Assessing the impact of the environmental contamination on the transmission of Ebola virus disease (EVD)

Abstract

This paper deals with the following biological question: how influential is the environmental contamination on the transmission of EVD? Based on the works in (Bibby et al., Environ Sci Technol Lett 2:2–6, 2015; Leroy et al., Nature 438: 575–576, 2005; World Health Organization. Unprecedented number of medical staff infected with Ebola), we design a new mathematical model to address this question by assessing the effect of the Ebola virus contaminated environment on the dynamical transmission of EVD. The formulated model captures two infection pathways through both direct human-to-human transmission and indirect human-to-environment-to-human transmission by incorporating the environment as a transition and/or reservoir of Ebola viruses. We compute the basic reproduction number ${\mathcal {R}}^{env}_0$ for the model with environmental contamination and prove that the disease-free equilibrium is globally asymptotically stable (GAS) whenever ${\mathcal {R}}^{env}_0 \le 1$. When ${\mathcal {R}}^{env}_0 > 1$, we show that the said model has a unique endemic equilibrium which is GAS. Similar results hold for the free environmental contamination sub-model (without the incorporation of the indirect transmission). More precisely, for the latter model, calculate the corresponding basic reproduction number ${\mathcal {R}}^{h}_0$ and establish the GAS of the disease-free and endemic equilibria, whenever ${\mathcal {R}}^{h}_0 \le 1$ and ${\mathcal {R}}^{h}_0 > 1$, respectively. At the endemic level, we show that the number of infected individuals for the full model with the environmental contamination is greater than the corresponding number for the free environmental contamination sub-model. In conjunction with the inequality ${\mathcal {R}}^{h}_0 < {\mathcal {R}}^{env}_0$, our finding suggests a negative answer to the biological question under investigation, i.e. the contaminated environment plays a detrimental role on the transmission dynamics of EVD by increasing the endemic level and/or the severity of the outbreak. Therefore, it is natural to implement a control strategy which aim at reducing the severity of the disease by providing adequate hygienic living conditions, educate populations at risk to follow rigorously those basic hygienic conditions as well as ask them avoid contact with suspected contaminated objects. Further, we perform numerical simulations to support the theory.

Appendix: Proof of Theorem 3.8

To establish Theorem 3.8 when ${\mathcal {R}}^{env}_0 >1$ which shows the local stability of subsystem (2.9), we used the center manifold theory proposed in [15]. To this end, we introduce the following change of variables: $x_1=S_h,\, x_2= E_h, \, x_3= I_h, \, x_4= R_h,\, x_5 = V$. Therefore, $x'_1 = S'_h, \, x'_2 = E'_h, \,x'_3 = I'_h, \,x'_4 = R'_h, \,x'_5 = V'$. The disease-free equilibrium of subsystem (2.9) becomes $x^*_0 = ({\varLambda }_h/\mu _h,0,0,0,0)$.

Let $\sigma _v > 0$ be the non-negative real numbers such that $\beta _{hv}=\sigma _v\beta _{hh}$, , then the basic reproduction number ${\mathcal {R}}^{env}_0$ becomes

$$\begin{aligned} {\mathcal {R}}^{env}_0 = \dfrac{\beta _{hh}\left( {\varPhi }_h\omega K\mu _h\mu _v + \alpha _h\sigma _v{\varLambda }_h\omega \right) }{K\mu _h\mu _v(\mu _h+\omega )(\mu _h+\gamma )}. \end{aligned}$$

Let $\beta _{hh}=\phi$   be the bifurcation parameter, then solving the equation   ${\mathcal {R}}^{env}_0 =1$ for $\beta _{hh}$ yields

$$\begin{aligned} \beta _{hh} = \phi = \beta ^*_{hh} = \dfrac{K\mu _h\mu _v(\mu _h+\omega )(\mu _h+\gamma )}{{\varPhi }_h\omega K\mu _h\mu _v + \alpha _h\sigma _v{\varLambda }_h\omega }. \end{aligned}$$

Notice that ${\mathcal {R}}^{env}_0 > 1$ if and only if $\beta _{hh} > \beta ^*_{hh}$.

