Within-Host Stochastic Emergence Dynamics of Immune-Escape Mutants

However, the majority of existing studies have focussed on emergence at the population level, and not within a host. In particular, the coexistence of preexisting 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 longterm. 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.

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 deathrate. Our model can be readily applied to existing within-host data, as demonstrated with application to HIV, HCV, and cancer dynamics.

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].

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.

For instance, HIV is known to successfully f1x 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 with-in-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.

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.

In their model, a faster-rep-licating strain emerged via mutation from a preexisting 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.

We use ‘im-mune-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.

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 longterm 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.

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]: Table 1. Glossary of notation. Symbol Usage t Time (measured as number of generation since start of process) (P1, (P2 Growth rate of initial, mutated infection x1, x2 Size of initial, mutated infection y Size of immune response K Maximum size of immune response r Unscaled growth rate of immune response R* ‘Effective’ initial reproductive ratio in the presence of immunity, R — yo p Scaled immune growth rate, r/o1 ,u Individual mutation rate from first to second infection I'I Emergence probability of second strain in an non-equilibrium population P Overall emergence probability of second strain P2X, Extinction probability, given an ‘effective’ reproductive ratio

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:

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.

If (p1/ 01 < 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 nonzero size. However, if (pl/01 2 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:

We define the reproductive rate of the infection, where there is a single immune cell ()1 = 1) equals R1 = (pl/01. We also set p = r/ 01 (this can be formally shown by rescaling time by T = 01 t). 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: Equation 4 formally shows that the infected cell line increases in size (dxl/dy > 0) if y < R1 = (pl/01, 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 81 Text), and yields the following function for x1(y):

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:

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 p, 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 81 Text).

That is, the initial strain has R1 < K so is guaranteed to go extinct in the longterm. 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 [,4 per time step, the overall emergence probability P is given for yM the maximum immune size for which it is possible for immune escape to arise, and H(y) the emergence probability of an escape mutation were it to appear. The formulation of each of these will be discussed in turn.

The emergence probability of the new strain has to account for both the increase in immune response before the mutation occurs, which reduces the ‘effec-tive’ growth-rate of a mutated strain when it appears due to an heightened immune size (greater y in the model); and the continuing growth of immunity once it has appeared. Both these points factor in how the overall immune strength affects emergence probabilities, but this strength can increase over time. In theory, this emergence probability can be calculated from first principles using a time-inhomogeneous branching-process equation, which accounts for the birth and death of pathogens as the lymphocyte population size is changing (see [4, 21] for examples of branching-process formulations in epidemiology). However, due to the complexity of the model, specifically the nonlinear form of the change in immune response (Equation 2), it is not possible to derive an exact analytical solution for H (Section 1 of 81 Text). We proceed by comparing a previously-found solution to H in a changing susceptible population to our model, and adapting the result accordingly. When investigating a second strain emerging from a preexisting epidemic, Hartfield and Alizon [21] found that a good approximation for H is a function of the form:

Equation 8 arose during analysis of pathogen emergence in an epidemiological SIR framework [21], by noting how for a single strain, the ratio of the rate of change of infected and susceptible individuals over time reflected the emergence probability. This logic was then carried over to a two-strain scenario, so it details how the ongoing spread of the first pathogen affected emergence of mutated strains. To give an example, when applied to a model of pathogen emergence in an SIR setting, the effective reproductive ratio Rl< would equal the standard reproductive ratio R0, reduced by a factor So/N, if there initially existed So susceptible individuals out of a total population of size N. There would also be a term in the denominator that is proportional to the rate of spread of the initial, unmutated strain.

The rational behind Equation 8 is that the p(x1 + x2)y(K — y) accounts for the reduction in emergence probability due to the continuing onset of immunity, given a total infection size (x1 + x2). If it reaches a steady-state [that is, p(x1 + x2)y(K — y) —> 0] then H collapses mathematically to (1 — ijt). Hence Equation 8 accounts for how the presence of infection triggers immune responses that restrict the emergence probability of mutated infections; this is the feedback not yet considered in previous within-host evolution models.

