Vector Preference Annihilates Backward Bifurcation and Reduces Endemicity

Abstract

We propose and analyze a mathematical model of a vector-borne disease that includes vector feeding preference for carrier hosts and intrinsic incubation in hosts. Analysis of the model reveals the following novel results. We show theoretically and numerically that vector feeding preference for carrier hosts plays an important role for the existence of both the endemic equilibria and backward bifurcation when the basic reproduction number ${\mathcal {R}}_0$ is less than one. Moreover, by increasing the vector feeding preference value, backward bifurcation is eliminated and endemic equilibria for hosts and vectors are diminished. Therefore, the vector protects itself and this benefits the host. As an example of these phenomena, we present a case of Andean cutaneous leishmaniasis in Peru. We use parameter values from previous studies, primarily from Peru to introduce bifurcation diagrams and compute global sensitivity of ${\mathcal {R}}_0$ in order to quantify and understand the effects of the important parameters of our model. Global sensitivity analysis via partial rank correlation coefficient shows that ${\mathcal {R}}_0$ is highly sensitive to both sandflies feeding preference and mortality rate of sandflies.

Appendices

Appendix 1

Proof of Theorem 1

1.a)

As $f(\lambda _h)$ in Eq. (10) is a quadratic function with $A>0$, it follows that the minimum of f occurs at $\hat{\lambda }_h= -\frac{B}{2A}>0$ with $f(\hat{\lambda _h})=-\frac{B^{2}-4AC}{4A}$. Thus if $B^2-4AC=0$, then system (1) has a unique endemic equilibrium.

1.b)

As $f(\lambda _h)$ in Eq. (10) is a quadratic function with $B<0$, $A>0$, $C>0$, $B^{2}-4AC>0$, it follows that the minimum of f occurs at $\hat{\lambda }_h= -\frac{B}{2A}>0$ with $f(\hat{\lambda _h})=-\frac{B^{2}-4AC}{4A}$. Thus if $B^{2}-4AC>0$, then system (1) has two endemic equilibria.

1.c)

For $\alpha _v<\alpha _v^{*}$ and ${\mathcal {R}}_{0}<\sqrt{H}<1$. By hypothesis, $A>0$, $B>0$ and $C>0$. Then, Eq. (10) does not have any positive root. Thus, conclusion 1.c) holds.

1.d)

For $\alpha _v<\alpha _v^{*}$ and ${\mathcal {R}}_{0} \ge 1$ we get $B<0$ and $C\le 0$. Then conclusion 1.d holds.

2.a)

For $\alpha _v \ge \alpha _v^{*}$ and ${\mathcal {R}}_{0} >1$ we get $C<0$. Then system (1) has a unique endemic equilibrium.

2.b)

For $\alpha _v \ge \alpha _v^{*}$ and ${\mathcal {R}}_{0} \le 1$ we get $C\ge 0$ and $H>{\mathcal {R}}_{0}^2$. Then system (1) has no endemic equilibrium. $\square$

Appendix 2

Proof of Theorem 2

The Jacobian matrix of system (1), computed at $E_{0}$ for $b_1^*$, is given by :

$$\begin{aligned} J(E_0,b_1^{*}) = \begin{bmatrix} -\mu _h&\quad 0&\quad \omega _h&\quad 0&\quad -b_1^*\beta _1 \\ 0&\quad -(\sigma _h+\mu _h)&\quad 0&\quad 0&\quad b_1^*\beta _1\\ 0&\quad \sigma _h&\quad -(\mu _h+\delta _h+\omega _h)&\quad 0&\quad 0 \\ 0&\quad 0&\quad -\frac{b_2\beta _2\alpha _v\varLambda _v\mu _h}{\mu _v\varLambda _h}&\quad -\mu _v&\quad 0\\ 0&\quad 0&\quad \frac{b_2\beta _2\alpha _v\varLambda _v\mu _h}{\mu _v\varLambda _h}&\quad 0&\quad -\mu _v\\ \end{bmatrix}. \end{aligned}$$

