A West Nile Virus Model with Vertical Transmission and Periodic Time Delays

Abstract

Seasonal change has played a critical role in the evolution dynamics of West Nile virus transmission. In this paper, we formulate and analyze a novel delay differential equation model, which incorporates seasonality, the vertical transmission of the virus, the temperature-dependent maturation delay and the temperature-dependent extrinsic incubation period in mosquitoes. We first introduce the basic reproduction ratio $R_0$ for this model and then show that the disease is uniformly persistent if $R_0>1$. It is also shown that the disease-free periodic solution is attractive if $R_0<1$, provided that there is only a small invasion. In the case where all coefficients are constants and the disease-induced death rate of birds is zero, we establish a threshold result on the global attractivity in terms of $R_0$. Numerically, we study the West Nile virus transmission in Orange County, California, USA.

Appendix: A Periodic Delay Differential Equation

In this Appendix, we consider a scalar differential equation with periodic delay:

$$\begin{aligned} \frac{{\hbox {d}} u(t)}{{\hbox {d}}t}=a(t)e^{-\alpha (t)u(t-\tau _A(t))}u(t-\tau _A(t))-d(t)u(t), \end{aligned}$$

(25)

where all coefficients are positive $\omega$-periodic functions. Let $Z:=C([-{\hat{\tau }}_A,0],{\mathbb {R}})$, $Z^+:=C([-{\hat{\tau }}_A,0],{\mathbb {R}}_+)$. For any $\varphi \in Z^+$, define

$$\begin{aligned} {\tilde{f}}(t,\varphi )=a(t)e^{-\alpha (t)\varphi (-\tau _A(t))}\varphi (-\tau _A(t))-d(t)\varphi (0). \end{aligned}$$

It is easy to see that for any $\varphi \in Z^+$ with $\varphi (0)=0$, we have ${\tilde{f}}(t,\varphi )\ge 0$. It then follows from Smith (1995, Theorem 5.2.1) that for any $\varphi \in Z^+$, system (25) has a unique nonnegative solution $w(t, \varphi )$ on its maximal interval $[0, \sigma _\varphi )$ of existence with $w_0=\varphi$. Since

$$\begin{aligned} \frac{{\hbox {d}}u(t)}{{\hbox {d}}t} \le \frac{a(t)}{\alpha (t)}e^{-1}-d(t)u(t), \end{aligned}$$

system (25) is dominated by the following cooperative system:

$$\begin{aligned} \frac{{\hbox {d}}{\bar{u}}(t)}{{\hbox {d}}t} = \frac{a(t)}{\alpha (t)}e^{-1}-d(t){\bar{u}}(t). \end{aligned}$$

(26)

Clearly, system (26) has a globally attractive positive $\omega$-periodic solution ${\bar{u}}^*(t)$; that is, $\lim _{t \rightarrow \infty }({\bar{u}}(t)-{\bar{u}}^*(t))=0$. Thus, solutions of system (26) are bounded and ultimately bounded. By the comparison principle, solutions of system (25) exist globally on $[0,\infty )$ and are ultimately bounded.

Let $P_1(t)$ be the solution maps of system (25); that is, $P_1(t)\psi =w_t(\psi )$, $\forall t \ge 0$, where $w(t,\psi )$ is the unique solution of system (25) satisfying $w_0=\psi \in Z^+$. Then $P_1:=P_1(\omega )$ is the Poincaré map associated with system (25) on $Z^+$. Thus, $\{P_1^n\}_{n=0}^\infty$ is point dissipative on $Z^+$. Let $\rho (D P_1(0))$ be the spectral radius of the Frechét derivative of $P_1$ at zero. Note that the linearized system (25) at zero is

$$\begin{aligned} \frac{{\hbox {d}}u(t)}{{\hbox {d}}t}=a(t)u(t-\tau _A(t))-d(t)u(t). \end{aligned}$$

(27)

Let ${\tilde{P}}$ be the Poincaré map associated with system (27) on Z. By the continuity and differentiability of solutions with respect to initial values, it follows that $P_1$ is differentiable at zero and $D P_1(0)={\tilde{P}}$. Define

$$\begin{aligned} {\mathcal {Z}}:=C([-\tau _A(0),0],{\mathbb {R}}),~ {\mathcal {Z}}^+:=C([-\tau _A(0),0],{\mathbb {R}}_+). \end{aligned}$$

By the method of steps (see, e.g., Li and Zhao 2019; Lou and Zhao 2017), we have the following result.