With these notations, system (2.9) takes the form:

$$\begin{aligned} \left\{ \begin{array}{l} f_1 := x'_1(t) = {\varLambda }_h - \dfrac{\phi {\varPhi }_hx_1x_3}{ x_1+ x_2+x_3+x_4} - \dfrac{\phi \sigma _v x_1 x_5}{K + x_5}- \mu _h x_1, \\ f_2 := x'_2(t) = \dfrac{\phi {\varPhi }_hx_1x_3}{ x_1+ x_2+x_3+x_4} + \dfrac{\phi \sigma _v x_1 x_5}{K + x_5} - \left( \mu _h + \omega \right) x_2, \\ f_3 := x'_3(t) = \omega x_2 - \left( \mu _h+\gamma \right) x_3, \\ f_4 := x'_4(t) = \gamma (1-f)x_3 - \mu _hx_4, \\ f_5 := x'_5(t) = \alpha _h x_3 - \mu _v x_5.\\ \end{array}\right. \end{aligned}$$

(6.1)

The Jacobian matrix of subsystem (6.1) at the disease-free equilibrium $x^*_0$ when $\phi = \phi ^*$ is

$$\begin{aligned} J_{\phi ^*} = \begin{pmatrix} -\mu _h &{} 0 &{} -\phi ^*{\varPhi }_h &{} 0 &{} -\dfrac{\phi ^*\sigma _v{\varLambda }_h}{\mu _hK}\\ 0 &{} -(\mu _h+\omega ) &{} \phi ^*{\varPhi }_h &{} 0 &{} \dfrac{\phi ^*\sigma _{v}{\varLambda }_h}{\mu _hK} \\ 0 &{} \omega &{} -(\mu _h+\gamma ) &{} 0 &{} 0 \\ 0 &{} 0 &{} \gamma (1-f) &{} -\mu _{h} &{} 0 \\ 0 &{} 0 &{} \alpha _h &{} 0 &{} -\mu _v \\ \end{pmatrix}. \end{aligned}$$

It is straightforward that the transformed system (6.1), with $\phi =\phi ^*$ has a hyperbolic equilibrium point (i.e., the Jacobian matrix $J_{\phi ^*}$ has a simple eigenvalue with zero real part (here, zero is a simple eigenvalue), and the remaining eigenvalues have negative real parts). Therefore the Center Manifold Theory [15] can applied to analyze the dynamics of subsystem (6.1) near the bifurcation parameter $\phi =\phi ^*$. It is easy to see that a corresponding right-eigenvector of $J_{\phi ^*}$ associated to the zero eigenvalue is $\mathbf{w} = (w_1, w_2,w_3,w_4,w_5)^T$ , and a corresponding non-negative left-eigenvector associated to zero is given by $\mathbf{v} = (v_1, v_2, v_3, v_4, v_5)$, where

$$\begin{aligned} \left\{ \begin{array}{ll} w_1= &{}-\mu _v(\mu _h+\omega )(\mu _h+\gamma ),\\ w_2=&{} \mu _h\mu _v(\mu _h+\gamma ),\\ w_3= &{}\mu _h\mu _v\omega , \\ w_4=&{} \mu _v\gamma (1-f)\omega ,\\ w_5=&{} \mu _h\alpha _h\omega , \end{array}\right. \quad \text {and} \quad \left\{ \begin{array}{ll} v_1=&{}0,\\ v_2=&{} \mu _h\mu _v\omega K, \\ v_3=&{} \mu _h\mu _v K(\mu _h+\omega ),\\ v_4=&{}0,\\ v_5=&{}\sigma _{v}{\varLambda }_h\omega . \end{array}\right. \end{aligned}$$

