Modeling the 2014–2015 Ebola Virus Disease Outbreaks in Sierra Leone, Guinea, and Liberia with Effect of High- and Low-risk Susceptible Individuals

Abstract

Ebola virus disease (EVD) is a rare but fatal disease of humans and other primates caused by Ebola viruses. Study shows that the 2014–2015 EVD outbreak causes more than 10,000 deaths. In this paper, we propose and analyze a deterministic model to study the transmission dynamics of EVD in Sierra Leone, Guinea, and Liberia. Our analyses show that the model has two equilibria: (1) the disease-free equilibrium (DFE) which is locally asymptotically stable when the basic reproduction number (${\mathcal {R}}_{0}$) is less than unity and unstable if it is greater than one, and (2) an endemic equilibrium (EE) which is globally asymptotically stable when ${\mathcal {R}}_0$ is greater than unity. Furthermore, the backward bifurcation occurs, a coexistence between a stale DFE and a stable EE even if the ${\mathcal {R}}_{0}$ is less than unity, which makes the disease control more strenuous and would depend on the initial size of subpopulation. By fitting to reported Ebola cases from Sierra Leone, Guinea, and Liberia in 2014–2015, our model has captured the epidemic patterns in all three countries and shed light on future Ebola control and prevention strategies.

Appendices

Appendices

The Proof of Theorem 3.5 Using Direct Lyapunov Function Theory

Proof

Here, we adopt previous technique to prove Theorem 3.5 (see, for instance, Roop-O et al. 2015; Sun et al. 2017; Yang et al. 2017).

Firstly, we define a Lyapunov function given below:

$$\begin{aligned} \begin{aligned} V(t)&=n_1\bigg (S_1-S_1^{*}-S_1^{*}\ln \frac{S_1}{S_1^{*}}\bigg )+ n_2\bigg (S_2-S_2^{*}-S_2^{*}\ln \frac{S_2}{S_2^{*}}\bigg )\\&\quad + n_3\bigg (E-E^{*}-E^{*}\ln \frac{E}{E^{*}}\bigg )\\&\quad + n_4\bigg (I-I^{*}-I^{*}\ln \frac{I}{I^{*}}\bigg )+ n_5\bigg (D-D^{*}-D^{*}\ln \frac{D}{D^{*}}\bigg ). \end{aligned} \end{aligned}$$

(A-1)

Therefore, the derivative of the Lyapunov function (A-1) calculated along solutions of model (1) is given by

$$\begin{aligned} \begin{aligned} {\dot{V}}(t)&=n_1\bigg (1-\frac{S_1^{*}}{S_1}\bigg ) \dot{S_1}+n_2\bigg (1-\frac{S_2^{*}}{S_2}\bigg )\dot{S_2}+ n_3\bigg (1-\frac{E^{*}}{E}\bigg ){\dot{E}} \\&\quad +n_4\bigg (1-\frac{I^{*}}{I}\bigg ) {\dot{I}}+n_5\bigg (1-\frac{D^{*}}{D}\bigg ){\dot{D}}. \end{aligned} \end{aligned}$$

(A-2)

By direct computation from Eq. (A-2), we have

$$\begin{aligned} \begin{aligned} n_1\bigg (1-\frac{S_1^{*}}{S_1}\bigg )\dot{S_1}&=\bigg (1-\frac{S_1^{*}}{S_1}\bigg )\bigg (\rho \pi -\lambda S_1-\mu S_1\bigg )\\&=n_1\bigg (1-\frac{S_1^{*}}{S_1}\bigg )\bigg (\lambda ^{*}S_1^*+\mu S_1^*-\lambda S_1-\mu S_1\bigg )\\&=n_1\lambda ^* S_1^*\bigg (1-\frac{S_1^*}{S_1}\bigg )\bigg (1-\frac{\lambda S_1}{\lambda ^* S_1^*}\bigg )-\mu \frac{(S_1-S_1^*)^2}{S_1}\\&\le n_1\lambda ^* S_1^* \bigg (1-\frac{\lambda S_1}{\lambda ^* S_1^*}-\frac{S_1^*}{S_1}+\frac{\lambda }{\lambda ^*}\bigg ), \end{aligned} \end{aligned}$$