Lemma A.1

For any $\varphi \in {\mathcal {Z}}^+$, system (25) has a unique nonnegative solution $w(t,\varphi )$ with $w_0=\varphi$ for all $t\ge 0$.

For any given $t \ge 0$, let $S_1(t)$ be the solution maps of system (25) on ${\mathcal {Z}}^+$. Let ${\tilde{S}}(t)$ be the solution maps of linear system (27) on ${\mathcal {Z}}$. Then ${\tilde{S}}:={\tilde{S}}(\omega )$ is the Poincaré map associated with linear system (27) and $D S_1(0)={\tilde{S}}$. It is easy to see that $r({\tilde{P}})=r({\tilde{S}})$ (see, e.g., Lou and Zhao 2017). Further, we have the following observations.

Remark A.2

By the uniqueness of solutions, it follows that for any $\psi \in Z^+$ and $\phi \in {\mathcal {Z}}^+$ with $\psi (\theta )=\phi (\theta )$ for all $\theta \in [-\tau _A(0), 0]$ we have $w(t, \psi )=\nu (t, \phi )$ for all $t \ge 0$, where $w(t, \psi )$ and $\nu (t, \phi )$ are solutions of system (25) satisfying $w_0=\psi$ and $\nu _0=\phi$, respectively.

Lemma A.3

$S_1(t)$ is an $\omega$-periodic semiflow on ${\mathcal {Z}}^+$ in the sense that (i) $S_1(0)=I;$ (ii) $S_1(t+\omega )=S_1(t) \circ S_1(\omega )$ for all $t\ge 0;$ and (iii) $S_1(t)\psi$ is continuous in $(t, \psi ) \in [0, \infty ) \times {\mathcal {Z}}^+.$

To obtain the global dynamics of system (25), we need the following assumptions:

• (A1) $r({\tilde{P}})>1$, where $r({\tilde{P}})$ is the spectral radius of ${\tilde{P}}$.

• (A2) ${\bar{u}}^*(t-\tau _A(t))<\frac{1}{\alpha (t)}$ for all $t \in [0,\omega ]$.

Theorem A.4

Let (A1) and (A2) hold. Then system (25) admits a unique positive $\omega$-periodic solution $u^*(t)$ which is globally attractive in ${\mathcal {Z}}^+ \setminus \{0\}$.

Proof

Define $W:=[0,{\bar{u}}^*_0]_{\mathcal {Z}}$, where ${\bar{u}}^*_0 \in {\mathcal {Z}}$, and ${\bar{u}}^*_0(\theta )={\bar{u}}^*(\theta )$ for all $\theta \in [-\tau _A(0),0]$. For any $\psi \in {\mathcal {Z}}^+$, we have $0\le u(t,\psi )\le {\bar{u}}(t,\psi )$, $\forall t \ge 0$. Then $0\le u_t(\psi )\le {\bar{u}}_t(\psi )$, $\forall t \ge 0$. Letting $t=n\omega$, we have

$$\begin{aligned} 0\le u_{n\omega }(\psi )\le {\bar{u}}_{n\omega }(\psi ), \forall n \ge 0; \end{aligned}$$

that is,

$$\begin{aligned} 0\le S_1^n(\psi )\le {\bar{S}}_1^n(\psi ), \forall n \ge 0, \end{aligned}$$

(28)

where ${\bar{S}}_1$ is the Poincaré map associated with linear system (26) on ${\mathcal {Z}}^+$.