Note we use yim-t instead of y(t) in this argument, since the R* term should describe the baseline replication rate if the second strain is only present. For the emergence probability, we first note that as with Equations 3 and 4, a single mutated cell has an effective birth rate of R2 and death rate of y. Standard results from birth-death models states that the mean growth rate is equal to R2 — y, with variance equal to R2 + y [33]. We therefore determine the baseline emergence probability by substituting these terms into the classic result by Feller [34]: Note that Equation 9 is not constant over time since it is a function of y. Overall, we obtain the following approximation for H: Also note that we use x1(y)+1 in the denominator, since the total pathogen population is accounted for by the initial pathogen size when the mutant appears, x1(y), plus a single mutated indiVidual.

The emergence of mutated infections can be arrested in one of two ways. Either the immune response becomes high enough so as to drive the first infection to extinction (x101) = O), or so that any further emergence is impossible (H(y) = 0). Therefore, yM can be easily found by finding the smallest solution out of x1(y) = 0 or H(y) = 0. These equations have to be solved numerically (Section 1 of 81 Text).

x1 exhibits an inverted-U shape, with the maximum size decreasing if p increases, as expected from the form of Equation 5. From a biological standpoint, it illustrates the fact that the parasite population grows until the immune response is activated, at which point it decreases. H, on the other hand, demonstrates a U shape. Emergence is initially high due to fewer immune cells being present, and again as the first strain dies out, since the immune population is not proliferating as fast and is less likely to remove mutated strains as they appear. Results are qualitatively similar for different parameter values, With H further decreasing With higher values of p (Section 1 of 81 Text).

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: Ar = 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.

That is, the birth rate per time step for each pathogen is Poisson-distributed with mean (R,- - x,- - AT), and death rate with mean (x,- - y - AT) 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 pb(x1 + x2)y, for pg, is the birth rate parameter, and the mean number of deaths equalled y(x1 + x2)(pd + p(y/K)), where pd is the death rate and p 2 pb — pd. Since there are no distinct pb and pd terms in the model (just p), then we set pd = 1 and varied p1,, so p = p], — 1. The net growth rate p was varied, generally between 0.5 and 19 depending on the size of the other parameters.

In order to maintain this assumption, we set yim-t = 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.

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].

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 H (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]).

3 compares the full analytical solution (Equation 7, with H 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.

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.

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 (81 Fig).

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 81 Text). Similar results apply if feedbacks only affect the mutant after it has arose (that is, H = (1 — ijt) 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.

Immune escape can either be achieved by overgrowing the immune response (increasing the intrinsic pathogen growth rate (p), or by tolerating it (reducing the immune-mediated death rate a). 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:

In order to augment its spread, the infection can either increase its intrinsic growth rate by a certain factor (i.e. change (p2 —> (p2 c, Where c > 1 is a numerical constant), or instead tolerate the immune response (mathematically equivalent to the transformation 02 —> az/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 (p2 —> (p2 c, the pathogen rate of change becomes:

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 02 —> 02 / c, a different result emerges:

Mathematically, this is a simple consequence of the fact that the growth rate is scaled by 1/01, so any rescaling of 02 also affects p through its effect on the 02/01 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 (p2 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.

Hence, one eXpects that it is more advantageous for an infection to increase (p2 and outgrow the immune response, instead of reducing 02 and tolerating it. Furthermore, note that the emergence probability H in the presence of epidemiological feedbacks contains a term of order Up (as with dxz/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 (p2 is increased by a set factor (so that only R2 is changed by this rescaling), to outcomes where 02 is scaled (which will not just affect R2 but will also increase p by the same factor, as outlined above).

5(a) demonstrates how, for large parameter values, increasing (p2 only produces a higher overall emergence probability. Qualitatively similar results arise if using different parameter values.

These results appear to contradict a previous analysis by Alizon [27]. Using nested models, it was shown that in order to maXimise the epidemiological (between-host) reproductive ratio R0, increasing the growth rate might not always be the best option for a pathogen because it shortens the duration of the infection.

Principally, [27] aimed to determine the longterm R0 across a population, while our model concerns the emergence of new strains intra-host. In addition, [27] used deterministic equations, and did not investigate the stochastic emergence of new strains (that is, whether those with a smaller R0 are likely to emerge if they appear at a low frequency). However, one major difference that is worthy of further investigation concerns the different immune-response functions used in each model.