(A-3)

and

$$\begin{aligned} \begin{aligned} n_2\bigg (1-\frac{S_2^{*}}{S_2}\bigg )\dot{S_2}&=\bigg (1-\frac{S_2^{*}}{S_2}\bigg )\bigg ((1-\rho )\pi -v\lambda S_2-\mu S_2\bigg )\\&=n_2\bigg (1-\frac{S_2^{*}}{S_2}\bigg )\bigg (v\lambda ^{*}S_2^*+\mu S_2^*-v\lambda S_2-\mu S_2\bigg )\\&=n_2v\lambda ^* S_2^*\bigg (1-\frac{S_2^*}{S_2}\bigg )\bigg (1-\frac{\lambda S_2}{\lambda ^* S_2^*}\bigg )-\mu \frac{(S_2-S_2^*)^2}{S_2}\\&\le n_2v\lambda ^* S_2^* \bigg (1-\frac{\lambda S_2}{\lambda ^* S_1^*}-\frac{S_2^*}{S_2}+\frac{\lambda }{\lambda ^*}\bigg ), \end{aligned} \end{aligned}$$

(A-4)

and

$$\begin{aligned} \begin{aligned} n_3\bigg (1-\frac{E^{*}}{E}\bigg ){\dot{E}}&=\bigg (1-\frac{E^{*}}{E}\bigg )\bigg (\lambda S_1+v\lambda S_2-g_1 E\bigg )\\&=n_3\bigg (1-\frac{E^{*}}{E}\bigg )\bigg (\lambda S_1+v\lambda S_2-(\lambda ^* S_1^*+v\lambda ^* S_2^*) \frac{E}{E^*}\bigg )\\&=n_3\lambda ^* S_1^* \bigg (1-\frac{E^*}{E}\bigg )\bigg (\frac{\lambda S_1}{\lambda ^* S_1^*}-\frac{E}{E^*}\bigg ) \\&\quad +v\lambda ^* S_2^* \bigg (1-\frac{E^*}{E}\bigg )\bigg (\frac{\lambda S_2}{\lambda ^* S_2^*}-\frac{E}{E^*}\bigg )\\&=n_3\lambda ^* S_1^* \bigg (\frac{\lambda S_1}{\lambda ^* S_1^*}-\frac{E}{E^*}-\frac{\lambda S_1 E^*}{\lambda ^* S_1^* E}+1\bigg ) \\&\quad +v\lambda ^* S_2^* \bigg (\frac{\lambda S_2}{\lambda ^* S_2^*}-\frac{E}{E^*}-\frac{\lambda S_2 E^*}{\lambda ^* S_2^* E}+1\bigg ), \end{aligned} \end{aligned}$$

(A-5)

and

$$\begin{aligned} \begin{aligned} n_4\bigg (1-\frac{I^{*}}{I}\bigg ){\dot{I}}&=\bigg (1-\frac{I^{*}}{I}\bigg )\bigg (\sigma E-g_2 I\bigg )\\&=n_4\bigg (1-\frac{I^{*}}{I}\bigg )\bigg (\sigma E-\sigma E^*\frac{I}{I^*}\bigg )\\&=n_4\sigma E^*\bigg (1-\frac{I^{*}}{I}\bigg )\bigg (\frac{E}{E^*}-\frac{I}{I^*}\bigg )\\&=n_4\sigma E^* \bigg (\frac{E}{E^*}-\frac{I}{I^*}-\frac{I^* E}{I E^*}+1\bigg ), \end{aligned} \end{aligned}$$

(A-6)

and

$$\begin{aligned} \begin{aligned} n_5\bigg (1-\frac{D^{*}}{D}\bigg ){\dot{D}}&=\bigg (1-\frac{D^{*}}{D}\bigg )\bigg (f\gamma I -\theta D\bigg )\\&=n_5\bigg (1-\frac{D^{*}}{D}\bigg )\bigg (f\gamma I -f\gamma I^*\frac{D}{D^*}\bigg )\\&=n_5f\gamma I^*\bigg (1-\frac{D^{*}}{D}\bigg )\bigg (\frac{I}{I^*}-\frac{D}{D^*}\bigg )\\&=n_5f\gamma I^* \bigg (\frac{I}{I^*}-\frac{D}{D^*}-\frac{D^* I}{D I^*}+1\bigg ). \end{aligned} \end{aligned}$$

(A-7)