The characteristic polynomial of the Jacobian matrix is:

$$\begin{aligned} (\lambda _h+\mu _h)(\lambda _h+\mu _v)(\lambda _h^3+\lambda _h^2 d_1+\lambda _hd_2+ d_3) \end{aligned}$$

where

$$\begin{aligned} d_1= & {} \mu _v+\mu _h+\delta _h+\omega _h+\sigma _h+\mu _h\\ d_2= & {} (\mu _h+\delta _h+ \omega _h)\mu _v+(\sigma _h+\mu _h)(\mu _h+\delta _h+ \omega _h)+(\sigma _h+\mu _h)\mu _v\\ d_3= & {} (\sigma _h+\mu _h)(\mu _h+\delta _h+\omega _h)\mu _v +\frac{b_1^*\beta _1\sigma _h\beta _2b_2\alpha _v\mu _h\varLambda _v}{\mu _v\varLambda _h} \end{aligned}$$

We replace the value of $b_1^*$ in $d_3$, and we get

$$\begin{aligned} \lambda _h(\lambda _h+\mu _h)(\lambda _h+\mu _v)(\lambda _h^2+\lambda _h d_1+ d_2)=0 \end{aligned}$$

The Jacobian matrix admits a zero eigenvalue and the other eigenvalues are real and negative. Thus, the disease-free equilibrium $E_0$ is a non-hyperbolic equilibrium and assumption (A1) of Theorem (Castillo-Chavez 2004) is demonstrated. We indicate by $v=(v_1, v_2, v_3,v_4,v_5)$ and $w=(w_1,w_2,w_3,w_4,w_5)^T$, a right and a left eigenvector associated with the zero eigenvalue, respectively, such that their dot product is one $v.w=1$. Multiplying vJ and Jw and setting each of them equal to zero yields:

$$\begin{aligned}&-\mu _h w_1+\omega _h w_3-b_1^*\beta _1 w_5=0\quad -(\sigma _h+\mu _h) w_2+b_1^*\beta _1 w_5=0\\&\sigma _h w_2-(\mu _h+\delta _h+\omega _h) w_3=0 \quad -\frac{b_2\beta _2 \alpha _v\varLambda _v\mu _h }{\mu _v\varLambda _h}w_3-\mu _vw_4=0\\&\frac{b_2\beta _2 \alpha _v\varLambda _v\mu _h }{\mu _v\varLambda _h}w_3-\mu _vw_5 =0\quad -\mu _h v_1=0\quad -(\sigma _h+\mu _h)v_2+\sigma _hv_3=0 \\&-(\mu _h +\delta _h+\omega _h)v_3+\frac{b_2\beta _2\alpha _v\varLambda _v\mu _h}{\mu _v\varLambda _h}=0 -\mu _v v_4=0 \quad b_1^{*}\beta _1v_2-\mu _vv_5=0 \\ \end{aligned}$$

then,

$$\begin{aligned} v= & {} \left( 0,\sigma _h,\sigma _h+\mu _h,0,\frac{\mu _v\varLambda _h(\sigma _h+\mu _h)(\mu _h+\delta _h+\omega _h)}{\beta _1b_2\beta _2\alpha _v\varLambda _v\mu _h}\right) \\ w= & {} \left( -\frac{((\mu _h+\delta _h)(\mu _h+\sigma _h)+\mu _h\omega _h) \beta _1\mu _v}{\mu _h\sigma _h\theta }, \frac{(\mu _h+\delta _h+\omega _h)\beta _1\mu _v}{\theta },\right. \\&\left. \frac{\beta _1\mu _v}{\theta },-\frac{b_2\beta _2\alpha _v\mu _h\beta _1\varLambda _v}{\mu _v\varLambda _h\theta },\frac{b_2\beta _2\alpha _v\mu _h\beta _1\varLambda _v}{\mu _v\varLambda _h\theta }\right) ^T \end{aligned}$$

where

$$\begin{aligned} \theta =(\mu _h+\delta _h+\omega _h)(\sigma _h+\mu _h+\sigma _h\beta _1\mu _v)+\beta _1\mu _v(\sigma _h+\mu _h) \end{aligned}$$