Hence, the immune response was not just stimulated by the presence of an infection, but is also dependent on pathogen replication. The reason for this assumption is that the eXpression of pathogen peptides on the infected cell surface is strongly dependent on the replication activity [40]. This is Why latent Viruses, such as herpes Virus, can go unnoticed from the immune system and persist for long periods of time. Interestingly, if b = 1 then, according to this form of dy / dt, overgrowing and tolerating the immune response Will have equal effects on pathogen emergence. This can be seen by forming dxz / dy as before, and after substituting either (p2 —> (p2 c or 02 —> az/c, one sees that each transformation results in the same rescaled differential equation (Section 3 of 81 Text):

However, due to the ongoing spread of immunity in our model, increasing immune tolerance might still cause the greatest negative impact on the emergence of mutated strains. This is due to its ensuing effect of increasing the ‘effective’ immune growth, which can restrict new strains from emerging (as reflected in the l/p term in H). This is indeed what is observed; an example of this behaviour is given in Fig. 5(b). Note that increasing p in this case leads to opposing behav-iour than to what is seen for our model, in the sense that higher p reduces the advantage of increased (pz. Presumably, this is due to the (p2 term affecting the immune growth-rate, which can offset the reduction in emergence probability caused by the ongoing immune growth.

While this question has been extensively studied at the epidemiological level, the question of evolutionary emergence within hosts has received less scrutiny. Furthermore, few existing models have accounted for population dynamics feedbacks that can impact and restrict the emergence of mutated strains.

To this end, we tailor a broadly-applicable population dynamics model to the interaction between pathogens and immune cells, a system which share similarities with predator-prey models [29]. As with previous work studying feedbacks at the epidemiological level [21] , we highlight how ongoing population spread (in this case, of immune cells) can strongly limit the emergence of new strains. This result is reflected in the fact that the emergence probability is greatly reduced due to ongoing immune proliferation (Equation 10); accounting for this feedback shows how it strongly restricts emergence of mutated strains (Section 3 of 81 Text).

We demonstrate that, rather surprisingly, it always pays in the short-term for a pathogen to increase its replication rate, and to try and outgrow the immune response. The rationale behind this behaviour is that if immune tolerance rises by a certain factor, it will not cause an equal reduction on pathogen death since immunity is still present at a high frequency. Intuitively, the presence of multiple strains will cause indirect competition between them, since the presence of both causes a higher spread of immunity which reduces the mutated strain’s growth rate when rare. This phenomenon manifests itself mathematically as an increase in the scaled immune growth rate, p. Since increasing effective immunity would also reduce the emergence probability of new strains (Equation 10), it is evolutionary advantageous for pathogens to evolve a higher growth rate instead (Fig. 5).

It is a concern in epidemiological modelling that key results are heavily dependent on the specific mathematical functions used [41], and several models have been previously used to reflect immune growth [29]. However, most of these functions approximate to eXponential growth when immunity initially appears; furthermore, the form of the emergence probability (Equation 8) makes clear that any immune growth will feedback onto the emergence of the mutated strain, limiting its appearance. For this reason, our conclusion that infectious diseases would benefit more from higher growth rates should also be robust to how immunity spreads over time.

However, the observed increase is small. A slight increase might only be necessary if too large a growth rate would trigger a larger immune response, or kill off the host too quickly and thus limit the epidemiological spread of the pathogen (both effects have been previously investigated by [27] ). It has been proposed that immune escape for HIV becomes less efficient over time, which might be the cause of a low increase in replication rate [15]. Hence it might be mechanistically more feasible for a pathogen to instead evade the immune response, by mutating in or close to a virus epitope, so as to avoid recognition by lymphocytes that would otherwise neutralise it (a variant on the ‘immune tolerance’ scenario outlined above). These type of escape mutations have been widely documented in HIV infections, which is one reason why the virus is able to persist for long periods of time, and also why vaccination is so difficult [43, 44]. Similarly, although not a chronic illness, evidence exists demonstrating that influenza A/H1N1 evades immunity via evolution causing increased virus affinity to cell receptors, which enlarges its replication rate [20]. To further investigate this issue, future models need to be created that combine short-term effects of pathogen emergence, as used here, with longterm forecasts of the pathogenic steady-state (since our model only deals with the initial, emergence stage).

