Evolution of parasite transmission dispersion

Abstract

An open question in epidemiology is why transmission is often overdispersed, meaning that most new infections are driven by few infected individuals. For example, around 10% of COVID-19 cases cause 80% of new COVID-19 cases. This overdispersion in parasite transmission is likely driven by intrinsic heterogeneity among hosts, i.e. variable SARS-CoV-2 viral loads. However, host heterogeneity could also indirectly increase transmission dispersion by driving parasite adaptation. Specifically, transmission variation among hosts could drive parasite specialization to highly infectious hosts. Adaptation to rare, highly infectious hosts could amplify transmission dispersion by simultaneously decreasing transmission from common, less infectious hosts. This study considers whether increased transmission dispersion can be, in part, an emergent property of parasite adaptation to heterogeneous host populations. We develop a mathematical model using a Price equation framework to address this question that follows the epidemiological and evolutionary dynamics of a general host-parasite system. The results predict that parasite adaptation to heterogeneous host populations drives high transmission dispersion early in epidemics. Furthermore, parasite adaptation can maintain increased transmission dispersion at endemic equilibria if virulence differs between hosts in a heterogeneous population. More broadly, this study provides a framework for predicting how parasite adaptation determines transmission dispersion for emerging and re-emerging infectious diseases.

Keywords: epidemiology; evolution; parasite; superspreading.

Figure 1.

Large differences in host quality drive increased transmission dispersion ($vmr(R_e)$). Transmission dispersion is high when the host population is roughly equally split between high- and low-yield hosts (% ${\displaystyle S_L\approx 0.5}$, where % $S_L=s_L(t)/(s_H(t)+s_L(t))$) and equal to zero when the host population is entirely high- or low-yield hosts. In the case where the high-yield host is slightly more productive from the perspective of the parasite ($R_{eH}>R_{eL}$), $c_H=0.1,c_L=0,y_H=0.1,y_L=0.2$, while in the case where the high-yield host is much more productive from the perspective of the parasite (${\displaystyle R_{eH}\gg R_{eL}}$), $c_H=1,c_L=0,y_H=0.1,y_L=1$. For all cases: ${\displaystyle x=0.5,\lambda =50,\delta =0.02,\gamma =0.6,\rho ={10}^{-2},ϵ=0.25}$.

Figure 2.

The modelling framework follows the epidemiological dynamics of the host population (using a SI model [22]) and the evolutionary dynamics of the parasite within-host parasite growth rate (using the Price equation [23]). The epidemiological dynamics impact selection on the within-host parasite growth rate. The value of the within-host parasite growth rate impacts how quickly infected hosts transmit the parasite and die from the infection, thus impacting the epidemiological dynamics. The form of the Price equation used here ignores the impact of mutation. The plot in the bottom left shows an example of positive selection on the within-host growth rate as the trait is positively correlated with fitness. The plot in the bottom right shows an example of the epidemiological dynamics for high- and low-yield susceptible and infectious hosts.

Figure 3.

Epidemiological dynamics of a heterogeneous host population early (a,c) and late (b,d), with (a,b) and without evolution (c,d). (a,b) The density of susceptible high-yield ($S_H$) and low-yield hosts ($S_L$) and infectious high-yield ($I_H$) and low-yield hosts ($I_L$). (c,d) The total density of susceptible ($S$) and infectious ($I_H$) hosts with and without parasite evolution. In the absence of parasite evolution, the parasite within-host growth rate ($ϵ$) is set to 0.25 and does not change. Note that high- and low-yield susceptible host densities are identical when they have equal proportions in the host population ($p=0.5$ in equations (2.1a)–(2.1d)). $c_H=1,c_L=0.1,y_H=0.1,y_L=1,x=0.5,\lambda =50,\delta =0.02,\gamma =0.6,\rho ={10}^{-2},va{r}_H(ϵ)=va{r}_L(ϵ)=1$.

Figure 4.

Evolutionary dynamics of the within-host parasite growth rate ($ϵ$) early (a) and late (b). The trait value of the within-host parasite growth rate in high-yield (${ϵ}_H$) and low-yield hosts (${ϵ}_L$). Dotted black line shows the value of the within-host growth rate in the absence of adaptation ($ϵ=0.25$). $c_H=1,c_L=0.1,y_H=0.1,y_L=1,x=0.5,\lambda =50,\delta =0.02,\gamma =0.6,\rho ={10}^{-2},va{r}_H(ϵ)=va{r}_L(ϵ)=1$.

Figure 5.

Parasite adaptation drives higher parasite fitness and transmission dispersion. Both parasite fitness and transmission dispersion are highest early in epidemics when susceptible host density is also high. Parasite fitness of high- and low-yield hosts ($R_{eH},R_{eL}$) and transmission dispersion (${\displaystyle vmr(R_e)}$) over time. $c_H=1,c_L=0.1,y_H=0.1,y_L=1,x=0.5,\lambda =50,\delta =0.02,\gamma =0.6,\rho ={10}^{-2},va{r}_H(ϵ)=va{r}_L(ϵ)=1$.

Figure 6.

