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## Within-host population dynamics and the evolution of microparasites in a heterogeneous host population, (2002)

Venue: | Evolution |

Citations: | 33 - 2 self |

### BibTeX

@ARTICLE{Ganusov02within-hostpopulation,

author = {Vitaly V Ganusov and Carl T Bergstrom and Rustom Antia},

title = {Within-host population dynamics and the evolution of microparasites in a heterogeneous host population,},

journal = {Evolution},

year = {2002},

pages = {213--223}

}

### OpenURL

### Abstract

Abstract. Why do parasites harm their hosts? The general understanding is that if the transmission rate and virulence of a parasite are linked, then the parasite must harm its host to maximize its transmission. The exact nature of such trade-offs remains largely unclear, but for vertebrate hosts it probably involves interactions between a microparasite and the host immune system. Previous results have suggested that in a homogeneous host population in the absence of super-or coinfection, within-host dynamics lead to selection of the parasite with an intermediate growth rate that is just being controlled by the immune system before it kills the host Key words. Evolution of microparasites, host heterogeneity, immune system, virulence, within-host models. Received April 30, 2001. Accepted September 14, 2001 Microparasites depend critically on their hosts to ensure both their livelihood and transmission, yet many are virulent, that is, they cause harm to their hosts. Why do microparasites do this? In some cases virulence is not selected but is simply coincidental; for example, virulence observed in polio may arise due to ''short-sighted'' evolution of the virus resulting in the infection of neurons even though this does not increase transmission Broadly speaking, theoretical models of virulence evolution have typically employed several different conceptual approaches (for a comprehensive review see Frank 1996), two of which we would like to emphasize in particular. The first approach is essentially epidemiological (Anderson and May 1991). In this approach parasites evolve to maximize the net reproductive rate R 0 , the average number of new infections arising from a single infected host introduced into a wholly susceptible population (May and Anderson 1983): where  is the transmissibility; ␣, b, and v are the rate constants for the parasite-induced and natural host mortality and recovery, respectively; and N is the density of susceptible hosts. In the absence of superinfection, selection in the parasite population will act to maximize R 0 that influence this within-host dynamics of the parasite will affect both virulence of the parasite and its transmission. The immunological defenses of vertebrates have evolved to combat parasites and may be expected to be one of the most important factors in determining the within-host dynamics of the parasites. Several models have explicitly considered how the interaction between the parasite and the immune response affects the within-host dynamics and the consequences for the evolution of virulence of parasites The epidemiological and within-host approaches each have their advantages and shortcomings. The major advantage of the epidemiological approach lies in its generality. By painting nature with a broad brush, these models are typically not restricted to single particular infections. This approach has been used in a number of ways. First, this approach can help determine the nature of the trade-offs between ␣,  and v from the epidemiology of spread of specific infections such as the myxoma virus Within-host models typically make much stronger assumptions about the details of infection and host response, and as such are necessarily more narrow in applicability. However, in contrast to many epidemiological models, the trade-offs between transmissibility and virulence naturally emerge from the within-host dynamics of the parasite and immune system Introducing Heterogeneity The evolution of virulence is, at its core, a coevolutionary process between parasite and host. As such, we might expect that a proper understanding of this process will require consideration of heterogeneity in both the parasite and host populations. Most previous studies have focused on how heterogeneous parasites will evolve in a population of homogeneous hosts (e.g., Nowak and May 1994). More complex extensions of these models also incorporate host heterogeneity with differences in host susceptibility, host recovery rates, and the ability to transmit the parasite. (These and other forms of host heterogeneity have been observed in a number of host-parasite systems; e.g., The addition of heterogeneity to the within-host models of acute infections is particularly important because the simple model predicts that to maximize transmission, the growth rate of the parasite will evolve to be sufficiently high that maximum parasite density falls just short of the lethal density (the density of the parasite at which it kills the host) before it is cleared by the immune response In this paper we introduce a simple type of heterogeneity, random heterogeneity in the parameters describing the host response, and explore how it affects parasite evolution. We expect random heterogeneity to be a virtually ubiquitous aspect of the host-parasite interactions. For example, there may be stochastic variation in the initial parasite inoculum, and hosts may, as a result of being exposed to different antigenic and nutritional environments, have small differences in their specific immune responses following infection the evolution of virulence in heterogeneous populations. We find that contrary to conventional expectations RESULTS AND DISCUSSION Formulation of the Mathematical Model We employ a model introduced by (1) infection is initiated in a new host by a fixed inoculum, P 0 , and the parasite population grows exponentially in the absence of a specific immune response; (2) the presence of the parasite induces a specific immune response in the host, which grows by clonal expansion in a parasite-dependent manner and kills the parasite at a rate proportional to the product of the parasite and immune cells densities; (3) there is a lethal, or threshold, density of parasite, D, at which infection kills the host; and (4) the rate of transmission of the parasite from an infected host is proportional to the parasite density within the host, and we assume that the parasite is selected