A model for epidemic dynamics in a community with visitor subpopulation

Abstract

With a five dimensional system of ordinary differential equations based on the SIR and SIS models, we consider the dynamics of epidemics in a community which consists of residents and short-stay visitors. Taking different viewpoints to consider public health policies to control the disease, we derive different basic reproduction numbers and clarify their common/different mathematical natures so as to understand their meanings in the dynamics of the epidemic. From our analyses, the short-stay visitor subpopulation could become significant in determining the fate of diseases in the community. Furthermore, our arguments demonstrate that it is necessary to choose one variant of basic reproduction number in order to formulate appropriate public health policies.

Keywords: Basic reproduction number; Epidemic dynamics; Mathematical model; Ordinary differential equations; Public health.

Fig. 1

The scheme of the model for the epidemic dynamics of a community with short-stay visitor subpopulation.

Fig. 2

Numerical examples of temporal variation of system (2). (a) $({\mathcal{R}}_{rr},{\mathcal{R}}_{vv},{\mathcal{R}}_{vr},{\mathcal{R}}_{rv})=(0.75,0.50,0.05,32.73)$; (b) $({\mathcal{R}}_{rr},{\mathcal{R}}_{vv},{\mathcal{R}}_{vr},{\mathcal{R}}_{rv})=(0.75,1.50,0.08,56.69)$. Commonly, $N_r=100000.00,N_v=100.00,\rho =0.14,M=20.00,$$(x_r\left(0\right),y_r\left(0\right),y_v\left(0\right))=(99990.0,10.0,0.0)$. ${\mathcal{R}}_{vr}:={\beta }_{vr}{N}_v/\rho ,$${\mathcal{R}}_{rv}:={\beta }_{rv}{N}_r{N}_v/M$.

Fig. 3

The dependence of the final size of susceptible resident population $x_r^{*}$ on the initial size of infective resident population y_r(0) and on β_rr. (a) ${\beta }_{rr}=2.0\times {10}^{-5},$${\mathcal{R}}_{rr}=10.0,$$(x_r\left(0\right),y_r\left(0\right),y_v\left(0\right))=(N_r-y_{r0},y_{r0},0.0)$; (b) $(x_r\left(0\right),y_r\left(0\right),y_v\left(0\right))=(99990.0,10.0,0.0)$. The horizontal axis in (b) shows the value of ${\mathcal{R}}_{rr}$ which is a function of β_rr as given by (4). Commonly, $N_r=100000.0,$$N_v=100.0,$$\rho =0.2,$$M=0.5,$${\beta }_{vr}=1.6\times {10}^{-4},$${\beta }_{rv}=1.0\times {10}^{-5},$${\beta }_{vv}=4.0\times {10}^{-5},$${\mathcal{R}}_{vv}=0.8,$${\mathcal{R}}_{vr}=0.08,$${\mathcal{R}}_{rv}=200.0$.

Fig. 4

The dependence of the final size of susceptible resident population $x_r^{*}$ on β_vv and on M. (a) ${\beta }_{rr}=2.0\times {10}^{-5},$${\mathcal{R}}_{rr}=10.0$; (b) ${\beta }_{rr}=3.0\times {10}^{-6},$${\mathcal{R}}_{rr}=1.50$; (c) ${\beta }_{rr}=1.5\times {10}^{-6},$${\mathcal{R}}_{rr}=0.75$. The horizontal axes show the values of ${\mathcal{R}}_{vv}$ which is a function of β_vv for the upper figures with $M=0.5$ and that of M for the lower ones with ${\beta }_{vv}=4.0\times {10}^{-5}$ as given by (5). Commonly, $N_r=100000.0,$$N_v=100.0,$$\rho =0.2,$${\beta }_{vr}=1.6\times {10}^{-4},$${\beta }_{rv}=1.0\times {10}^{-5},$${\mathcal{R}}_{vr}=0.08,$${\mathcal{R}}_{rv}=200.0,$$(x_r\left(0\right),y_r\left(0\right),y_v\left(0\right))=(99990.0,10.0,0.0)$.

Fig. 5

Decomposition of the basic reproduction numbers R_0∣r and R_0∣v defined by (8) and (9).

Fig. 6

Classification of the region $({\mathcal{R}}_{rr},{\mathcal{R}}_{vv})$ in terms of the values of R_0∣r and R_0∣v. (a) $\mathcal{B}<1$; (b) $\mathcal{B}=1$; (c) $\mathcal{B}>1$. The boundary corresponds to the set of $({\mathcal{R}}_{rr},{\mathcal{R}}_{vv})=({\mathcal{R}}_{rr}^{*},{\mathcal{R}}_{vv}^{*})$ defined in Corollary 5.1.1 with Theorem 5.1.

Fig. 7

Differences in the values of R_0∣r, R_0∣v and R_0∣c given by (8), (9) and (10) with $\mathcal{B}=2.0$. (a) ${\mathcal{R}}_{vv}=0.6$; (b) ${\mathcal{R}}_{rr}=0.1$. The three curves intersect when they take the value of unity.

Fig. 8

Numerical examples of the temporal variation of the system (2). (a) $(x_r\left(0\right),y_r\left(0\right),y_v\left(0\right))=(99990.0,10.0,0.0)$; (b) $(x_r\left(0\right),y_r\left(0\right),y_v\left(0\right))=(100000.0,0.0,1.0)$. Commonly, $\mathcal{B}=2.0,$${\mathcal{R}}_{rr}=0.4,$${\mathcal{R}}_{vv}=0.5,$${\mathcal{R}}_{rv}=16.90,$${\mathcal{R}}_{vr}=0.02,$$R_{0\mid r}=1.200,$$R_{0\mid v}=1.167,$$R_{0\mid c}=1.084$.