Predicting the emergence of new pathogenic strains is a key goal of evolutionary epidemiology. However, the majority of existing studies have focussed on emergence at the population level, and not within a host. In particular, the coexistence of pre-existing and mutated strains triggers a heightened immune response due to the larger total pathogen population; this feedback can smother mutated strains before they reach an ample size and establish. Here, we extend previous work for measuring emergence probabilities in non-equilibrium populations, to within-host models of acute infections. We create a mathematical model to investigate the emergence probability of a fitter strain if it mutates from a self-limiting strain that is guaranteed to go extinct in the long-term. We show that ongoing immune cell proliferation during the initial stages of infection causes a drastic reduction in the probability of emergence of mutated strains; we further outline how this effect can be accurately measured. Further analysis of the model shows that, in the short-term, mutant strains that enlarge their replication rate due to evolving an increased growth rate are more favoured than strains that suffer a lower immune-mediated death rate (‘immune tolerance’), as the latter does not completely evade ongoing immune proliferation due to inter-parasitic competition. We end by discussing the model in relation to within-host evolution of human pathogens (including HIV, hepatitis C virus, and cancer), and how ongoing immune growth can affect their evolutionary dynamics.
Parasites and pathogens pose a continuous threat to human, livestock, and plant health since new strains can readily emerge, via mutation or recombination, from pre-existing strains. Generally, the focus has been on detection of emerging diseases at the population level, in order to track and control their spread [1,2]. Modelling approaches to predicting emergence have therefore primarily concentrated on detecting infections arising between individual hosts [3,4], and the contribution of within-host processes to pathogen emergence has often been overlooked. It is now well known that within-host evolution has strong effects on the epidemiology of many pathogens (reviewed in [5]), and can substantially affect the course of an infection, as illustrated by the cases of HIV [6] and hepatits C virus (HCV) [7]. Each step of the within-host evolutionary dynamics consists of the emergence of a rare mutant that takes over the pathogen population. Since mutated infections always initially appear as a few copies, they are prone to extinction so their emergence is best captured using stochastic dynamics (as opposed to deterministic approaches). Only a handful of previous models have investigated this stochastic within-host process. A widespread use of within-host models in static populations is to calculate the probability that infections will evolve drug resistance, and what regime is needed to avoid treatment failure [8–11]. One exception [12] studied viral emergence within a host, if most mutations were deleterious, in order to determine how different viral replication mechanisms affected the establishment of beneficial alleles. Parasite and pathogen evolution can radically affect the course of infections in hosts able to mount immune responses. For instance, HIV is known to successfully fix mutations within individuals which enable it to evade immune pressures [6]. This is in line with evidence that target cell limitation cannot account for HIV dynamics, and that immune limitation also needs to be present [13]. Arguably, this continual evasion of immunity prompts the chronic nature of HIV infections (see [14,15] for an illustration of these ‘Red Queen’ dynamics). In the case of HCV, it has also been found that chronic infections were associated with higher rates of within-host evolution [16]. Concerning a different chronic disease, it is now well-known that the ability of malignant cells to rapidly expand in size, ignoring biological signals to arrest growth (e. g. through mutating the p53 tumour suppression gene), and escaping the patient’s immune system are key steps in cancer development [17]. All these scenarios can be analysed in the larger framework of evolutionary rescue [18,19], where a change in the environment (in this case, the activation of an immune response) will cause the population to go extinct unless it evolves (develops an increased replicative ability, then subsequently rise to a large enough size to avoid stochastic loss). Although chronic infections are those most likely to evolve strains that evade immune pressures, evidence exists that such immune escape also plays a role in acute infections, like influenza [20]. Note that the latter evidence arises from serial passage experiments; although these tend to maximise selection at the within-host level, they still include a slight selective pressure for increased transmission. Modelling mutant emergence in an immune evasion context raises a technical challenge, as it is a non-equilibrium process. In most of the within-host models presented above [8–10,12], it is sufficient to know the initial state of the system to calculate emergence