Substituting $n_1=n_2=n_3=1$, $n_4=\frac{\lambda ^*}{\sigma E^*}(S_1^*+vS_2^*)$, and $n_5=\frac{\lambda ^*S_2^*}{f\gamma I^*}$, and Eqs. (A-3)–(A-7) into Eq. (A-2), we have

$$\begin{aligned} \begin{aligned} {\dot{V}}(t) \le&\, \lambda ^* S_1^*\bigg (2-\frac{S_1^*}{S_1}-\frac{E}{E^*}-\frac{\lambda S_1 E^*}{\lambda ^* S_1^* E}+\frac{\lambda }{\lambda ^*}\bigg )\\&\quad +v\lambda ^* S_2^*\bigg (2-\frac{S_2^*}{S_2}-\frac{E}{E^*}-\frac{\lambda S_2 E^*}{\lambda ^* S_2^* E}+\frac{\lambda }{\lambda ^*}\bigg )\\&\quad +\lambda ^* S_1^*\bigg (\frac{E}{E^*}-\frac{I}{I^*}-\frac{I^* E}{I E^*}+1\bigg )\\&\quad +v\lambda ^* S_2^*\bigg (\frac{E}{E^*}-\frac{I}{I^*}-\frac{I^* E}{I E^*}+1\bigg )\\&\quad +v\lambda ^*S_2^*\bigg (\frac{I}{I^*}-\frac{D}{D^*}-\frac{D^* I}{D I^*}+1\bigg ). \end{aligned} \end{aligned}$$

(A-8)

Suppose a function is define as $u(x)=1-x+\ln x$, then, if $x>0$ it leads to $u(x) \le 0$. Also, if $x=1$, then $u(x)=0$. This implies that $x-1\ge \ln (x)$ for any $x>0$ (Musa et al. 2019; Roop-O et al. 2015; Yang et al. 2017; Musa et al. 2020; Sun et al. 2017).

By using the above definition, direct calculation from Eq. (A-8), and conditions (i) and (ii), we have

$$\begin{aligned} \begin{aligned}&\bigg (2-\frac{S_1^*}{S_1}-\frac{E}{E^*}-\frac{\lambda S_1 E^*}{\lambda ^* S_1^* E}+\frac{\lambda }{\lambda ^*}\bigg )\\&\quad =\bigg (-\left( 1-\frac{\lambda }{\lambda ^*}\right) \left( 1-\frac{I \lambda ^*}{I^*\lambda }\right) +3-\frac{S_1^*}{S_1}-\frac{\lambda S_1 E^*}{\lambda ^* S_1^*E}-\frac{I\lambda ^*}{I^* \lambda }-\frac{E}{E^*}+\frac{I}{I^*}\bigg )\\&\quad \le \bigg (-\left( \frac{S_1^*}{S_1}-1\right) -\left( \frac{\lambda S_1 E^*}{\lambda ^* S_1^*E}-1\right) -\left( \frac{I\lambda ^*}{I^*\lambda }-1\right) -\frac{E}{E^*}+\frac{I}{I^*}\bigg )\\&\quad \le \bigg (-\ln \left( \frac{S_1^*}{S_1}\frac{\lambda S_1 E^*}{\lambda ^* S_1^* E}\frac{I\lambda ^*}{I^* \lambda }\right) -\frac{E}{E^*}+\frac{I}{I^*}\bigg )\\&\quad =\bigg (\frac{I}{I^*}-\ln \left( \frac{I}{I^*}\right) +\ln \left( \frac{E}{E^*}\right) -\frac{E}{E^*}\bigg ). \end{aligned} \end{aligned}$$

(A-9)

By using the above definition, direct calculation from Eq. (A-8), and conditions (i) and (ii), we have

$$\begin{aligned} \begin{aligned}&\bigg (2-\frac{S_2^*}{S_2}-\frac{E}{E^*}-\frac{\lambda S_2 E^*}{\lambda ^* S_2^* E}+\frac{\lambda }{\lambda ^*}\bigg )\\&\quad =\bigg (-\left( 1-\frac{\lambda }{\lambda ^*}\right) \left( 1-\frac{D \lambda ^*}{D^*\lambda }\right) +3-\frac{S_2^*}{S_2}-\frac{\lambda S_2 E^*}{\lambda ^* S_2^*E}-\frac{D\lambda ^*}{D^* \lambda }-\frac{E}{E^*}+\frac{D}{D^*}\bigg )\\&\quad \le \bigg (-\left( \frac{S_2^*}{S_2}-1\right) -\left( \frac{\lambda S_2 E^*}{\lambda ^* S_2^*E}-1\right) -\left( \frac{D\lambda ^*}{D^*\lambda }-1\right) -\frac{E}{E^*}+\frac{D}{D^*}\bigg )\\&\quad \le \bigg (-\ln \left( \frac{S_2^*}{S_2}\frac{\lambda S_2 E^*}{\lambda ^* S_2^* E}\frac{D\lambda ^*}{D^* \lambda }\right) -\frac{E}{E^*}+\frac{D}{D^*}\bigg )\\&\quad =\bigg (\frac{D}{D^*}-\ln \left( \frac{D}{D^*}\right) +\ln \left( \frac{E}{E^*}\right) -\frac{E}{E^*}\bigg ). \end{aligned} \end{aligned}$$

