Severe population collapses and species extinctions in multihost epidemic dynamics

Most infectious diseases, including more than half of known human pathogens, are not restricted to just one host, yet much of the mathematical modeling of infections has been limited to a single species. We investigate consequences of a single epidemic propagating in multiple species and compare and contrast it with the endemic steady state of the disease. We use the two-species susceptible-infected-recovered model to calculate the severity of postepidemic collapses in populations of two-host species as a function of their initial population sizes, the times individuals remain infectious, and the matrix of infection rates. We derive the criteria for a very large, extinction-level, population collapse in one or both of the species. The main conclusion of our study is that a single epidemic could drive a species with high mortality rate to local or even global extinction provided that it is coinfected with an abundant species. Such collapse-driven extinctions depend on factors different than those in the endemic steady state of the disease.

Figure 1

Top panel summarizes the SIR model [Eqs. (1)] in which susceptible populations of two species with lower curve $S_1$(blue) and upper curve $S_2$(red) are infected by a single pathogen with (cross)infection rates ${\beta }_{11}={\beta }_{12}={\beta }_{22}=5$, ${\beta }_{21}=0$. Infected individuals $I_{1,2}$ die (transition to $D_{1,2}$) at rates ${\gamma }_1={\gamma }_2=1$. The bottom panel shows the simulated time courses of $S_1\left(t\right)$ (blue) and $S_2\left(t\right)$ (red) with initial conditions $S_1\left(0\right)=S_2\left(0\right)=1$ and $I_1\left(0\right)=0,I_2\left(0\right)={10}^{-6}$. In order for equations to have a long-term endemic steady state of the pathogen, we added a small birth and natural death terms as described in the text. Right and left red arrows, respectively, point to the predicted steady state of the second population $S_2\left(\text{steady}\text{state}\right)=1/5$, and its population, $S_2\left(\text{collapse}\right)\simeq exp(-5)$, immediately after the initial epidemic ran its course. The blue arrow points to a much more severe postepidemic collapse of the first species: $S_1\left(\text{collapse}\right)\simeq exp(-10)$.

Figure 2

Decimal logarithm of the collapse ratio ${log}_{10}[S_1\left(0\right)/S_1\left(\text{collapse}\right)]={\mathrm{\Gamma }}_1/ln\left(10\right)$ in the population 1 as a function of the two species population sizes (a) and infections rates (b). Panel (a) shows the collapse ratio as a function of the initial populations $S_1\left(0\right)$ and $S_2\left(0\right)$ in a system where ${\gamma }_1={\gamma }_2=1$ and ${\beta }_{11}={\beta }_{12}=0.3$, and ${\beta }_{22}={\beta }_{21}=3$. White line is the predicted epidemic threshold at which the largest eigenvalue of the collapse matrix $\widehat{C}$ is equal to 1. Yellow-to-red colors indicate likely extinction of the species 1 with the initial population of ${10}^5$ susceptible individuals. White dot marks initial population sizes $S_1\left(0\right)=1$ and $S_2\left(0\right)=10$ used in panel (b), which shows the decimal logarithm of the collapse ratio calculated for these initial population sizes and variable infection rates ${\beta }_{11}={\beta }_{12}$ and ${\beta }_{22}={\beta }_{21}$. White line marks the predicted epidemic threshold.

Figure 3

Collapse ratio $S_1\left(0\right)/S_1(\text{collapse}$ of the population of the species 1 in the case of uni-directional transmission: $C_{21}=0$, following an epidemic started with a very small number of infected individuals $[I_1\left(0\right)=I_2\left(0\right)={10}^{-6}]$. (a) Collapse ratio in the population of the species 1 with fixed intraspecies collapse factor $C_{11}=0.1$ as a function of the species 2 collapse number $C_{22}$ and cross-species collapse number $C_{12}$ quantifying disease transmission from species 2 to 1. White line is the predicted epidemic threshold below which the overall reproduction number falls below 1. (b) Collapse ratio the population of species 1, with a fixed value of $C_{22}=2$ and variable $C_{11}$ and $C_{12}$. There is no epidemic threshold in this case as the basic reproduction number in the species 2 is selected to be larger than 1 so that the epidemic would always be able to start.