probability of immune or treatment escape. A non-equilibrium model was studied by Alexander and Bonhoeffer [11], who accounted for the reduction in available target cells when determining the emergence of drug resistance using an evolutionary rescue framework. With immune-escape however, the problem is that the immune state when the mutant infection appears is only temporary. In this case, there will be an initial strain present within the host, which triggers immunity. This strain can then mutate into a faster-replicating form, but if the mutated strain arises at a low frequency, immunity can destroy it before it has a chance to spread. This effect can be exacerbated by the fact that the mutated strain also prompts an increased immune response, so the emerging infection has a stronger defence to initially compete with (assuming immune growth is proportional to the total size of the pathogen population). This feedback, where increased immune growth prevents emergence of mutated strains with higher replication rates, can strongly affect the appearance of mutated strains within-host, and needs to be accounted for. Existing emergence models have not yet accounted for such within-host population feedbacks, especially those arising from the immune system. Recently, Hartfield and Alizon [21] tackled a related problem, regarding how epidemiological feedbacks affect disease emergence at the host population level. In their model, a faster-replicating strain emerged via mutation from a pre-existing infection; however, the continuing outbreak caused by the initial strain removed susceptible individuals from the population, which limited the initial spread of the mutated strain. It was shown that the ongoing depletion of susceptible individuals due to the initial strain spreading has a stronger effect on reducing pathogen emergence, than assumed by just scaling down the reproductive ratio by the frequency of susceptible individuals present when the mutated infection appears. That is, the feedback produced by the first strain in removing susceptible individuals caused a drastic decrease in the emergence probability of the mutated strain. Building on this previous study, we derive here an analytical approximation for the probability that an immune-escape mutant will emerge and maintain itself within a host. We use ‘immune-escape’ in the sense that while the mutated strain can be killed by immunity, it can ultimately outgrow immune growth and chronically persist. Furthermore, we use an acute infection setting, which are commonly used to study the within-host dynamics of ‘flu-like’ diseases [22–28]. Analysis of the model demonstrates how the ongoing proliferation of immune cells acts to decrease the emergence probability of mutated strains. Acute infection models are also useful in studying the first stages of chronic infections, where one observes exponential growth of the virus population, followed by a decline in the first weeks of infection. In addition, there are two questions that are worth investigating with this model from a biological standpoint. First, what is the fittest evolutionary strategy for an escape mutant: is it to overgrow the immune response (that is, increase its inherent replication rate to enable its persistence, even when immune cells are at capacity), or to tolerate it (prevent immunity from killing as many pathogens per immune cell)? Both these actions will lead to an increase in the pathogen’s net reproduction rate and can thus be described as immune evasion, but it is unclear if one process is favoured over the other. In addition, note that disabling the immune response is not possible, since the wild-type infection activates it. Second, to what extent do we need to account for the ongoing proliferation of the immune response? In other words, if we calculate the emergence probability based on the system state when the mutation occurs, how inaccurate would this estimate be? We end by discussing our results in the light of what is known about within-host evolution for several human infections.
In order to find an analytical solution for the within-host emergence of a mutated strain, we follow the approach of Hartfield and Alizon [21], which investigated the appearance of a new infectious pathogen from a pre-existing strain at the population level. To consider the within-host case, we construct and subsequently analyse a specific scenario of acute, immunising infections. Here, the first strain will go extinct because it does not replicate at a high enough rate. However, before vanishing, it can mutate into a form that grows unboundedly; we are interested in calculating the probability of this event occurring. We focus on this case to cover a general range of within-host evolution scenarios, which is important since there is yet no consensus on how to best model within-host infections (reviewed in [29]). Although it is feasible that the mutated strain has a maximum population size, mathematical analysis of pathogen emergence only needs to consider the dynamics of mutants when they are present in a few copies, rather than the long-term behaviour once they have already established. Therefore, the general results outlined in