Source-sink dynamics in a two-patch SI epidemic model with life stages and no recovery from infection

Abstract

This study presents a comprehensive analysis of a two-patch, two-life stage SI model without recovery from infection, focusing on the dynamics of disease spread and host population viability in natural populations. The model, inspired by real-world ecological crises like the decline of amphibian populations due to chytridiomycosis and sea star populations due to Sea Star Wasting Disease, aims to understand the conditions under which a sink host population can present ecological rescue from a healthier, source population. Mathematical and numerical analyses reveal the critical roles of the basic reproductive numbers of the source and sink populations, the maturation rate, and the dispersal rate of juveniles in determining population outcomes. The study identifies conditions for disease-free, endemic, and extinction equilibria in sink populations, emphasizing the potential for ecological and evolutionary mechanisms to facilitate coexistence or recovery. These findings provide insights into managing natural populations affected by disease, with implications for conservation strategies, such as the importance of maintaining reproductively viable refuge populations and considering the effects of dispersal and maturation rates on population recovery. The research underscores the complexity of host-pathogen dynamics in spatially structured environments and highlights the need for multi-faceted approaches to biodiversity conservation in the face of emerging diseases.

Figures (8)

• Figure 1: Transfer diagram for model (\ref{['eq:baseModel']}).
• Figure 2: Final infected (red) and susceptible (blue) proportions attained in both patches, varying values of $R_{0,1}$. Keeping the value of $R_{0,2}$ constant, with $R_{0,2}>1$. Note that when $R_{0,1}<1$, the disease-free equilibrium for the first patch is stable. When $R_{0,1}>1$, this equilibrium is not stable; this leads to an endemic equilibrium, given by the theoretical value of $I_1^*$ in proposition (\ref{['prop:endemic_equilibrium_patch_1']}). However, for larger values of $R_{0,1}$, the value of $I_1^*$ becomes negative, so the system converges to the extinction equilibrium.
• Figure 3: Final infected proportion in second patch for each $(R_{0,1}, R_{0,2})$ combination and an illustration of the three different types of equilibria regions for each case.
• Figure 4: (a): Maximum eigenvalue of $\mathcal{J}_1$ for each $R_{0,1}$ along with the infected population attained at the corresponding equilibrium point. We can see that all equilibria are stable in their corresponding $R_{0,1}$ regions. (b): Maximum eigenvalue of $\mathcal{J}_2$ for each $R_{0,2}$, with different curves for three values of $R_{0,1}$. Each curve corresponds to a vertical cut from Figure (\ref{['fig:r1-r2-effect-infected-patch-2']}).
• Figure 5: Infected at equilibria for both patches, depending on $\alpha$ (left) and $p$ (right). Other parameters fixed: $r=1, k=1, \beta_1=0.3, \beta_2=0.6, \mu_S=0.1, \mu_I=0.1$, $\alpha=0.6$ (for right panel) and $p=0.5$ (for left panel). Each graph is shown to be a horizontal cut of the corresponding heatmaps in Figure (\ref{['fig:ppalphagraph_matrix']}).

Theorems & Definitions (13)

• Remark 1
• Proposition 1
• Remark 2
• Proposition 2
• Remark 3
• Proposition 3
• Proposition 4
• Example 1: First Patch Analysis
• Example 2: Second Patch Analysis
• Example 3: Stability of the general endemic equilibrium