In this case, it has been postulated that the faster-replicating strain could persist in the longterm by sheer force of numbers [46], as predicted with our model. Several empirical studies exist that support this intuition, as observed with malaria [47, 48] and schistosomes [49] (although competition for another resource, such as red-blood cells for malaria infections, could have partly eXplained these results).

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 sub-clades [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 em} 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.

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 p 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.

Cancer growth is characterised by evading preprogrammed 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.

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.

points if they cannot be seen. (EPS)

Contains additional information on derivations, and further comparisons against simulation results (Mathematica NB format). The file is separated into three sections: Section 1 Setting up the mathematical model. In-depth mathematical analyses of the differential equations used, and how to derive the emergence probability if affected by immune growth (Equation 10 in the main text).

Full list of analytical comparisons With simulation outcomes, Which contributed to Fig. 3. Section 3 Mathematical analysis of analytical solution. Further outline of mathematical analysis and numerical computation, demonstrating (i) how immune feedbacks affect population growth, and (ii) how different (p and avalues affect mutated pathogen emergence. (PDF)

Same as Text 51, but in PDF format. (PDF)

We would like to thank Helen Alexander for helpful comments on the manuscript.

Performed the experiments: MH SA. Analyzed the data: MH SA. Contributed reagents/materials/analysis tools: MH SA. Wrote the paper: MH SA.

Appears in 39 sentences as: immune response (36) immune responses (4)

In *Within-Host Stochastic Emergence Dynamics of Immune-Escape Mutants*

- In particular, the coexistence of preexisting 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.Page 1, “Abstract”
- 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 .Page 1, “Author Summary”
- 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.Page 1, “Author Summary”
- Parasite and pathogen evolution can radically affect the course of infections in hosts able to mount immune responses .Page 2, “Introduction”
- 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).Page 2, “Introduction”
- 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).Page 3, “Introduction”
- 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)?Page 3, “Introduction”
- In addition, note that disabling the immune response is not possible, since the wild-type infection activates it.Page 3, “Introduction”
- Second, to what extent do we need to account for the ongoing proliferation of the immune response ?Page 3, “Introduction”
- (P1, (P2 Growth rate of initial, mutated infection x1, x2 Size of initial, mutated infection y Size of immune responsePage 4, “Model outline”
- K Maximum size of immune response r Unscaled growth rate of immune responsePage 4, “Model outline”

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Appears in 30 sentences as: Growth rate (1) growth rate (26) growth rates (4)

In *Within-Host Stochastic Emergence Dynamics of Immune-Escape Mutants*

- 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.Page 1, “Abstract”
- 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 deathrate.Page 2, “Author Summary”
- (P1, (P2 Growth rate of initial, mutated infection x1, x2 Size of initial, mutated infection y Size of immune responsePage 4, “Model outline”
- K Maximum size of immune response r Unscaled growth rate of immune responsePage 4, “Model outline”
- R* ‘Effective’ initial reproductive ratio in the presence of immunity, R — yo p Scaled immune growth rate , r/o1Page 4, “Model outline”
- Here, gal is the growth rate of the infection, 01 is the rate of destruction of the pathogen per immune cell, and y(t) is the number of immune cells (i.e.Page 5, “Model outline”
- Since it affects the growth rate of the immune response (and therefore also of the pathogen), it does determine the maximum value itself.Page 6, “Model outline”
- As this maximum is inversely proportional to p, smaller immune growth rates lead to larger peaks.Page 6, “Model outline”
- We use Equation 8 in our model by setting R* = R2 — yim-t, which is the rescaled growth rate of the mutated strain, corrected for the fact that the baseline immunity rate will reduce its initial selective advantage.Page 7, “Formulating emergence probability”
- Standard results from birth-death models states that the mean growth rate is equal to R2 — y, with variance equal to R2 + y [33].Page 7, “Formulating emergence probability”
- 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.Page 8, “Simulation methods”

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Appears in 23 sentences as: immune cell (11) immune cells (14)