this paper are also broadly applicable to cases where the infection population sizes are bounded. Our analytical approach involves using a set of deterministic differential equations to ascertain pathogen spread in a stochastic birth-death process, where an infection (or immune cell) can only either die or produce 1 offspring. This process is one of the most common ways of investigating stochastic disease spread [30]. A list of nomenclature used in the model is outlined in Table 1. Assume there exists an initial pathogen, or infected cell-line, the size of which at time t is denoted x1 (t). This line grows in size over time according to the following equation, which is well-used for within-host infection models [29]: d x 1 d t = x 1 (φ 1 - σ 1 y) (1) Here, φ1 is the growth rate of the infection, σ1 is the rate of destruction of the pathogen per immune cell, and y (t) is the number of immune cells (i. e. lymphocytes). For simplicity, we assume that there is complete cross-immunity between the various pathogen strains, so it is not necessary to model immune cell diversity. The growth of the immune cell population is modelled using a logistic-growth curve: d y d t = r x 1 y 1 - y K (2) Here, r is the proliferation rate of immune cells, and K is the maximum population size they can achieve. This formulation is an extension of the model developed by Gilchrist and Sasaki [24] to study acute infections. The main difference in that in their model, the density of immune cells is allowed to reach any value in order to clear the infection. Here, we impose that immune density does not go above a maximum threshold K, which correspond to an intrinsic limitation in the host resources allocated to immunity. Profile plots of typical infection responses are shown in Section 1 of S1 Text. If φ1/σ1 < K, then the first strain will increase in size until the immune-cells reach a maximum. After this point, the infection will decrease towards eventual extinction, while the immune response will be maintained at a non-zero size. However, if φ1/σ1 ≥ K then the first strain will continue to expand. A formal rational for this behaviour will be shown below. To proceed with finding an analytical solution for the emergence probability, we proceed as in previous analyses [21,24,31], and note that since y is monotonically increasing, we can use the immune cell population size as a surrogate measure of time. By dividing Equation 1 by Equation 2, we obtain a differential equation for x1 as a function of y: d x 1 d y = (φ 1 - σ 1 y) x 1 r x 1 y (1 - y K) (3) To simplify subsequent analyses, we make the following substitutions. We define the reproductive rate of the infection, where there is a single immune cell (y = 1) equals R1 = φ1/σ1. We also set ρ = r/σ1 (this can be formally shown by rescaling time by τ = σ1t). We use the notation R1 to draw parallels between the scaled pathogenic replication rate, and the reproductive ratio R0 in population-level, epidemiological models [32]. After making the required substitutions, Equation 3 can be rewritten as: d x 1 d y = K (R 1 - y) ρ y (K - y) (4) Equation 4 formally shows that the infected cell line increases in size (dx1/dy > 0) if y < R1 = φ1/σ1, and decreases if y > R1. A corollary of this result is that if R1 of a infection exceeds K, then it cannot go extinct in the long term. This differential equation is straightforward to solve (Section 1 of S1 Text), and yields the following function for x1 (y): x 1 (y) = x i n i t + 1 ρ log y y i n i t R 1 K - y K - y i n i t (K - R 1) (5) where xinit and yinit are, respectively, the number of pathogen and immune cells at the start of the process (time t = 0), and log is the natural logarithm. It is clear from Equation 4 that the maximum value of x1 occurs for y = R1. By substituting this value into Equation 5, we obtain the maximum value of x1 (denoted xM) as: x M = x i n i t + 1 ρ log R 1 y i n i t R 1 K - R 1 K - y i n i t (K - R 1) (6) Note that ρ has no effect on the position of the peak (that is, the value of y leading to the maximal infection level). Since it affects the growth rate of the immune response (and therefore also of the pathogen), it does determine the maximum value itself. As this maximum is inversely proportional to ρ, smaller immune growth rates lead to larger peaks. Finally, the maximum infection time (as a function of y) needed for the first infection to go extinct can be determined by solving x1 (y) = 0 numerically (Section 1 of S1 Text). Our goal is to calculate the probability of ‘evolutionary rescue’. That is, the initial strain has R1 < K so is guaranteed to go extinct in the long-term. However, a mutated form (with reproductive ratio R2) could arise with R2 > K, and if it does not go extinct when rare, it can outgrow immune proliferation. Previous theory on the emergence of novel pathogenic strains [21] showed that if mutated strains arise at rate μ per time step, the overall emergence probability P is given by: P = 1 - exp - μ ∫ y i n i t y M x 1 (y) Π (y) d y (7) for yM the maximum immune size for which it is possible for immune escape to arise, and Π (y) the emergence probability of an escape mutation were it to appear. The formulation of each of these will be discussed in turn. We verified our analytical solution by comparing it to simulation data. Simulations were written in C and based on the Gillespie Algorithm with tau-leaping [35,36]; source code has been deposited online in the Dryad data depository (doi: 10. 