The functions $f_k$, $k=1,\ldots ,5$ are the right side of the differential equations in (1a)-(1e). We define two quantities important for verification of the subcritical bifurcation

$$\begin{aligned} a=\sum _{k,i,j=1}^{5}v_kw_iw_j\frac{\partial ^2f_k}{\partial x_i\partial x_j}(E_0,b_1^*)\quad b=\sum _{k,i=1}^{5}v_kw_i\frac{\partial ^2f_k}{\partial x_i\partial b_1}(E_0,b_1^*) \end{aligned}$$

It can be checked that:

$$\begin{aligned} \frac{\partial ^2f_5}{\partial x_1 \partial x_3}= & {} \frac{\partial ^2f_5}{\partial S_h \partial I_h}=-\frac{b_2\beta _2\alpha _v\varLambda _v\mu _h^2}{\mu _v\varLambda _h^2} \frac{\partial ^2f_2}{\partial x_3 \partial x_5}=\frac{\partial ^2f_2}{\partial I_h \partial I_v}\\= & {} -\frac{\mu _v^2(\sigma _h+\mu _h)(\mu _h+\delta _h+\omega _h)}{b_2\sigma _h\beta _2\varLambda _v}\\ \frac{\partial ^2f_2}{\partial x_3 \partial x_5}= & {} \frac{\partial ^2f_2}{\partial I_h \partial I_v}=-\frac{\mu _v^{2}(\sigma _h+\mu _h)(\mu _h+\delta _h+\omega _h)}{b_2\sigma _h\beta _2\varLambda _v} \quad \frac{\partial ^2f_5}{\partial x_3^2}=\frac{\partial ^2f_5}{\partial I_h^2}\\= & {} -\frac{2b_2\beta _2\alpha _v^2\varLambda _v\mu _h^2}{\mu _v\varLambda _h^2}\\ \frac{\partial ^2f_5}{\partial x_3 \partial x_4}= & {} \frac{-b_2\beta _2\alpha _v \mu _h}{\varLambda _h} \quad \frac{\partial ^2f_5}{\partial x_5 \partial b_1}= \frac{\partial ^2f_5}{\partial I_v \partial b_1}=\beta _1. \end{aligned}$$

Accordingly to the coefficients a and b described in Theorem 4.1 of (Castillo-Chavez 2004), it follows:

$$\begin{aligned} b=\beta _1\;\hbox {is}\;\hbox {positive}, \end{aligned}$$

and

$$\begin{aligned} a= & {} \frac{2(\sigma _h+\mu _h)(\mu _h+\delta _h+\omega _h) \beta _1\mu _v(2\mu _v+b_2\beta _2)}{\theta ^2}\\&\times (\frac{\mu _v((\mu _h+\delta _h)(\mu _h+\sigma _h)+\mu _h\omega _h}{\mu _h\sigma _h(2\mu _v+b_2\beta _2)}-\alpha _v). \end{aligned}$$

Then, a is positive when $\alpha _v<\frac{\mu _v((\mu _h+\delta _h)(\mu _h+\sigma _h)+\mu _h\omega _h)}{\mu _h\sigma _h(2\mu _v+b_2\beta _2)} =\alpha _v^*$. Consequently, system (1) shows backward bifurcation at ${\mathcal {R}}_0$ when $\alpha _v< \alpha _v^*$.

On the other hand, a is always negative when

$$\begin{aligned} \alpha _v>\frac{\mu _v((\mu _h+\delta _h)(\mu _h+\sigma _h)+\mu _h\omega _h)}{\mu _h\sigma _h(2\mu _v+b_2\beta _2)} =\alpha _v^*. \end{aligned}$$

Therefore, system (1) exhibits a forward bifurcation at ${\mathcal {R}}_{0}=1$ when $\alpha _v>\alpha _v^*$. $\square$