Modeling spillover dynamics: understanding emerging pathogens of public health concern

Abstract

The emergence of infectious diseases with pandemic potential is a major public health threat worldwide. The World Health Organization reports that about 60% of emerging infectious diseases are zoonoses, originating from spillover events. Although the mechanisms behind spillover events remain unclear, mathematical modeling offers a way to understand the intricate interactions among pathogens, wildlife, humans, and their shared environment. Aiming at gaining insights into the dynamics of spillover events and the outcome of an eventual disease outbreak in a population, we propose a continuous time stochastic modeling framework. This framework links the dynamics of animal reservoirs and human hosts to simulate cross-species disease transmission. We conduct a thorough analysis of the model followed by numerical experiments that explore various spillover scenarios. The results suggest that although most epidemic outbreaks caused by novel zoonotic pathogens do not persist in the human population, the rising number of spillover events can avoid long-lasting extinction and lead to unexpected large outbreaks. Hence, global efforts to reduce the impacts of emerging diseases should not only address post-emergence outbreak control but also need to prevent pandemics before they are established.

Keywords: Disease modeling; Emerging infectious diseases; Spillover.

Figure 1

Stochastic realizations and the analytical mean-field solution for the infected reservoir population. The baseline values of the model parameters are shown in Table 1 in the SM. (a) Shows the full spectrum for the infected reservoir population, whereas (b) provides a closer look into the stochastic fluctuations (thin lines) observed around the endemic equilibrium (bold line). The baseline parameter values used in these computations are listed in Supplementary Table  S1.

Figure 2

A typical stochastic realization for the dynamics of the infection prevalence (adding the asymptomatic A and hospitalized H classes) in a human population. For different spillover rates (a) $\tau ={10}^{-3}$, (b) $\tau ={10}^{-4}$, (c) $\tau ={10}^{-5}$, (d) $\tau ={10}^{-6}$, the outbreak in the human population is fully driven by spillover events (with $\beta =0$ and $R_0^h=0$). The baseline parameter values used in these computations are listed in Supplementary Table  S1.

Figure 3

Stochastic realizations (bold lines) and the analytic mean-field solution (thin lines) for the asymptomatic A class (blue) and the hospitalized H class (orange). In (a) the spillover rate is $\tau ={10}^{-5}$ and in (b) is $\tau ={10}^{-4}$. The dynamics correspond to stage III of pathogen emergence with $0<R_0^h\approx 0.9<1$, close to criticality. The baseline parameter values used in these computations are listed in Supplementary Table  S1.

Figure 4

(First row): One hundred stochastic realizations and the analytic mean-field solution for the asymptomatic class (blue) and the hospitalized class (orange). Different spillover rates are investigated. In (a–d) $\tau ={10}^{-5}$, in (b–e) $\tau ={10}^{-4}$ and in (c–f) $\tau ={10}^{-3}$. (Second Row): Peak time distribution for the corresponding stochastic realizations based on an ensemble of ${10}^4$ stochastic realizations. The vertical black line corresponds to the peak time of the outbreak obtained from the mean-field solution for each case, respectively. The dynamics correspond to stage IV of pathogen emergence with $1<R_0^h\approx 1.4$. The baseline parameter values used in these computations are listed in Supplementary Table  S1.

Figure 5

Distribution of the overall infected population ($H+A$), based on an ensemble of ${10}^4$ stochastic realizations for different times considering different values of the spillover transmission rate. Left column: The time $t_1$ is 50 days before the mean-field peak time. Middle column: The time $t_2$ is the day of the peak. Right column: The time $t_3$ is 50 days post-peak. The spillover transmission rate is $\tau =10e^{-5}$ for the top row, $\tau =10e^{-4}$ for the middle row, and $\tau =10e^{-4}$ for the bottom row. The vertical black line corresponds to the mean field prevalence for each case, respectively. Other parameters used in these computations are listed in Supplementary Table  S1.

Figure 6

Stochastic realizations and the analytic mean-field solution for the asymptomatic class (blue) and the hospitalized class (orange). In (a) the average duration of natural immunity is 180 days and in (b) is 360 days. Other parameters are as shown in Supplementary Table  S1.

Figure 7

Weekly laboratory-confirmed monkeypox cases in (a) Spain, (b) France, (c) Germany, and (d) the United Kingdom, from May to September 2022. The data is represented by bar plots. The mean-field solution (solid line) together with an ensemble of 500 stochastic realizations is shown for each country.