Since $\lim _{t \rightarrow \infty }({\bar{u}}(t)-{\bar{u}}^*(t))=0$, $\lim _{t \rightarrow \infty }\Vert {\bar{u}}_t(\psi )-{\bar{u}}^*_t\Vert =0$. Letting $t=n\omega$, we have ${\bar{u}}^*_{n\omega }={\bar{u}}^*_0$, and hence,

$$\begin{aligned} \lim _{n \rightarrow \infty }\Vert {\bar{u}}_{n\omega }(\psi )-{\bar{u}}^*_{n\omega }\Vert =\lim _{n \rightarrow \infty }\Vert {\bar{S}}_1^n(\psi )-{\bar{u}}^*_0\Vert =0. \end{aligned}$$

Combining with (28), it follows that the omega limit set $\omega (\psi ) \subseteq W$ for all $\psi \in {\mathcal {Z}}^+$.

Let ${\tilde{S}}_{\epsilon }(t)$ be the solution maps of the following perturbed linear periodic system on ${\mathcal {Z}}$:

$$\begin{aligned} \frac{{\hbox {d}}u(t)}{{\hbox {d}}t}=a(t)(1-\epsilon )u(t-\tau _A(t))-d(t)u(t), \end{aligned}$$

(29)

and ${\tilde{S}}_{\epsilon }:={\tilde{S}}_{\epsilon }(\omega )$. Since $\lim _{\epsilon \rightarrow 0^+}r({\tilde{S}}_\epsilon )=r({\tilde{S}})>1$, we can fix a sufficiently small number $\epsilon _0>0$ such that $r({\tilde{S}}_{\epsilon _0})>1.$ It follows from Hale and Verduyn Lunel (1993, Theorem 3.6.1) and Remark A.2 that the linear operator ${\tilde{S}}_{\epsilon _0}$ is compact on ${\mathcal {Z}}$. By the same arguments as in Li and Zhao (2019, Lemma 6), it follows that $P_\epsilon (t)$ is also strongly monotone on ${\mathcal {Z}}$ for each $t\ge 2{\hat{\tau }}_A$. Choose an integer $n_0>0$ such that $n_0 \omega \ge 2{\hat{\tau }}_A$. Since ${\tilde{S}}^{n_0}_{\epsilon _0}={\tilde{S}}_{\epsilon _0}(n_0\omega )$, Liang and Zhao (2007, Lemma 3.1) implies that $r({\tilde{S}}_{\epsilon _0})$ is a simple eigenvalue of ${\tilde{S}}_{\epsilon _0}$ having a strongly positive eigenvector. By Wang and Zhao (2017a, Lemma 1), there is a positive $\omega$-periodic function $v^*(t)$ such that $w^*_{\epsilon _0}(t)=e^{\frac{\ln r({\tilde{S}}_{\epsilon _0})}{\omega }t}v^*(t)$ is a positive solution of system (29).

For the above fixed $\epsilon _0>0$, there exists a sufficiently small positive number $\delta _0=\delta _0({\epsilon _0})<\epsilon _0$ such that $e^{-\alpha (t)x}\ge 1-\epsilon _0, \forall t \ge 0, 0\le x \le \delta _0.$ Since $\lim _{\phi \rightarrow 0}S_{1t}(\phi )=0$ uniformly for $t \in [0,\omega ]$, there exists $\delta _1>0$ such that $\Vert S_{1t}(\phi )\Vert \le \delta _0, \forall t\in [0,\omega ], \Vert \phi \Vert \le \delta _1.$ We further have the following observation.

Claim 1.$\limsup _{n\rightarrow \infty }\Vert S_1^n(\psi )\Vert \ge \delta _1$ for all $\psi \in {\mathcal {Z}}^+ \setminus \{0\}$.

