A Population Dynamics Model of Mosquito-Borne Disease Transmission, Focusing on Mosquitoes’ Biased Distribution and Mosquito Repellent Use

Abstract

We present an improved mathematical model of population dynamics of mosquito-borne disease transmission. Our model considers the effect of mosquito repellent use and the mosquito’s behavior or attraction to the infected human, which cause mosquitoes’ biased distribution around the human population. Our analysis of the model clearly shows the existence of thresholds for mosquito repellent efficacy and its utilization rate in the human population with respect to the elimination of mosquito-borne diseases. Further, the results imply that the suppression of mosquito-borne diseases becomes more difficult when the mosquitoes’ distribution is biased to a greater extent around the human population.

Appendices

Derivation of the Basic Reproduction Number ${\mathscr {R}}_0$

At first we rearrange the system (30) as follows in the order according to the relation to the disease transmission:

$$\begin{aligned} \begin{array}{cl} \dfrac{\mathrm{d}f_{\mathrm I}}{\mathrm{d}t} &{}= (1-\xi \omega )\sigma _h \dfrac{f_{\mathrm V}f_{\mathrm S} }{1+\alpha f_{\mathrm I}}\, \eta _\omega - (\rho + \mu _h) f_{\mathrm I}\\ \dfrac{\mathrm{d}f_{\mathrm V}}{\mathrm{d}t} &{}= (1-\xi \omega )\sigma _m\dfrac{(1+\alpha ) f_{\mathrm I} (1-f_{\mathrm V})}{1+\alpha f_{\mathrm I}} - \mu _m f_{\mathrm V}\\ \dfrac{\mathrm{d}f_{\mathrm S}}{\mathrm{d}t} &{}= \mu _h - (1-\xi \omega )\sigma _h\dfrac{f_{\mathrm V} f_{\mathrm S}}{1+\alpha f_{\mathrm I}}\, \eta _\omega - \mu _h f_{\mathrm S} + \nu (1-f_{\mathrm S}-f_{\mathrm I}). \end{array} \end{aligned}$$

(45)

Next, we decompose the dynamical terms into two classes in which one shows the new infection process, and the other does show the other processes of the population dynamics:

$$\begin{aligned} \dfrac{\mathrm{d}\varvec{\varphi }}{\mathrm{d}t}={\mathscr {F}} (f_{\mathrm I}, f_{\mathrm V}, f_{\mathrm S})-{\mathscr {V}}(f_{\mathrm I}, f_{\mathrm V}, f_{\mathrm S}), \end{aligned}$$

(46)

where $\varvec{\varphi }:= {}^{{\mathsf {T}}} \big [ f_{\mathrm I}\ f_{\mathrm V} \ f_{\mathrm S} \big ]$;

$$\begin{aligned} {\mathscr {F}}(f_{\mathrm I}, f_{\mathrm V}, f_{\mathrm S})&:= \begin{bmatrix} (1-\xi \omega )\sigma _h \frac{f_{\mathrm V}f_{\mathrm S}}{1+\alpha f_{\mathrm I}}\,\eta _\omega \\ 0\\ 0 \end{bmatrix};\\ -{\mathscr {V}}(f_{\mathrm I}, f_{\mathrm V}, f_{\mathrm S})&:=\begin{bmatrix} - (\rho + \mu _h) f_{\mathrm I}\\ (1-\xi \omega )\sigma _m\frac{(1+\alpha ) f_{\mathrm I} (1-f_{\mathrm V})}{1+\alpha f_{\mathrm I}} - \mu _m f_{\mathrm V}\\ \mu _h - (1-\xi \omega )\sigma _h \frac{f_{\mathrm V}f_{\mathrm S} }{1+\alpha f_{\mathrm I}}\,\eta _\omega - \mu _h f_{\mathrm S} + \nu (1-f_{\mathrm S}-f_{\mathrm I}) \end{bmatrix}. \end{aligned}$$

The vector ${\mathscr {F}}$ is for the terms of new infection process, while $-{\mathscr {V}}$ is for the other. The Jacobian matrices of ${\mathscr {F}}$ and ${\mathscr {V}}$ about the disease-free equilibrium $\varvec{\varphi }_0: = {}^{{\mathsf {T}}}\big [0\ 0\ 1\big ]$ are given by

$$\begin{aligned} D{\mathscr {F}}(\varvec{\varphi }_0)&= \begin{bmatrix} 0&\quad (1-\xi \omega )\sigma _h\eta _\omega&\quad 0\\ 0&\quad 0&\quad 0\\ 0&\quad 0&\quad 0 \end{bmatrix};\\ D{\mathscr {V}}(\varvec{\varphi }_0)&= \begin{bmatrix} \rho + \mu _h&\quad 0&\quad 0\\ - (1-\xi \omega )\sigma _m(1+\alpha )&\quad \mu _m&\quad 0\\ \nu&\quad (1-\xi \omega )\sigma _h\eta _\omega&\quad \mu _h+\nu \end{bmatrix}. \end{aligned}$$