5061/dryad. df1vk). The time step was set to be very low: Δτ = 0. 00005. This is because the tau-leaping algorithm is accurate only if the expected number of events per time step is small [37]; since the growth rates of the pathogen strains and the lymphocytes are both large, a small time step is needed to make the simulation valid. The growth of both the original and mutated strains are simulated using scaled parameter rates. That is, the birth rate per time step for each pathogen is Poisson-distributed with mean (Ri ⋅ xi ⋅ Δτ), and death rate with mean (xi ⋅ y ⋅ Δτ) for i = 1,2. This is done to reduce the number of parameters in the model, and also enables accurate comparison with the scaled results. The change in size of the immune response is determined by standard logistic-growth dynamics for the Gillespie algorithm. That is, the Poisson mean number of births per time step equaled ρb (x1 + x2) y, for ρb is the birth rate parameter, and the mean number of deaths equalled y (x1 + x2) (ρd + ρ (y/K) ), where ρd is the death rate and ρ = ρb − ρd. Since there are no distinct ρb and ρd terms in the model (just ρ), then we set ρd = 1 and varied ρb, so ρ = ρb − 1. The net growth rate ρ was varied, generally between 0. 5 and 19 depending on the size of the other parameters. Note that in the analytical solution, it is assumed that the immune response does not die off. In order to maintain this assumption, we set yinit = 20 in the simulations. This also makes intuitive sense, because it is unlikely that the initial immune response is limited to just one cell. If the immune response goes extinct before both infected strains go extinct, or the mutated strain emerges, then that run is discarded, the simulation is reset and restarted. In simulations, R1 is either set to 60 if K = 100, R1 = 100 if K = 250 or 1000, or R1 = 2,000 for K = 10,000. R2 was varied, ensuring that R2 > K so the infected strain can outgrow the immune response (see Equation 4). The mutation rate was also varied over several orders of magnitude. The first strain is reintroduced (from an initial frequency of 1 cell) 10,000,000 times as separate replication runs. Since the emergence probabilities were predicted to be low, a large number of runs were needed in order to produced a meaningful estimate of emergence probability. The second strain is said to have emerged once it exceeded 20 cells; since meaningful values of R2 were large, only a handful of cells would have been needed to guarantee emergence, hence a low outbreak threshold could be used [38].
The first results we checked were individual profiles of the stochastic simulations, since these can be used to demonstrate the behaviour of within-host emergence. In the majority of cases, the first strain increase in size, but once immune proliferation also reaches its maximum then the first strain goes extinct soon after (Fig. 2 (a) ). However, emergence of the mutated strain can occur even if the immune cells are at their maximum size (Fig. 2 (b) ). This is to be expected, given the form of Π (Equation 10), which demonstrates that emergence is likeliest once the immune cells reach a steady-state, so ongoing proliferation does not restrict their establishment. Conversely, emergence probability is reduced if the immune response is spreading; this is because when the mutated infection is rare, it is less likely to reproduce as immune cells increase in number. Since the mutated infection population is only present in a few copies, this negative effect this immune growth has on reproductive ability will be drastic (see also equivalent population genetics results by Otto and Whitlock [39]). Fig. 3 compares the full analytical solution (Equation 7, with Π given by Equation 10) against simulation data. If K = 100, R1 = 60, we see that there is an accurate overlap between the two for a variety of mutation rates that span several orders of magnitude (Fig. 3 (a) and (b) ). This match demonstrates how our analytical solutions can be used to accurately predict emergence probabilities in the face of different scenarios, including antibiotic resistance and tumour formation, as both these processes are characterised by high mutation rates. We also tested a parameter set where the carrying capacity and initial growth rate was much higher (K = 1,000 and R1 = 100, or K = 10,000 and R1 = 2,000). Fig. 3 (c) - (f) demonstrates that the analytical results slightly underestimate the simulation results to a small degree, especially if the mutated strain’s growth rate is high and R2 is close to K, but becomes more accurate as R2 increases and generally provides a good approximation. These inaccuracies probably arise due to our analytical solution not fully accounting for the increased variance in both infection and immune growth rates that can arise if parameters are large, as in this model [30]. However, the error does not appear to be great, so the model can still be used to