To apply Theorem 4.1 in [15] and determine the nature and the direction of the bifurcation at ${\mathcal {R}}^{env}_0 = 1$, we must compute the following quantities:

$$\begin{aligned} \mathbf{a}= \sum _{k,j,i=1}^5 {v_kw_iw_j\frac{\partial ^2f_k}{\partial x_i\partial x_j}}\left( x_0^*,\phi ^* \right) ; \quad \mathbf{b}= \sum _{k,i=1}^5 {v_kw_i\frac{\partial ^2f_k}{\partial x_i\partial \phi }}\left( x_0^*,\phi ^* \right) \end{aligned}$$

The only non-vanishing second partial derivatives of f corresponding to the non-zero components of $\mathbf v$ evaluated at $\left( x_0^*,\phi ^*\right)$ are:

$$\begin{aligned}&\dfrac{\partial ^2f_2}{\partial x_2\partial x_3}\left( x_0^*,\phi ^* \right) = \dfrac{\partial ^2f_2}{\partial x_3\partial x_4}\left( x_0^*,\phi ^* \right) = \dfrac{\partial ^2f_2}{\partial x_3\partial x_5}\left( x_0^*,\phi ^* \right) = - \phi ^*{\varPhi }_h\dfrac{\mu _h}{{\varLambda }_h}\\&\dfrac{\partial ^2f_2}{\partial x_3^2}\left( x_0^*,\phi ^* \right) = -2\phi ^*{\varPhi }_h\dfrac{\mu _h}{{\varLambda }_h} ; \quad \dfrac{\partial ^2f_2}{\partial x_1\partial x_5}\left( x_0^*,\phi ^* \right) = \dfrac{\phi ^*\sigma _{v}}{K}; \\&\dfrac{\partial ^2 f_2}{\partial x_3\partial \phi ^*}\left( x_0^*,\phi ^* \right) =\sigma _v. \end{aligned}$$

Thus,

$$\begin{aligned} \mathbf{a}= & {} v_2\left[ 2w_1w_5\dfrac{\partial ^2f_2}{\partial x_1\partial x_5} + 2w_3\dfrac{\partial ^2f_2}{\partial x_2\partial x_3}\left( x_0^*,\phi ^* \right) \left( w_2 + w_4 + w_5\right) + w_3^2\dfrac{\partial ^2f_2}{\partial x_3^2}\left( x_0^*,\phi ^* \right) \right] \\= & {} - 2v_2 \bigg \{-w_1w_5\dfrac{\phi ^*\sigma _{v}}{K} + w_3\phi ^*{\varPhi }_h\dfrac{\mu _h}{{\varLambda }_h} \left( w_2 + w_3 + w_4 + w_5\right) \bigg \} \\= & {} - 2v_2\mu _h\mu _v\omega \bigg \{\dfrac{(\mu _h+\omega ) (\mu _h+\gamma )\phi ^*\alpha _h\sigma _{v}}{K} \bigg \}\\&- 2v_2\mu _h\mu _v\omega \bigg \{ \dfrac{\phi ^*{\varPhi }_h\mu _h}{{\varLambda }_h} \left[ \mu _h\mu _v(\mu _h+\gamma ) + \mu _h\mu _v\omega + \mu _v\gamma (1-f) \!+\! \mu _h\alpha _h\omega \right] \bigg \} \!<\! 0 \\ \mathbf{b}= & {} v_2w_3\dfrac{\partial ^2 f_2}{\partial x_3\partial \phi }\left( x_0^*,\phi ^* \right) = \sigma _v\mu _h^2\mu _v^2\omega ^2 K >0. \end{aligned}$$

Thanks to item 4 of Theorem 4.1 in [15], the endemic $P^*_h$ of sub-model (2.9) is locally asymptotically stable when ${\mathcal {R}}^{env}_0>1$, but near to 1. Moreover, the bifurcation of the subsystem (2.9) around ${\mathcal {R}}^{env}_0=1$ is trans-critical. The proof is complete.