Suppose, by contradiction, that $\limsup _{n\rightarrow \infty }\Vert S_1^n(\phi )\Vert < \delta _1$ for some $\phi \in {\mathcal {Z}}^+ \setminus \{0\}$. Then there exists an integer $N_0\ge 1$ such that $\Vert S_1^n(\phi )\Vert < \delta _1$ for all $n\ge N_0$. For any $t\ge N_0\omega$, we have $t=n\omega +t'$ with $n\ge N_0$, $t' \in [0, \omega ]$ and $\Vert S_{1t}(\phi )\Vert =\Vert S_1(t')S_1(n\omega )\phi \Vert \le \delta _0$. Then for all $t\ge N_0\omega +{\hat{\tau }}_A$, we have $\Vert u(t-\tau _A(t),\phi )\Vert \le \delta _0$. Then

$$\begin{aligned} \frac{{\hbox {d}}u(t)}{{\hbox {d}}t} \ge a(t)(1-\epsilon _0)u(t-\tau _A(t))-d(t)u(t), \end{aligned}$$

for all $t\ge N_0\omega +{\hat{\tau }}_A$. Since $\phi \in {\mathcal {Z}}^+ \setminus \{0\}$, there exists $t_0 \in [0,\tau _A(0)]$ such that $u(t_0,\phi )>0$. It then follows that $u(t,\phi )>0$ for all $t\ge t_0$, and hence, $u(t,\phi )>0$ for all $t\ge \tau _A(0)$. We can choose a sufficiently small number $k>0$ such that $u(t,\phi )\ge kw^*_{\epsilon _0}(t)$, $\forall t \in [N_0\omega +{\hat{\tau }}_A, N_0\omega +2{\hat{\tau }}_A]$. By Smith (1995, Theorem 5.1.1), we have $u(t,\phi )\ge kw^*_{\epsilon _0}(t)$, $\forall t \ge N_0\omega +2{\hat{\tau }}_A$. Thus, $\lim _{t \rightarrow 0}u(t, \phi )=\infty$, which is a contradiction.

For any $\phi \in W$, we have $\phi \le {\bar{u}}^*_0$. Since system (25) is dominated by system (26), it follows that

$$\begin{aligned} u(t,\phi )\le {\bar{u}}^*(t), \forall t \ge 0, \phi \in W. \end{aligned}$$

For any given $\varphi , \psi \in W$ with $\varphi \ge \psi$, let $v(t, \varphi )$ and $v(t, \psi )$ be the unique solutions of system (25) with $v_0=\varphi$ and $v_0=\psi$, respectively. Define $B(t,x):=a(t)e^{-\alpha (t)x}x.$ In view of (A2), we see that for any $\phi \in W$,

$$\begin{aligned} \frac{\partial B}{\partial x}(t,u(t-\tau _A(t),\phi ))=&a(t)e^{-\alpha (t)u(t-\tau _A(t),\phi )} [1-\alpha (t)u(t-\tau _A(t),\phi )]\\ \ge&a(t)e^{-\alpha (t)u(t-\tau _A(t),\phi )} [1-\alpha (t){\bar{u}}^*(t-\tau _A(t),\phi )] >0. \end{aligned}$$

By the arguments similar to those in Li and Zhao (2019), it is easy to show that we have $S_1(t): W\rightarrow W$ is monotone; that is, $v(t,\varphi )\ge v(t,\psi )$ for all $t\ge 0$. Next we prove that $S_1(t): W\rightarrow W$ is eventually strongly monotone. Let $\varphi , \psi \in W$ satisfy $\varphi > \psi$. Define ${\bar{v}}(t)=v(t,\varphi )$ and $v(t)=v(t,\psi )$. By the arguments similar to those in Li and Zhao (2019, Lemma 6), we have the following observation.

Claim 2. There exists $t_0 \in [0, {\hat{\tau }}_A]$ such that ${\bar{v}}(t)>v(t)$ for all $t\ge t_0$.