In *Within-Host Stochastic Emergence Dynamics of Immune-Escape Mutants*

- 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.Page 1, “Abstract”
- Analysis of the model demonstrates how the ongoing proliferation of immune cells acts to decrease the emergence probability of mutated strains.Page 3, “Introduction”
- 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 )?Page 3, “Introduction”
- 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.Page 4, “Model outline”
- Here, gal is the growth rate of the infection, 01 is the rate of destruction of the pathogen per immune cell, and y(t) is the number of immune cells (i.e.Page 5, “Model outline”
- 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.Page 5, “Model outline”
- The growth of the immune cell population is modelled using a logistic-growth curve:Page 5, “Model outline”
- Here, r is the proliferation rate of immune cells , and K is the maximum population size they can achieve.Page 5, “Model outline”
- 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.Page 5, “Model outline”
- 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.Page 5, “Model outline”
- We define the reproductive rate of the infection, where there is a single immune cell ()1 = 1) equals R1 = (pl/01.Page 5, “Model outline”

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Appears in 6 sentences as: differential equation (3) differential equations (2) differential equations: (1)

In *Within-Host Stochastic Emergence Dynamics of Immune-Escape Mutants*

- 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.Page 4, “Model outline”
- By dividing Equation 1 by Equation 2, we obtain a differential equation for x1 as a function of y:Page 5, “Model outline”
- This differential equation is straightforward to solve (Section 1 of 81 Text), and yields the following function for x1(y):Page 5, “Model outline”
- 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:Page 12, “Comparing pathogen growth against death rate”
- This can be seen by forming dxz / dy as before, and after substituting either (p2 —> (p2 c or 02 —> az/c, one sees that each transformation results in the same rescaled differential equation (Section 3 of 81 Text):Page 15, “Comparing pathogen growth against death rate”
- In-depth mathematical analyses of the differential equations used, and how to derive the emergence probability if affected by immune growth (Equation 10 in the main text).Page 18, “Supporting Information”

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Appears in 6 sentences as: immune system (6)

In *Within-Host Stochastic Emergence Dynamics of Immune-Escape Mutants*

- through mutating the p53 tumour suppression gene), and escaping the patient’s immune system are key steps in cancer development [17].Page 2, “Introduction”
- Existing emergence models have not yet accounted for such within-host population feedbacks, especially those arising from the immune system .Page 3, “Introduction”
- This is Why latent Viruses, such as herpes Virus, can go unnoticed from the immune system and persist for long periods of time.Page 15, “Comparing pathogen growth against death rate”
- More generally, the huge challenge exerted by the immune system on pathogen populations defacto generates apparent competition between different infection strains (or even species) co-infecting the same host [45].Page 16, “Discussion”
- Chronic infections accumulate much more diversity, in line with the hypothesis that they continuously evolve to evade the immune system [16].Page 17, “Implications of the model for different cases of immune escape”
- Yet it is known that tumours can mutate immune-checkpoint networks to generate protection from the immune system .Page 17, “Implications of the model for different cases of immune escape”

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Appears in 5 sentences as: R0 (5)

In *Within-Host Stochastic Emergence Dynamics of Immune-Escape Mutants*

- 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].Page 5, “Model outline”
- To give an example, when applied to a model of pathogen emergence in an SIR setting, the effective reproductive ratio Rl< would equal the standard reproductive ratio R0 , reduced by a factor So/N, if there initially existed So susceptible individuals out of a total population of size N. There would also be a term in the denominator that is proportional to the rate of spread of the initial, unmutated strain.Page 7, “Formulating emergence probability”
- Using nested models, it was shown that in order to maXimise the epidemiological (between-host) reproductive ratio R0 , increasing the growth rate might not always be the best option for a pathogen because it shortens the duration of the infection.Page 13, “Comparing pathogen growth against death rate”
- Principally, [27] aimed to determine the longterm R0 across a population, while our model concerns the emergence of new strains intra-host.Page 13, “Comparing pathogen growth against death rate”
- In addition, [27] used deterministic equations, and did not investigate the stochastic emergence of new strains (that is, whether those with a smaller R0 are likely to emerge if they appear at a low frequency).Page 13, “Comparing pathogen growth against death rate”

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Appears in 5 sentences as: time step (6)