to maximize its total transmission from an infected host during the course of the acute infection. Given these assumptions, the rates of change in the densities of the parasite (P) and specific immune cells (X) will be given by the following equations: dP ϭ rP Ϫ hPX and where r and s are the maximum growth rates of the parasite and immune cells, respectively; h is the rate constant for clearance of the parasite by the immune response; and k is the parasite density that stimulates immune cells to grow at half their maximum rate. Because we are primarily concerned with the dynamics of parasites during acute infections, we ignore the contraction and memory phases of the immune response that occur following control and clearance of the parasite. Biologically the relative magnitudes of various parameters are 0 where X 0 ϭ X(0) is the precursor frequency (the initial number of immune cells specific to the parasite). The within-host dynamics of the parasites with different growth rates are illustrated in where e is the base of the natural logarithm. What consequences does this have for the evolution of virulence? The answer depends on how virulence is defined. We consider two commonly used measures for virulence, the LD 50 and the case mortality. (The LD 50 , or lethal dose 50, is the initial dose of the parasite required to kill 50% of infected hosts in a fixed period of time Evolution in a Heterogeneous Environment In the simplest model we assumed that, except for the parasite growth rate, r, all the parameters describing the parasite-host interaction are fixed. In this section we introduce random heterogeneity of the parameters describing the host response, and ask how the optimal growth rate, total transmission, and virulence level of the parasite change with the amount of heterogeneity present in the host population. Heterogeneity in host response to parasite pressure may take a number of (not mutually exclusive) forms. For example, hosts may vary in immune-cell activation rates (different k) , frequencies of precursor immune cells (different X 0 ), immunecell specificity to the parasite (h), and threshold densities at which parasite load becomes lethal (D). For simplicity, we will begin the analysis by introducing heterogeneity in only one parameter, the threshold density, D, at which the parasite kills the host. We do so by generating a distribution of threshold densities for a particular host population described by the probability density function f(D) such that f(D) dD is the probability that a given host has lethal density in a range (D, D ϩ dD). We consider two simple distributions of the probability density function f: uniform and gamma distributions. We characterize the degree of host heterogeneity by the coefficient of variation, CV ϭ /͗D͘, where ͗D͘ is the mean 2 ͙ and 2 is the variance of the threshold density. As in the simple model, we assume that different parasite strains differ only in their growth rate, r, and that r evolves so as to maximize the total transmission over the course of an infection. The average (expected) total transmission of a parasite with growth rate r in this heterogeneous host population is given by the integral of the product of the probability density function, f(D), and the instantaneous transmission rate, l(r, D) over the course of the infection: ͵ 0 Substituting l(r, D) from equation ͵ r 0 (This approximation has been used to generate the curves in Transmission and the optimal growth rate The first two panels of Evolution of virulence We now examine the effects of host heterogeneity on virulence evolution. We measure virulence in two different ways, as the case mortality, M, and the LD 50 . how the case mortality changes with increasing levels of host heterogeneity. For both uniform and gamma distributions of threshold density values, the case mortality increases from zero as heterogeneity increases. When threshold density, D, is uniformly distributed, there is an initial period when the heterogeneity is low during which there is no increase in the case mortality. When threshold densities are given by a gamma distribution, the case mortality increases almost linearly 218 VITALY V. GANUSOV ET AL. with increasing heterogeneity. The increase in case mortality is determined by two factors. The optimal growth rate, r opt , decreases, reducing the maximum density of the parasite within the host. However, heterogeneity in the lethal density, D, results in some fraction of the host population having a sufficiently low lethal density that they are killed before the immune response controls the parasite. In other words, the case mortality increases with increasing heterogeneity as a consequence of the following two changes. First, increasing heterogeneity in D (when the parasite growth rate, r, is fixed and is below the optimal value, r*, given by eq. 4) results in an increasing fraction of hosts having a peak parasitemia above the lethal density, D; this leads to increasing case mortality. Second, introducing heterogeneity changes the optimal growth rate r opt (r opt ϭ r* when heterogeneity is zero). All else being equal, decreases in r opt lead to a reduction in the peak parasitemia; this leads to decreasing case mortality. As heterogeneity increases from zero to higher values, these two changes pull the case mortality in opposite directions, and for nonlinear equations we find it difficult to intuit the net result. The results of our analysis show that the net effect is an increase in case mortality with increasing heterogeneity. Taken together, Heterogeneity in other parameters We now briefly describe the consequences of introducing heterogeneity in the other parameters describing the interaction of the parasite with the immune response. We do so by looking at how the introduction of heterogeneity in the other parameters changes the total transmission, the optimal growth rate of the parasite, and its virulence. (In this paper we restrict our analysis to the addition of heterogeneity in one parameter at a time and do not consider addition of heterogeneity in several parameters simultaneously.) As it can be seen in The addition of heterogeneity in the size of initial inoculum, P 0 , leads to much smaller changes in the total transmission, the optimal growth rate of the parasite, and its virulence. Although these changes are too small to be seen when plotted with changes in the other parameters, they follow the same trend (results not shown). For example, total