$R_0$ May Not Tell Us Everything: Transient Disease Dynamics of Some SIR Models Over Patchy Environments

Abstract

This paper examines the short-term or transient dynamics of SIR infectious disease models in patch environments. We employ reactivity of an equilibrium and amplification rates, concepts from ecology, to analyze how dispersals/travels between patches, spatial heterogeneity, and other disease-related parameters impact short-term dynamics. Our findings reveal that in certain scenarios, due to the impact of spatial heterogeneity and the dispersals, the short-term disease dynamics over a patch environment may disagree with the long-term disease dynamics that is typically reflected by the basic reproduction number. Such an inconsistence can mislead the public, public healthy agencies and governments when making public health policy and decisions, and hence, these findings are of practical importance.

Appendix

1.1 The Dependence of $\lambda _{\max }$ and $\lambda _{\min }$ for SIR Epidemic Patch Model

Taking partial derivatives of (11) with respect to $\Gamma _0^{(i)}$ for $i \in \{1, \, 2\}$, we have

$$\begin{aligned} \dfrac{\partial \lambda _{\max }}{\partial \Gamma _0^{(1)}} = \dfrac{\partial \lambda _{\min }}{\partial \Gamma _0^{(2)}} = \dfrac{M+\sqrt{M^2+D^2}}{2\sqrt{M^2+D^2}} \quad \text {and} \quad \dfrac{\partial \lambda _{\max }}{\partial \Gamma _0^{(2)}} = \dfrac{\partial \lambda _{\min }}{\partial \Gamma _0^{(1)}} = \dfrac{-M+\sqrt{M^2+D^2}}{2\sqrt{M^2+D^2}} \end{aligned}$$

where $M:=m_1-m_2=\Gamma _0^{(1)}-d_{21}^I-\Gamma _0^{(2)}+d_{12}^I$ and $D=d_{12}^I+d_{21}^I>0$. All of these partial derivatives are positive since

$$\begin{aligned} \sqrt{M^2+D^2} > |M|. \end{aligned}$$

(17)

With respect to travel rates, the partial derivative of $\lambda _{\max }$,

$$\begin{aligned} \dfrac{\partial \lambda _{\max }}{\partial d_{21}^I} = \dfrac{D-M-\sqrt{M^2+D^2}}{2\sqrt{M^2+D^2}}, \end{aligned}$$

is positive if $M<0$ and is negative if $M>0$, and,

$$\begin{aligned} \dfrac{\partial \lambda _{\max }}{\partial d_{12}^I} = \dfrac{D+M-\sqrt{M^2+D^2}}{2\sqrt{M^2+D^2}}, \end{aligned}$$

is positive if $M>0$ and is negative if $M<0$. As for $\lambda _{\min }$, the partial derivatives are

$$\begin{aligned} \dfrac{\partial \lambda _{\min }}{\partial d_{21}^I} = \dfrac{-D+M-\sqrt{M^2+D^2}}{2\sqrt{M^2+D^2}} \quad \text {and} \quad \dfrac{\partial \lambda _{\min }}{\partial d_{12}^I} = \dfrac{-D-M-\sqrt{M^2+D^2}}{2\sqrt{M^2+D^2}}, \end{aligned}$$

which are both negative by the inequality (17).