Letting $g(t,y)=B(t,v(t-\tau _A(t))-d(t)y.$ Since $\frac{\partial B}{\partial x}(t,u(t-\tau _A(t),\phi ))>0$, we then have

$$\begin{aligned} \frac{{\hbox {d}}{\bar{v}}(t)}{{\hbox {d}}t}=&B(t,{\bar{v}}(t-\tau _A(t))-d(t){\bar{v}}(t)\\>&B(t,v(t-\tau _A(t))-d(t){\bar{v}}(t) =g(t,{\bar{v}}(t)), \forall t>t_0+{\hat{\tau }}_A, \end{aligned}$$

and hence,

$$\begin{aligned} \frac{{\hbox {d}}{\bar{v}}(t)}{{\hbox {d}}t}-g(t,{\bar{v}}(t))>0=\frac{{\hbox {d}}v(t)}{{\hbox {d}}t}-g(t,v(t)), \forall t>t_0+{\hat{\tau }}_A. \end{aligned}$$

Since ${\bar{v}}(t_0+{\hat{\tau }}_A)> v(t_0+{\hat{\tau }}_A)$, it follows from Walter (1997, Theorem 4) that ${\bar{v}}(t)>v(t)$ for all $t > t_0+{\hat{\tau }}_A$. Since $t_0 \in [0,{\hat{\tau }}_A]$, it follows that $S^n_1(t): W\rightarrow W$ is strongly monotone for any $n\omega > 3{\hat{\tau }}_A$.

For any given $\phi \gg 0$ in W and $\lambda \in (0,1)$, let $z(t, \phi )$ and $z(t, \lambda \phi )$ be the solutions of system (25) satisfying $z_0=\phi$ and $z_0=\lambda \phi$, respectively. Denote $w(t)=\lambda z(t, \phi )$ and $v(t)=z(t, \lambda \phi )$. By the method of steps, $w(t)>0$ and $v(t)>0$ for all $t\ge 0$. For all $\theta \in [-\tau _A(0),0]$, we have $w(\theta )=\lambda \phi (\theta )=v(\theta )$. For any $t \in [0, \bar{\tau }_A]$, we have $-\tau _A(0)=0-\tau _A(0)\le t-\tau _A(t)\le \bar{\tau }_A-\tau _A(\bar{\tau }_A)\le \bar{\tau }_A-\bar{\tau }_A=0$, and hence, $w(t-\tau _A(t))=\lambda \phi (t-\tau _A(t))=v(t-\tau _A(t))$. Thus,

$$\begin{aligned} \frac{{\hbox {d}}w(t)}{{\hbox {d}}t}\mid _{t=0}&= a(0)e^{-\alpha (0)z(-\tau _A(0))}w(-\tau _A(0))-d(0)w(0)\\&<a(0)e^{-\alpha (0)w(-\tau _A(0))}w(-\tau _A(0))-d(0)w(0)\\&=a(0)e^{-\alpha (0)v(-\tau _A(0))}v(-\tau _A(0))-d(0)v(0)\\&=\frac{{\hbox {d}}v(t)}{{\hbox {d}}t}\mid _{t=0}. \end{aligned}$$

Since $w(0)=v(0)>0$, there must be an $\xi \in (0, \bar{\tau }_A)$ such that $0< w(t)<v(t)$ holds for all $t \in (0,\xi )$. We further claim that $w(t)<v(t)$ for all $0<t\le \bar{\tau }_A$. Assume not, then there exists ${\bar{t}} \in (0, \bar{\tau }_A]$ such that $w(t)<v(t)$ for all $t \in (0, {\bar{t}})$ and $w({\bar{t}})=v({\bar{t}})$, which implies that $\frac{{\hbox {d}}w(t)}{{\hbox {d}}t} \mid _{t={\bar{t}}}\ge \frac{{\hbox {d}}v(t)}{{\hbox {d}}t}\mid _{t={\bar{t}}}$. However, we have