transmission decreases monotonically with increasing heterogeneity in P 0 . In contrast, the addition of heterogeneity in the growth rate, r, leads to larger changes (in total transmission, optimal growth rate, and virulence) in comparison with those observed following the addition of heterogeneity to D. Only in the case of heterogeneity in the maximum growth rate of immune cells, s, we find that whereas the behavior is qualitatively similar to that for the other parameters at low levels of heterogeneity, it is qualitatively different when heterogeneity is large. The difference arises because, when heterogeneity in s is large, there are some hosts with very small s, and in these hosts the immune response develops very slowly and allows for prolonged transmission of parasites with low rates of growth. Consequently at very high levels of heterogeneity in s, the r opt declines, the total transmission begins to increase, and the case mortality drops. We note that when s is small the infection ceases to be an acute infection of short duration. Estimating Epidemiological Parameters In the introduction, we contrasted the within-host approach taken in the sections Formulation of the Mathematical Model and Evolution in a Heterogeneous Environment with the epidemiological approach commonly employed to study virulence evolution. In this section, we derive the connection between these two modeling approaches and demonstrate that the epidemiological parameters of parasite transmission depend explicitly on the within-host dynamics. From the withinhost dynamics, we can calculate the epidemiological parameters which define the basic reproductive rate of the parasite (eq. 1), namely the rate of parasite transmission from infected to susceptible hosts, , and the rates of parasite-induced host mortality, ␣, and recovery, v, and examine the resultant tradeoffs between these epidemiological parameters. The epidemiological parameters can be obtained from the within-host dynamics as follows. First, the basic reproductive number, R 0 , is proportional to the average number of parasites transmitted from an infected host over the course of acute infection, that is, R 0 (r) ϭ uL(r), where u is a coefficient of proportionality. Second, the transmission rate of a parasite with the growth rate r from a host with a lethal density D equals the total transmission of the parasite over the course of acute infection, l(r, D), divided by the duration of acute infection, ⌬(r, D): where l(r, D) is given by equation Consequences of heterogeneity in other parameters on the evolution of microparasites. We compare the consequences of heterogeneity in the different parameters for the total transmission (A), the optimal growth rate (B), case mortality (C), and LD 50 (D). , and s (-· · -) is modeled by a gamma distribution. Parameters are the same as in The mortality rate, ␣, equals the rate of parasite-induced host death following infection. If hosts die following infection, then ␣ is inversely proportional to the average duration of the infection. If hosts survive the infection then ␣ is zero. If m(r, D) equals the case mortality (derived in the Appendix) and ⌬(r, D) equals the duration of infection, we get Similarly, the recovery rate of hosts with a lethal density D infected with a parasite with growth rate r is The average recovery and mortality rates of the host population with heterogeneity described by the distribution f(D) are found similarly to the average transmission rate (r): where D* ϭ k (r/(heX 0 )) r/s Using the derived expressions for the epidemiological parameters we illustrate how , ␣, and v change as a function of the growth rate of the parasite r when host heterogeneity is held at a fixed level and the trade-offs between these parameters for parasites with different growth rates Summary In this paper we have used a simple mathematical model to analyze how host heterogeneity affects the within-host dynamics and evolution of microparasites in vertebrate hosts. We introduce a simple form of heterogeneity, the random heterogeneity that arises inevitably in host populations due to factors such as stochastic variation in the initial density of the parasite; the precursor frequency of immune cells specific for the parasite; as well as the parameters h, k, and D, which determine the within-host dynamics of the parasite. We find that: (1) parasites evolve to an intermediate growth rate, which changes with increasing heterogeneity; (2) the total transmission of the evolved parasite decreases as heterogeneity increases; (3) the observed pattern of virulence evolution is sensitive to the measure of virulence employed (case mortality vs. LD 50 , the former of which provides the 221 EVOLUTION OF VIRULENCE preferable measure); (4) in contrast to the generally accepted view, virulence does not necessarily decrease with increasing heterogeneity, and indeed is likely to increase with increasing heterogeneity; and (5) the parameters for epidemiological spread of the disease can be estimated from the within-host dynamics, and in doing so we can examine the trade-offs between these epidemiological parameters that result from the interaction of the parasite and the immune response of the host. Our finding that parasites should cause more damage to the host population when hosts are heterogeneous seems to contradict the widespread belief that the evolution of parasites in heterogeneous populations should select for less virulent parasites In reality we could find both types of heterogeneity. Random heterogeneity (i.e., small differences in parameters governing the host-parasite interactions) is expected to be present in any host population due to phenotypic (or other) differences between individual hosts. When this happens, our model predicts that virulence, measured by the case mortality rate, will increase with increasing random heterogeneity. Trade-off heterogeneity may be found in other cases. The evidence, suggesting that trade-off heterogeneity exists includes serial passage experiments in which the increase in virulence of parasites when they are serially passaged through one host type is frequently linked with a decrease in the parasites virulence in another host type (Ebert 1998). Ultimately, our understanding of virulence evolution will benefit from incorporation of both random and trade-off forms of heterogeneity into evolutionary models, so as to assess the relative importance of these two sources of variation. ACKNOWLEDGMENTS