In *Within-Host Stochastic Emergence Dynamics of Immune-Escape Mutants*

- Previous theory on the emergence of novel pathogenic strains [21] showed that if mutated strains arise at rate [,4 per time step , the overall emergence probability P is given for yM the maximum immune size for which it is possible for immune escape to arise, and H(y) the emergence probability of an escape mutation were it to appear.Page 6, “Formulating emergence probability”
- The time step was set to be very low: Ar = 0.00005.Page 8, “Simulation methods”
- 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.Page 8, “Simulation methods”
- That is, the birth rate per time step for each pathogen is Poisson-distributed with mean (R,- - x,- - AT), and death rate with mean (x,- - y - AT) for i = 1, 2.Page 8, “Simulation methods”
- That is, the Poisson mean number of births per time step equaled pb(x1 + x2)y, for pg, is the birth rate parameter, and the mean number of deaths equalled y(x1 + x2)(pd + p(y/K)), where pd is the death rate and p 2 pb — pd.Page 8, “Simulation methods”

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Appears in 4 sentences as: population level (4)

In *Within-Host Stochastic Emergence Dynamics of Immune-Escape Mutants*

- However, the majority of existing studies have focussed on emergence at the population level , and not within a host.Page 1, “Abstract”
- Generally, the focus has been on detection of emerging diseases at the population level , in order to track and control their spread [1, 2].Page 2, “Introduction”
- Recently, Hartf1eld and Alizon [21] tackled a related problem, regarding how epidemiological feedbacks affect disease emergence at the host population level .Page 3, “Introduction”
- 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 preexisting strain at the population level .Page 4, “Model outline”

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Appears in 3 sentences as: analytical results (3)

In *Within-Host Stochastic Emergence Dynamics of Immune-Escape Mutants*

- Comparison of analytical results with simulations dataPage 9, “Comparison of analytical results with simulations data”
- 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.Page 9, “Comparison of analytical results with simulations data”
- We verified this intuition by comparing the analytical results for cases where (p2 is increased by a set factor (so that only R2 is changed by this rescaling), to outcomes where 02 is scaled (which will not just affect R2 but will also increase p by the same factor, as outlined above).Page 13, “Comparing pathogen growth against death rate”

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Appears in 3 sentences as: cell population (3)

In *Within-Host Stochastic Emergence Dynamics of Immune-Escape Mutants*

- The growth of the immune cell population is modelled using a logistic-growth curve:Page 5, “Model outline”
- 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.Page 5, “Model outline”
- In [27] , the model assumed that the dynamics of the immune cell population took the form dy/ dt = pr xy (Equation 3 in that paper).Page 15, “Comparing pathogen growth against death rate”

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Appears in 3 sentences as: mathematical model (2) mathematical modelling (1)

In *Within-Host Stochastic Emergence Dynamics of Immune-Escape Mutants*

- 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 longterm.Page 1, “Abstract”
- 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.Page 2, “Author Summary”
- Section 1 Setting up the mathematical model .Page 18, “Supporting Information”

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Appears in 3 sentences as: simulation data (2) simulations data (1)

In *Within-Host Stochastic Emergence Dynamics of Immune-Escape Mutants*

- We verified our analytical solution by comparing it to simulation data .Page 8, “Simulation methods”
- Comparison of analytical results with simulations dataPage 9, “Comparison of analytical results with simulations data”
- 3 compares the full analytical solution (Equation 7, with H given by Equation 10) against simulation data .Page 9, “Comparison of analytical results with simulations data”

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Appears in 3 sentences as: simulation results (3)

In *Within-Host Stochastic Emergence Dynamics of Immune-Escape Mutants*

- 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.Page 9, “Comparison of analytical results with simulations data”
- 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 (81 Fig).Page 10, “Comparison of analytical results with simulations data”
- Contains additional information on derivations, and further comparisons against simulation results (Mathematica NB format).Page 18, “Supporting Information”

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Appears in 3 sentences as: steady-state (3)

In *Within-Host Stochastic Emergence Dynamics of Immune-Escape Mutants*

- If it reaches a steady-state [that is, p(x1 + x2)y(K — y) —> 0] then H collapses mathematically to (1 — ijt).Page 7, “Formulating emergence probability”
- This is to be expected, given the form of H (Equation 10), which demonstrates that emergence is likeliest once the immune cells reach a steady-state , so ongoing proliferation does not restrict their establishment.Page 9, “Comparison of analytical results with simulations data”
- To further investigate this issue, future models need to be created that combine short-term effects of pathogen emergence, as used here, with longterm forecasts of the pathogenic steady-state (since our model only deals with the initial, emergence stage).Page 16, “Discussion”

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