Transmission dispersion is highest when parasites adapt to host populations that are mostly composed of low-yield hosts. The variance-to-mean ratio of $R_e(t)$ at the endemic equilibrium as the percentage of susceptible low-yield hosts ($s_L$) in the system varies. $c_H=1,c_L=0.1,y_H=0.1,y_L=1,x=0.5,\lambda =50,\delta =0.02,\gamma =0.6,\rho ={10}^{-2},va{r}_H(ϵ)=va{r}_L(ϵ)=1$.

Figure 7.

The difference between transmission dispersion ($vmr(R_e)$) with and without parasite evolution is greatest when most hosts are born low yield (high $p$) and either high-yield hosts are much higher quality than low-yield hosts from the parasite's perspective (${\displaystyle y_L\ll y_H,c_H\gg c_L}$). $y_H=0.1,y_L=1,c_H=0.5,c_L=0$. All other parameters are the same as in figure 3.

Figure 8.

Phenotypic variance in the population of the within-host growth rate (var $ϵ$) controls how quickly adaptation occurs. Plots show the dynamics of within-host growth rate ($ϵ$) adaptation for different values of var $ϵ$ and demonstrate that $ϵ$ adapts more quickly and reaches higher values when var $ϵ$ is high. All parameters are the same as in figure 3.

Figure 9.

Variance in the within-host growth rate (var $ϵ$) impacts parasite adaptation and transmission dispersion. (a–c) Plots show how var $ϵ$ impacts peak within-host growth rates ($ϵ$), $R_e$ and transmission dispersion (vmr$(R_e)$) that occur early in the epidemic (see figure 5). (d–f) Plots show how var $ϵ$ impacts the within-host growth rate ($ϵ$), $R_e$ and transmission dispersion (vmr$(R_e)$) at the endemic equilibrium (see figure 5). All parameters are the same as in figure 3.

Figure 10.

Transmission dispersion (vmr$(R_e)$) is greatest when most hosts are born low yield and when virulence is low in high-yield hosts (small $y_H$). Low virulence in high-yield hosts results in a large difference in transmission potential between the high- and low-yield hosts. The difference in vmr${\displaystyle (R_e)}$ with and without parasite evolution is also the greatest when most hosts are born low yield (high $p$) and virulence is low in high-yield hosts (small $y_H$). Note that the baseline value for $y_H$ in the main text is 0.1. All other parameters are the same as in figure 3.

Figure 11.

Transmission dispersion (vmr$(R_e)$) is greatest when most hosts are born low yield and when the transmission set point is low in low-yield hosts (small $c_L$). A low transmission set point in low-yield hosts results in a large difference in transmission potential between the high- and low-yield hosts. The difference in vmr${\displaystyle (R_e)}$ with and without parasite evolution is also the greatest when most hosts are born low yield (high $p$) and the transmission set point is low in low-yield hosts (small $c_L$). Note that the baseline value for $c_L$ in the main text is 0.1. All other parameters are the same as in figure 3.

Figure 12.

Transmission dispersion (vmr$(R_e)$) is greatest when most hosts are born low yield and when the transmission concavity parameter ($x$) is large. Transmission increases from increased within-host growth rates ($ϵ$) approach a linear increase as the concavity parameter in the transmission function approaches 1. Thus, large $x$ results in a large difference in transmission potential between the high- and low-yield hosts. The same holds true for the difference in vmr${\displaystyle (R_e)}$ with and without parasite evolution. Note that the baseline value for $x$ in the main text is 0.5. All other parameters are the same as in figure 3.

Figure 13.

Transmission dispersion (vmr$(R_e)$) is greatest when most hosts are born low yield. The rate that new hosts are born ($\lambda$) has no impact on transmission dispersion in this parameter range. The same holds true for the difference in vmr${\displaystyle (R_e)}$ with and without parasite evolution. We also investigated how changes in $\lambda$ impact the epidemiological dynamics but do not include these results in the text because they were negligible. Note that the baseline value for $\lambda$ in the main text is 50. All other parameters are the same as in figure 3.

Figure 14.

Transmission dispersion (${\displaystyle \mathrm{v}\mathrm{m}\mathrm{r}(R_e)}$) is greatest when most hosts are born low yield. The natural host mortality rate ($\delta$) has no impact on transmission dispersion in this parameter range. The same holds true for the difference in vmr${\displaystyle (R_e)}$ with and without parasite evolution. Note that the baseline value for $\delta$ in the main text is 0.02. All other parameters are the same as in figure 3.

Figure 15.

Transmission dispersion (vmr$(R_e)$) is greatest when most hosts are born low yield. The host recovery rate ($\gamma$) has a small impact on transmission dispersion such that intermediate values for high $p$ decrease transmission dispersion slightly. The same holds true for the difference in vmr${\displaystyle (R_e)}$ with and without parasite evolution. Note that the baseline value for $\gamma$ in the main text is 0.6. All other parameters are the same as in figure 3.

Figure 16.

Transmission dispersion (vmr$(R_e)$) is greatest when most hosts are born low yield. The transmission scaling parameter ($\rho$) has no impact on transmission dispersion in this parameter range. The same holds true for the difference in vmr${\displaystyle (R_e)}$ with and without parasite evolution. Note that the baseline value for $\rho$ in the main text is ${10}^{-2}$. All other parameters are the same as in figure 3.