, neutrophil evasion, and IFN evasion during the first phase of infection (Figs 3D and 3E, see also Fig C i and viii in S1 Text).

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Fig 3. Viral evolution during acute infections is most strongly affected by neutrophil and IFN deficiencies.

A) Percent change in each immunological variable (columns) resulting from an immunodeficiency in the indicated immune cell, cytokine, or antibody (rows) during acute infections (phase one and two; ). The percent change in each immunological variable is calculated as , where and are the integrals of the variable over the length of the phase in immunodeficient and healthy hosts, respectively; the percent change in susceptible cells is calculated using the integral of . Green boxes indicate target variables. Note that NaN values reflect static T cell and antibody responses during the first phase of infection. B-C) Dynamics of susceptible and infected cells (B) and virions (C) during acute infection in healthy (solid lines), and immunocompromised hosts with IFN (dotted dashed lines) or neutrophil (dashed lines) deficiencies. D-E) Difference in time-averaged selection coefficient between IFN (D) and neutrophil (E) deficient and immunocompetent hosts, normalized by its maximum (absolute) value. During acute infections, 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 Fig C i and viii in S1 Text). For acute infections, only mutations affecting cell entry, , virion evasion of IFN, , and virion evasion of neutrophils, , experience significant selection. B-E) White background: first phase of infection. Light grey background: second phase of infection. Dashed vertical grey line: 12 hours post-infection.

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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, triggering an earlier onset of the second phase of infection. Since virion mutations are neutral during this phase (see equation (6)), the normalized time-averaged selection coefficient is larger in immunocompetent hosts by the end of the second phase of infection (Fig 3D and 3E). Thus, if refractory cells do not revert to susceptibility (i.e., there is no third phase of infection), mutations are at a higher frequency in immunocompetent hosts by the end of the infection.

T cell concentrations play a pivotal role in speeding or slowing viral evolution during phase three.

If refractory cells do not revert to susceptibility (i.e.,  = 0), 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 by setting in both immunodeficient and immunocompetent hosts.

During the third phase of infection, T cells play an important role in infected cell killing, thus reducing the infection size and increasing the availability of susceptible cells. Accordingly, the value of virion and infected cell mutations increases and decreases, respectively. T cell production is suppressed by bound IL-6 concentrations [48,49] generated through the stimulation of unbound IL-6 by neutrophils and monocytes, with unbound IL-6 generated by infected cells, monocytes, and macrophages (Fig 1; see also Section 1 in S1 Text). 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. Decreased bound IL-6 concentrations increase 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 effects on bound IL-6 or T cell concentrations (Fig 4A); this agrees with local sensitivity analysis results which show that only antibody concentrations are sensitive to small variations in antibody related parameters (Fig F in S1 Text).

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Fig 4. Immunodeficiencies affecting T cell concentrations strongly affect viral evolution during persistent infections.

A) Percent change in each immunological variable (columns) resulting from an immunodeficiency in the indicated immune cell, cytokine, or antibody (rows) during persistent infections (i.e.,  > 0). The percent change in each immunological variable is calculated as , where and are the integrals of the variable over the length of the phase in immunodeficient and healthy hosts, respectively; the percent change in susceptible cells is calculated using the integral of . Green boxes indicate target variables. Note that NaN values reflect static T cell and antibody responses during the first phase of infection. B-C) Dynamics of susceptible and infected cells (B), and neutrophils, IFN, and T cells (C) during acute infection in healthy (solid lines), and immunocompromised hosts with IL-6 (dashed lines), monocyte (heavy dotted lines), and T cell (light dotted lines) deficiencies. D-F) Difference in time-averaged selection coefficient between T cell (D), IL-6 (E), and monocyte (F) deficient and immunocompetent hosts, normalized by its maximum (absolute) value. T cell killing of infected cells is a key determinant of the size of the infection and availability of susceptible cells during persistent infections. D) 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. E) A deficiency in IL-6 (either directly, or due to a macrophage deficiency; Fig C vi in S1 Text) will increase T cell concentrations and so speed the evolution of all mutations considered. F) 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 Fig C ii-iv in in S1 Text), will increase T cell concentrations, but will also reduce neutrophils and IFN. B-F) White background: first phase of infection. Light grey background: second phase of infection. Dark grey background: third phase of infection. Dashed vertical grey line: 12 hours post-infection.

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Because a T cell deficiency both reduces the direct effect of T cell evasion and 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 4D). IL-6 and macrophage deficiencies have the opposite effect, as they increase T cell concentrations, accelerating the rate of increase in frequency of each of the five mutations considered (Fig 4E). As lower neutrophil concentrations increase infection size, T cell counts are also increased by lower neutrophil concentrations arising through deficiencies in monocytes, G-CSF, GM-CSF, and/or neutrophils, which also reduce IFN (Fig 4A). Thus, these deficiencies 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 4F).

Speed of evolution of multiple mutations

Finally, we ask how the speed at which individual mutations increase in frequency is affected 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 affects the speed of evolution by altering the fitness of a mutation depending on its genetic background and produces linkage disequilibrium (LD), i.e., non-random assortment, between mutations [50]. The amount of LD in deterministic models tends to scale with the strength of epistasis. For beneficial mutations, such as those studied here, these combined effects mean positive epistasis speeds evolution, whereas negative epistasis slows evolution [50–52] (S1 Text Section 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 (S1 Text Section 4.5). From this approximation, two predictions emerge. First, if both mutations directly affect the fitness of virions (i.e., , , , or ) or both mutations directly affect the fitness of infected cells, epistasis is positive, speeding their joint evolution (S1 Text Section 4.5). If instead, one mutation directly affects virion fitness, whereas the other mutation directly affects the fitness of infected cells (e.g., evasion of T cells), epistasis is negative, slowing their joint evolution (S1 Text Section 4.5).

Simulation results suggest that positive epistasis between virion mutations can dramatically speed evolution by enhancing fitness and by the generation of large amounts of LD, however 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 affect the rate of infection of new cells, which will directly produce epistasis [53]. 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, . As this difference grows, negative epistasis between virion and infected cell mutations becomes increasingly weak (S1 Text Section 4.5). Consequently, as our model parameterization assumes virion turnover is relatively rapid, negative epistasis is comparatively weak.

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Fig 5. The sign of epistasis and the speed of viral evolution vary based upon the combination of mutations considered.

A) Difference in time-averaged selection coefficients between immunocompromised and immunocompetent hosts during persistent infection with multiple mutations (i.e., + and +) compared to persistent infection with single mutations (i.e., , , or alone). The difference in the time-averaged selection coefficients is normalized by their respective maximum (absolute) value. B) Linkage disequilibrium between + (solid yellow line) and + (dashed brown line). A-B) When different mutations target immune evasion by the same life history stage (solid lines), 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. Conversely, when one mutation targets T cell evasion, while the other targets virion evasion of the immune response (dashed lines) negative epistasis occurs. In contrast to positive epistasis, negative epistasis in our model is weak, and so is incapable of generating much linkage disequilibrium. That negative epistasis is weak arises due to the rapid turnover of virions relative to infected cells. White background: first phase of infection. Light grey background: second phase of infection. Dark grey background: third phase of infection.

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