Then, with the $2\times 2$ matrices

$$\begin{aligned} {\mathcal {F}}:= \begin{bmatrix} 0&\quad (1-\xi \omega )\sigma _h\eta _\omega \\ 0&\quad 0 \end{bmatrix} \quad \text{ and }\quad {\mathcal {V}}:= \begin{bmatrix} \rho + \mu _h&\quad 0\\ -(1-\xi \omega )\sigma _m (1+\alpha )&\quad \mu _m \end{bmatrix}, \end{aligned}$$

the next-generation matrix ${\mathcal {K}}$ is given by ${\mathcal {F}}{\mathcal {V}}^{-1}$, that is,

$$\begin{aligned} {\mathcal {K}}={\mathcal {F}}{\mathcal {V}}^{-1} = \begin{bmatrix} \dfrac{ (1-\xi \omega )^2\sigma _m\sigma _h\eta _\omega (1+\alpha )}{\mu _m (\mu _h +\rho )}&\quad \dfrac{(1-\xi \omega )\sigma _h \eta _\omega }{\mu _m}\\ 0&\quad 0 \end{bmatrix}. \end{aligned}$$

(47)

The theory by van den Driessche and Watmough (2002), van den Driessche and Watmough (2008) says that the spectrum radius, that is, the maximum absolute value of the eigenvalue of ${\mathcal {K}}$ gives the basic reproduction number ${\mathscr {R}}_0$. Therefore, from (47), we can derive the basic reproduction number (31).

Local Stability of the Endemic Equilibrium $E_+$

In this appendix, we consider the local stability of the endemic equilibrium $E_+$, $\left( f_{\mathrm S},f_{\mathrm I},f_{\mathrm V}\right) = \left( f_{\mathrm S}^*, f_{\mathrm I}^*, f_{\mathrm V}^*\right)$ uniquely determined by (34)–(37) when it exists, that is, when ${\mathscr {R}}_0 > 1$ as shown in Lemma 2. Setting $\left( f_{\mathrm S},f_{\mathrm I},f_{\mathrm V}\right) = \left( f_{\mathrm S}^*+x, f_{\mathrm I}^*+y, f_{\mathrm V}^*+z\right)$, we can get the following system of linear ordinary differential equations in terms of the perturbation ${}^{{\mathsf {T}}}\big [x\ y\ z\big ]$ around the endemic equilibrium $E_+$ for (30):

$$\begin{aligned} \dfrac{\mathrm{d}}{\mathrm{d}t} \begin{bmatrix} \, x\, \\ y\\ z \end{bmatrix} =\left\{ \begin{array}{lll} -(\mu _h+\rho )\,\frac{f_{\mathrm I}^*}{f_{\mathrm S}^*}-(\mu _h+\nu ) &{} \quad (\mu _h +\rho )\,\frac{\alpha f_{\mathrm I}^*}{1+\alpha f_{\mathrm I}^*}-\nu &{}\quad -(\mu _h +\rho )\,\frac{f_{\mathrm I}^*}{f_{\mathrm V}^*} \\ (\mu _h+\rho )\,\frac{f_{\mathrm I}^*}{f_{\mathrm S}^*} &{}\quad -(\mu _h +\rho )\,\frac{1+2\alpha f_{\mathrm I}^*}{1+\alpha f_{\mathrm I}^*} &{}\quad (\mu _h +\rho )\,\frac{f_{\mathrm I}^*}{f_{\mathrm V}^*} \\ 0 &{}\quad \mu _m\,\frac{f_{\mathrm V}^*/f_{\mathrm I}^*}{1+\alpha f_{\mathrm I}^*} &{}\quad -\frac{\mu _m}{1-f_{\mathrm V}^*} \end{array}\right\} \begin{bmatrix} \, x\, \\ y\\ z \end{bmatrix}, \end{aligned}$$

(48)

where we used the relations (34) about $E_+$.