(A-10)

Also from Eq. (A-8), we have

$$\begin{aligned} \begin{aligned} \frac{E}{E^*}-\frac{I}{I^*}-\frac{I^* E}{I E^*}+1&=\bigg (u\bigg (\frac{I^*E}{IE^*}\bigg )+\frac{E}{E^*}-\ln \bigg (\frac{E}{E^*}\bigg )-\frac{I}{I^*}+\ln \bigg (\frac{I}{I^*}\bigg )\bigg ) \\&\quad \le \frac{E}{E^*}-\ln \bigg (\frac{E}{E^*}\bigg )+\ln \bigg (\frac{I}{I^*}\bigg )-\frac{I}{I^*}. \end{aligned} \end{aligned}$$

(A-11)

Similarly,

$$\begin{aligned} \begin{aligned} \frac{I}{I^*}-\frac{D}{D^*}-\frac{D^* I}{D I^*}+1&=\bigg (u\bigg (\frac{D^*I}{DI^*}\bigg )+\frac{I}{I^*}-\ln \bigg (\frac{I}{I^*}\bigg )-\frac{D}{D^*}+\ln \bigg (\frac{D}{D^*}\bigg )\bigg ) \\&\quad \le \frac{I}{I^*}-\ln \bigg (\frac{I}{I^*}\bigg )+\ln \bigg (\frac{D}{D^*}\bigg )-\frac{D}{D^*}. \end{aligned} \end{aligned}$$

(A-12)

Hence,

$$\begin{aligned} \begin{aligned} {\dot{V}}(t) \,=\,&\lambda ^* S_1^*\bigg (\frac{I}{I^*}-\ln \bigg (\frac{I}{I^*}\bigg )+\ln \bigg (\frac{E}{E^*}\bigg )-\frac{E}{E^*}\bigg )\\&+ v\lambda ^* S_2^*\bigg (\frac{D}{D^*}-\ln \bigg (\frac{D}{D^*}\bigg )+\ln \bigg (\frac{E}{E^*}\bigg )-\frac{E}{E^*}\bigg )\\&+ \lambda ^* S_1^*\bigg (\frac{E}{E^*}-\ln \bigg (\frac{E}{E^*}\bigg )+\ln \bigg (\frac{I}{I^*}\bigg )-\frac{I}{I^*}\bigg )\\&+ v\lambda ^* S_2^*\bigg (\frac{E}{E^*}-\ln \bigg (\frac{E}{E^*}\bigg )+\ln \bigg (\frac{I}{I^*}\bigg )-\frac{I}{I^*}\bigg )\\&+ v\lambda ^* S_2^*\bigg (\frac{I}{I^*}-\ln \bigg (\frac{I}{I^*}\bigg )+\ln \bigg (\frac{D}{D^*}\bigg )-\frac{D}{D^*}\bigg ). \end{aligned} \end{aligned}$$

(A-13)

Equations (A-3)–(A-13) ensure that $\dot{V(t)}\le 0$. Further, the equality $\frac{\mathrm{d}V}{\mathrm{d}t}=0$ holds only if $S_1=S_1^{*}$, $S_2=S_2^{*}$$E=E^{*}$, $I=I^{*}$, and $D=D^{*}$. Thus, the endemic equilibrium state (5) is the only positive invariant set to the system (1) contained entirely in $\biggl \{(S_1, S_2, E, I, D) \in \Omega : S_1=S_1^{*}, S_2=S_2^{*}, E=E^{*}, I=I^{*}, D=D^{*} \biggr \}$. Hence, it follows from the LaSalle’s invariance principle (LaSalle 1976) that every solutions to the Eq. (1) with initial conditions in $\Omega$ converge to endemic equilibrium point, $\Gamma ^{*}$, as $t\rightarrow \infty$. Thus, the positive endemic equilibrium is globally asymptotically stable. $\square$