provide accurate estimates of emergence probability for this parameter set. Finally, we also tested how well analytical solutions work for cases where R1 < R2 < K. Although such a mutated strain replicates more quickly, Equation 4 shows that it will die out in the long-term. Hence our analytical solutions might not correctly reflect the emergence probability of these infections. Nevertheless, even in this case, Equation 7 accurately matches up with simulation results in this parameter range, although some inaccuracies arise for K = 1,000 (S1 Fig). To exemplify why it is important to account for ongoing immune growth, we compared our full solution for the emergence probability (Equation 7, with Π equal to Equation 10), to a ‘naive’ estimate that does not assume ongoing immune proliferation (Π = 1 − 1/R2). We see that our result leads to a greatly reduced emergence probability, with values from our model being ∼1,000 times lower than the naive estimate (Fig. 4, and Section 3 of S1 Text). Similar results apply if feedbacks only affect the mutant after it has arose (that is, Π = (1 − P e x t *) as given by Equation 9). Hence, as in previous non-equilibrium models [21], one needs to account for dynamical feedbacks, both before and after the mutated strain appears, in order to account for the reduced emergence probability in these scenarios. We next studied what process has a larger effect on pathogen emergence. Immune escape can either be achieved by overgrowing the immune response (increasing the intrinsic pathogen growth rate φ), or by tolerating it (reducing the immune-mediated death rate σ). One might expect that the two processes would lead to similar increase in escape probability, since both affect the effective reproductive rate R*. However, this intuition need not hold in the face of immune-mediated feedbacks. We commence with a heuristic analysis based on the deterministic model to predict general behaviour for a newly-emerging strain, then use numerical analyses to check this reasoning. If there are two strains spreading concurrently, the deterministic rate of change of immunity, y, and the second strain x2, is given by the following set of differential equations: d x 2 d t = x 2 (φ 2 - σ 2 y) (11a) d y d t = r (x 1 + x 2) y 1 - y K (11b) Further recall that that the basic pathogen growth rate with only one immune cell is R2 = φ2/σ2. In order to augment its spread, the infection can either increase its intrinsic growth rate by a certain factor (i. e. change φ2 → φ2 c, where c > 1 is a numerical constant), or instead tolerate the immune response (mathematically equivalent to the transformation σ2 → σ2/c). We can investigate the effect of both these rescaled variables on the rate of change of pathogen spread by substituting them into Equation 4. After making the substitution φ2 → φ2 c, the pathogen rate of change becomes: d x 2 d y = K (c R 2 - y) ρ y (K - y) x 2 x 1 + x 2 σ 2 σ 1 (12) The x2/ (x1 + x2) term arises due to the presence of the pre-existing initial strain x1, and the death-rate ratio σ2/σ1 appears since ρ was initially scaled by σ1; this term disappears if we assume equal death rates. Apart from these terms, Equation 12 is conceptually the same as Equation 4, but with R scaled by a factor c, as expected. However, if we instead scale σ2 → σ2 / c, a different result emerges: d x 2 d y = K (c R 2 - y) c ρ y (K - y) x 2 x 1 + x 2 σ 2 σ 1 (13) Here, not only is the reproductive ratio R2 scaled, but the immune growth term ρ is also altered. Mathematically, this is a simple consequence of the fact that the growth rate is scaled by 1/σ1, so any rescaling of σ2 also affects ρ through its effect on the σ2/σ1 ratio. Biologically, this outcome reflects the fact that a rescaling of immune-mediated death would not have the same effect on the growth rate than changing φ2 by the same ratio, since pathogen death is also a function of the immune population, y (see Equation 1). So although tolerating immunity might increase the infection growth rate, it comes at a cost of increasing the effective growth rate of immunity, as each immune cell would have a larger average impact on pathogen death. Therefore, while a rise in growth rate (φ2) would only affect R2, reducing the death rate (σ2) comes at a cost of increasing the effective proliferation rate of immune cells. Hence, one expects that it is more advantageous for an infection to increase φ2 and outgrow the immune response, instead of reducing σ2 and tolerating it. Furthermore, note that the emergence probability Π in the presence of epidemiological feedbacks contains a term of order 1/ρ (as with dx2/dy), so the effective rise in immunity will further lead to a negative overall effect on emergence probability. We verified this intuition by comparing the analytical results for cases where φ2 is increased by a set factor (so that only R2 is changed by this rescaling), to outcomes where σ2 is scaled (which will not just affect R2 but will also increase ρ by the same factor, as outlined above). Fig. 5 (a) demonstrates how, for large parameter values, increasing φ2 only produces a higher overall emergence probability. Qualitatively similar results arise if using different parameter values.