Next, let us consider the following function ${\mathscr {L}} = {\mathscr {L}}(x, y, z)$ constructed by the solution ${}^{{\mathsf {T}}}\big [x\ y\ z\big ]$ of the ordinary differential equations given by (48):

$$\begin{aligned} {\mathscr {L}}(x, y, z) := \dfrac{\, 1\, }{2}\, (x+y)^2 + \dfrac{\rho +2 (\mu _h +\nu )}{2(\mu _h +\rho )}\,\dfrac{f_{\mathrm S}^*}{f_{\mathrm I}^*}\, y^2 + \dfrac{\, Q\, }{2}\, z^2, \end{aligned}$$

(49)

where we will determine a positive constant Q appropriately in the following arguments. With a positive constant Q, the function ${\mathscr {L}}$ takes only nonnegative value, and becomes zero when and only when $x = y = z = 0$, which corresponds to the endemic state $E_+$.

Time derivative of ${\mathscr {L}}$ along the solution ${}^{{\mathsf {T}}}\big [x\ y\ z\big ]$ of (48) gives the following equation:

$$\begin{aligned} \left. \dfrac{\mathrm{d}{\mathscr {L}}}{\mathrm{d}t}\right| _{{(48)}}&= -(\mu _h +\nu )x^2 -(A_0y^2-A_1yz+A_2z^2) \nonumber \\&= -(\mu _h +\nu )x^2 -A_0\big (y-\dfrac{A_1}{2A_0}z\big )^2 + \dfrac{A_1^2-4A_0A_2}{4A_0}\, z^2 \end{aligned}$$

(50)

with positive constants given by

$$\begin{aligned} A_0&= \rho +\mu _h +\nu +\big \{ \rho + 2(\mu _h +\nu )\big \}\,\dfrac{f_{\mathrm S}^*/f_{\mathrm I}^*}{1+\alpha f_{\mathrm I}^*};\\ A_1&= \big \{ \rho + 2(\mu _h +\nu )\big \}\,\dfrac{f_{\mathrm S}^*}{f_{\mathrm V}^*} + \mu _m\,\dfrac{f_{\mathrm V}^*/f_{\mathrm I}^*}{1+\alpha f_{\mathrm I}^*}\, Q;\\ A_2&= \dfrac{\mu _m}{1-f_{\mathrm V}^*}\, Q. \end{aligned}$$

Hence, if we can choose a positive value of Q such that $A_1^2 - 4A_0A_2 < 0$, then we have the time derivative (50) which is always non-positive for any ${}^{{\mathsf {T}}}\big [x\ y\ z\big ]$ and becomes zero for ${}^{{\mathsf {T}}}\big [0\ 0\ 0\big ]$. The formula $A_1^2 - 4A_0A_2$ can be expressed as the quadratic function of Q, $G(Q): = B_2Q^2-2B_1Q+B_0$ with positive constants

$$\begin{aligned} B_2&= \mu _m^2 \left( \dfrac{f_{\mathrm V}^*/f_{\mathrm I}^*}{1+\alpha f_{\mathrm I}^*} \right) ^2;\\ B_1&= \mu _m\big \{\rho + 2(\mu _h+\nu )\big \}\, \dfrac{f_{\mathrm S}^*/f_{\mathrm I}^*}{1+\alpha f_{\mathrm I}^*}\, \dfrac{1+f_{\mathrm V}^*}{1-f_{\mathrm V}^*} + \dfrac{2\mu _m (\rho +\mu _h+\nu )}{1-f_{\mathrm V}^*};\\ B_0&= \big \{ \rho + 2(\mu _h +\nu )\big \}^2\left( \dfrac{f_{\mathrm S}^*}{f_{\mathrm V}^*}\right) ^2. \end{aligned}$$

Since $B_1>0$ and $B_1^2 -B_0B_2 > 0$, we find that the equation $G(Q) < 0$ for a positive finite range of Q. Therefore, if we choose a value of Q from the positive range, then the time derivative (50) is always non-positive for any ${}^{{\mathsf {T}}}\big [x\ y\ z\big ]$. Since the largest invariant set where the time derivative (50) becomes zero is the singleton consisting of only ${}^{{\mathsf {T}}}\big [0\ 0\ 0\big ]$, the function ${\mathscr {L}}$ becomes a Lyapunov function for the equilibrium ${}^{{\mathsf {T}}}\big [0\ 0\ 0\big ]$ of the dynamical system (48). Thus, by LaSalle’s invariance principle (LaSalle 1976), the equilibrium ${}^{{\mathsf {T}}}\big [0\ 0\ 0\big ]$ is asymptotically stable with respect to the dynamical system (48). Consequently, the endemic equilibrium $E_+$ is locally asymptotically stable whenever it exists.