An SIS epidemic model with mass action infection mechanism in a patchy environment

Abstract

In this paper, we study an SIS (susceptible–infected–susceptible) epidemic model with mass action infection mechanism in a patchy environment. We first analyze the long‐time dynamics of the model in terms of the basic reproduction number scriptR0$\mathcal {R}_0$, and prove under certain conditions that the disease‐free equilibrium (DFE) is globally attractive if R0≤1$\mathcal {R}_0\le 1$ whereas the endemic equilibrium (EE) is globally attractive if R0>1$\mathcal {R}_0>1$. Then we establish the existence and uniqueness of EE provided R0>1$\mathcal {R}_0>1$, which is achieved, among other ingredients, by reducing the equilibrium problem into the one involving only the component of infected population. We further investigate the asymptotic profile of EE as the motility rate of the susceptible and/or infected population becomes small. We also perform the theoretical analysis on the corresponding model with linear birth/death effect. By comparing the findings of the models considered here with those of other closely related ones, our results suggest that the variation of total population, individual motility, infection mechanism, and spatial heterogeneity all can play significant roles in the dynamics of disease transmission, which may shed light on the practical implications for predicting the patterns of disease outbreak and designing optimal control strategies.

On the dynamics of an epidemic patch model with mass‐action transmission mechanism and asymmetric dispersal patterns

“…This result seems to be new even for the case when  is symmetric. (iii) Theorem 7 extends and improves [18,Theorem 8], which is for the case when  is symmetric.…”

“…Since ∑ 𝑗∈Ω (𝑆 * 𝑗 + 𝐼 * 𝑗 ) = 𝑁, 𝑰 * = (𝑁 − ∑ 𝑗∈Ω 𝑟 𝑗 )𝜶. Since (𝑺 * , 𝑰 * ) is independent of the chosen subsequence, the claim holds.□ Theorems 8, 9, and 10 extend and improve[18, Theorem 9], which is for the case when  is symmetric.…”

“…Let  be defined by (18). To emphasize the dependence of  on 𝑁, we will write it as  (𝑙, 𝑁), which is defined if (𝑙𝑁 − (𝒓∕𝜶) 𝑀 )𝜶 is a subsolution of ( 16).…”

“…This model has been studied in Ref. 18 when the connectivity matrix  = (𝐿 𝑖𝑗 ) is symmetric. In particular, the authors defined a basic reproduction number  0 and studied the global dynamics of the model if 𝑑 𝑆 = 𝑑 𝐼 or (𝛽 1 , … , 𝛽 𝑛 ) is a multiple of (𝛾 1 , … , 𝛾 𝑛 ).…”

“…\end{cases}} \end{equation}$$This model has been studied in Ref. 18 when the connectivity matrix $$\mathcal {L}=(L_{ij})$$ is symmetric. In particular, the authors defined a basic reproduction number $$\mathcal {R}_0$$ and studied the global dynamics of the model if $$d_S=d_I$$ or $$(\beta _1, \dots, \beta _n)$$ is a multiple of $$(\gamma _1, \dots,\gamma _n)$$.…”