Our model can also be used to shed light on different processes of within-host emergence that have been observed in clinical studies. Different diseases show varying outcomes with regards to the production of immune-escape mutations. Two extensively-studied human diseases, HIV and HCV, are both characterised by a successive emergence of new strains over time. In particular, HIV is well-known to produce ‘escape mutations’ that evade T cell-mediated immunity [43,44]. On a within-host phylogeny, this behaviour is characterised by creation of new subclades [5–7]. This behaviour can be intuitively explained by noting that HIV has extremely high mutation rates, estimated at ∼0. 2 errors made per replication cycle and the ability to produce 1010 – 1012 virions per day [6] (although mutation creation might be limited by a lower Ne, which has been estimated to lie between 1,000 [50] and 100,000 [51]). Furthermore, a sizeable proportion of mutations are beneficial, with estimates of adaptive substitutions in the env gene placed at ∼55% [52] (although the actual proportion of spontaneous beneficial mutations will be less than this, as observed from mutagenesis studies with viruses [53,54]). What may seem surprising is that given this evolutionary potential, we do not see a more pronounced increase in HIV replication rate throughout the course of an infection. Yet, conversely, immune escape is very strongly selected for [14]. One possibility could be that given the constrains on RNA virus genomes [55], evading the immune response might be easier to achieve than increasing the replication rate. Furthermore, if the maximum immunity population size is high, it could be too complicated mechanistically to outgrow it, rather than than simply evading it. HCV also shows a similar propensity to produce immune-escape mutations, with an estimated substitution rate of 1. 2×10−4 per replication cycle and 1012 virions generated each day [7] (although, as with HIV, the effective population size is greatly lower than this value [7]). Acute hepatitis C infections are characterised by little diversity accumulating over time, except in one specific genetic region (NS5B; [56]). Chronic infections accumulate much more diversity, in line with the hypothesis that they continuously evolve to evade the immune system [16]. These infections are also characterised by different immune profiles: acute infections lead to a high, sustained immune response, which tends to be greatly lowered in chronic infections [57]. Our model suggests that if the immune response naturally increased rapidly (i. e. a higher intrinsic ρ in the model), it can greatly limit the emergence of new pathogens as it rises, preventing immune escape and a chronic illness. There exists evidence that mutations in hosts correlate with infection outcome, mainly due to SNPs at the IL28B loci, which could cause this effect [58–60]. However, it remains controversial as to what determines virus clearance, especially since there exist evidence for virus control over infection outcome [61]. Therefore, the effect on immune response on the creation of escape mutations requires further study. Other diseases are characterised by low probability of emergence despite frequent mutation, for which immune feedbacks could be the cause. Cancer growth is characterised by evading pre-programmed cell death and extremely rapid cell replication [17]. Furthermore, genomes present in cancer cells usually carry ‘mutator’ alleles, causing additional cell instability [62]. Therefore, cell mutation can be extremely common, but can generally be stopped by the immune response. Yet it is known that tumours can mutate immune-checkpoint networks to generate protection from the immune system. One prominent example is the up-regulation of ligands for the programmed cell death protein 1 (PD1) pathway, which can block antitumour immune responses [63]. Our model suggests that due to the rapid replication of tumour cells, they could also strongly trigger a heightened immune rate, the increased spread of which will greatly prevent cancer emergence. This mechanism could explain why tumour emergence is rare relative to the potential mutation rate. By accounting for the ongoing spread of immunity, we have quantified how this particular effect can feedback onto emergence of mutated pathogens intra-host, and inhibit the appearance of mutated strains. Further analysis of the model demonstrates how it is beneficial for a pathogen to increase its replication rate and attempt to outgrow immunity, as opposed to tolerate it. This model can therefore shed light on expected within-host evolutionary dynamics of infections, as well as determine why extremely rapidly replicating pathogens do not emerge as often as expected.
The ongoing evolution of infectious diseases provides a constant health threat. This evolution can either result in the production of new pathogens, or new strains of existing pathogens that escape prevailing drug treatments or immune responses. The latter process, also known as immune escape, is a predominant reason for the persistence of several viruses, including HIV and hepatitis C virus (HCV), in their human host. As a consequence, the within-host emergence of new strains has been the intense focus of modelling studies. However, existing models have neglected important feedbacks that affects this emergence probability. Specifically, once a mutated pathogen arises that spreads more quickly than the initial (resident) strain, it potentially triggers a heightened immune response that can eliminate the mutated strain before it spreads. Our study outlines novel mathematical modelling techniques that accurately quantify how ongoing immune growth reduces the emergence probability of mutated pathogenic strains over the course of an infection. Analysis of this model suggests that, in order to enlarge its emergence probability, it is evolutionary beneficial for a mutated strain to increase its growth rate rather than tolerate immunity by having a lower immune-mediated death-rate. Our model can be readily applied to existing within-host data, as demonstrated with application to HIV, HCV, and cancer dynamics.
Within-Host Stochastic Emergence Dynamics of Immune-Escape Mutants