$$\begin{aligned} \frac{{\hbox {d}}w(t)}{{\hbox {d}}t}\mid _{t={\bar{t}}}&= a({\bar{t}})e^{-\alpha ({\bar{t}})z({\bar{t}}-\tau _A({\bar{t}}))}w({\bar{t}}-\tau _A({\bar{t}}))-d({\bar{t}})w({\bar{t}})\\&<a({\bar{t}})e^{-\alpha ({\bar{t}})w({\bar{t}}-\tau _A({\bar{t}}))}w({\bar{t}}-\tau _A({\bar{t}}))-d({\bar{t}})w({\bar{t}})\\&=a({\bar{t}})e^{-\alpha ({\bar{t}})v({\bar{t}}-\tau _A({\bar{t}}))}v({\bar{t}}-\tau _A({\bar{t}}))-d({\bar{t}})v({\bar{t}})\\&=\frac{{\hbox {d}}v(t)}{{\hbox {d}}t}\mid _{t={\bar{t}}}, \end{aligned}$$

which is a contradiction. This shows that $w(t)<v(t)$ for all $0<t\le \bar{\tau }_A$. By the similar arguments for any interval $(n\bar{\tau }_A,(n+1)\bar{\tau }_A],$$n=1,2,3,\cdots ,$ we can get $w(t)<v(t)$ for all $t>0$; that is, $u_t(\lambda \phi ) \gg \lambda u_t(\phi )$ for all $t>\tau _A(0)$.

Now we fix an integer $n_0$ such that $n_0\omega >3{\hat{\tau }}_A$. It then follows that $S_1^{n_0}$ is a strongly monotone and strictly subhomogeneous map on W. Note that $DS_1^{n_0}(0)=DS_1(n_0\omega )(0)={\tilde{S}}(n_0\omega )=({\tilde{S}}(\omega ))^{n_0}={\tilde{S}}^{n_0}$ and $r({\tilde{S}}^{n_0})=(r({\tilde{S}}))^{n_0}$. Since (A1) holds, it follows from Zhao (2017b, Theorem 2.3.4) that there exists a unique positive $n_0\omega$-periodic solution ${\bar{w}}(t)=u^*(t)$ which is globally attractive for system (25) in $W\setminus \{0\}$.

Next we show that ${\bar{w}}(t)$ is also an $\omega$-periodic solution of system (25). Let ${\bar{w}}(t)=w(t,\psi ^*)$. By the properties of periodic semiflows, we have $S_1^{n_0}(S_1(\psi ^*))=S_1(S_1^{n_0}(\psi ^*))=S_1(\psi ^*)$, which implies that $S_1(\psi ^*)$ is also a positive fixed point of $S_1^{n_0}$. By the uniqueness of the positive fixed point of $S_1^{n_0}$, it follows that $S_1(\psi ^*)=\psi ^*$. Then ${\bar{w}}(t)=M^*(t)$ is an $\omega$-periodic solution of system (25).

For any $\psi \in {\mathcal {Z}}^+$, it follows from Zhao (2017b, Lemma 1.2.1) that $\omega (\psi )$ is an internally chain transitive set for $S_1: {\mathcal {Z}}^+ \rightarrow {\mathcal {Z}}^+$. Since $\omega (\psi ) \subseteq W$, by Zhao (2017b, Theorem 1.2.2), either $\omega (\psi )=0$ or $\omega (\psi )=\psi ^*$ for all $\psi \in {\mathcal {Z}}^+$. Claim 1 implies that $\omega (\psi )\ne 0$ for all $\psi \in {\mathcal {Z}}^+ \setminus \{0\}$. Thus, $\omega (\psi )=\psi ^*$ for all $\psi \in {\mathcal {Z}}^+ \setminus \{0\}$. Therefore, system (25) admits a unique positive $\omega$-periodic solution $w(t,\psi ^*)=u^*(t)$ which is globally attractive in ${\mathcal {Z}}^+ \setminus \{0\}$. $\square$