A simple epidemic model for semi-closed community reveals the hidden outbreak risk in nursing homes, prisons, and residential universities

Abstract

We develop a general SIS model to study the epidemic transmission in such semi-closed communities. The community population is divided into susceptible and infected in terms of the infection state, and concerning the physical structure of the crowd, they are classified into mobile and fixed individuals. The mobile individuals can be inside or outside the community, while the fixed individuals can be only inside the community. There are fixed infection sources outside the community, measuring the epidemic severity in society. We attribute the spreading to two reasons: (i) clustered infection among the community population and (ii) the epidemic in society spreading to the community population. We discuss the model in two cases. In the first case, the epidemic spreads in society, such that reasons (i) and (ii) work together. The results show that concerning fixed individuals (e.g. the elderly in nursing homes), a more closed community always promotes the infection. In the second case, there is no epidemic spreading in society, such that only reason (i) works. The results show that restricting all individuals to the community produces equivalent consequences as allowing them going outside the community. We should evenly distribute individuals inside and outside to form isolation. A counterexample is residential universities implementing closed management, where only students are restricted to campus. The model shows such management may lead to severe epidemics, and to prevent the epidemic outbreaks, students should have free access to being on or off campus.

Keywords: COVID-19; Closed management; Clustered infection; SIS model; Semi-closed community.

Fig. 1

The schematic diagram of population classification and epidemic transmission direction. The arrows indicate the direction of transmission of the epidemic. $I_0$, $I_1$, $S_0$, $S_1$ are independent variables. $N_0$, $N_1$ are auxiliary variables, calculated by $\theta$, N. $I_0+S_0$ is constrained by $N_0$; $I_1+S_1$ is constrained by $N_1$. I, S are dependent variables, calculated by $I_0$, $I_1$ or $S_0$, $S_1$. $I_c$ is an input parameter

Fig. 2

When $\alpha =0.1$, $\mu =0.05$, $N=1$, $\theta =0.5$, $\epsilon =0.5$, a the analytic curves of functions $I_1^{\ast }=g_1(I_0^{\ast })$ and $I_0^{\ast }=g_0(I_1^{\ast })$ with $I_c=0.1$ and $I_c=0.9$; b1 time evolution of indexes $p{I}_0(t)$, $p{I}_1(t)$ and pI(t) when $I_c=0.1$; b2 time evolution of indexes $p{I}_0(t)$, $p{I}_1(t)$ and pI(t) when $I_c=0.9$. The intersection coordinate of $g_1(I_0^{\ast })$ and $g_0(I_1^{\ast })$ is the equilibrium point. The theoretical prediction is consistent with numerical simulations under different $I_c$

Fig. 3

When $I_c=1.2$, $\alpha =0.1$, $\mu =0.05$, $N=1$, the heat maps depicting different indexes as binary functions of $\theta$ and $\epsilon$. The numerical solution of ${\theta }^{\ast }(\epsilon )$ and ${\epsilon }^{\ast }(\theta )$ are directly marked in the heat maps. a $p{I}^{\ast (1)}$. b1 $p{I}_0^{\ast (1)}$. b2 $p{I}_1^{\ast (1)}$. In this case, the epidemic in society is relatively severe

Fig. 4

When $I_c=0.5$, $\alpha =0.1$, $\mu =0.05$, $N=1$, the heat maps depicting different indexes as binary functions of $\theta$ and $\epsilon$. The numerical solution of ${\theta }^{\ast }(\epsilon )$ and ${\epsilon }^{\ast }(\theta )$ are directly marked in the heat maps. a $p{I}^{\ast (1)}$. b1 $p{I}_0^{\ast (1)}$. b2 $p{I}_1^{\ast (1)}$. In this case, the epidemic in society is mild

Fig. 5

When $I_c=0.1$, $\alpha =0.1$, $\mu =0.05$, $N=1$, the heat maps depicting different indexes as binary functions of $\theta$ and $\epsilon$. The numerical solution of ${\theta }^{\ast }(\epsilon )$ and ${\epsilon }^{\ast }(\theta )$ are directly marked in the heat maps. a $p{I}^{\ast (1)}$. b1 $p{I}_0^{\ast (1)}$. b2 $p{I}_1^{\ast (1)}$. In this case, the epidemic in society is not severe

Fig. 6

When $I_c=0$, $\mu =0.05$, $N=1$, $\theta =0.5$, $\epsilon =0.5$, a the analytic curves of functions $I_1^{\ast }=g_1(I_0^{\ast })$ and $I_0^{\ast }=g_0(I_1^{\ast })$ with $\alpha =0.1$; b1 time evolution of indexes $p{I}_0(t)$, $p{I}_1(t)$ and pI(t) when $\alpha =0.1$ (${\mathcal{R}}_0=1.3090>1$); b2 time evolution of indexes $p{I}_0(t)$, $p{I}_1(t)$ and pI(t) when $\alpha =0.05$ (${\mathcal{R}}_0=0.6545<1$). The intersection coordinates of $g_1(I_0^{\ast })$ and $g_0(I_1^{\ast })$ are the equilibrium points. The theoretical prediction is consistent with numerical simulations under different ${\mathcal{R}}_0$

Fig. 7

When $I_c=0$, $\alpha =0.1$, $\mu =0.05$, $N=1$, the heat maps depicting different indexes as binary functions of $\theta$ and $\epsilon$. The numerical solution of ${\theta }^{\ast }(\epsilon )$ and ${\epsilon }^{\ast }(\theta )$ are directly marked in the heat maps. a $p{I}^{\ast (1)}$ or $p{I}^{\ast (0)}$. b1 $p{I}_0^{\ast (1)}$ or $p{I}_0^{\ast (0)}$. b2 $p{I}_1^{\ast (1)}$ or $p{I}_1^{\ast (0)}$. In this case, there is no epidemic in society