Resonance of Periodic Combination Antiviral Therapy and Intracellular Delays in Virus Model

Abstract

There is a substantial interest in detailed models of viral infection and antiviral drug kinetics in order to optimize the treatment against viruses such as HIV. In this paper, we study within-viral dynamics under general intracellular distributed delays and periodic combination antiviral therapy. The basic reproduction number $R_0$ is established as a global threshold determining extinction versus persistence, and spectral methods are utilized for analytical and numerical computations of $R_0$. We derive the critical maturation delay for virus and optimal phase difference between sinusoidally varying drug efficacies under various intracellular delays. Furthermore, numerical simulations are conducted utilizing realistic pharmacokinetics and gamma-distributed viral production delays for HIV. Our results demonstrate that the relative timing of the key viral replication cycle steps and periodic antiviral treatment schedule involving distinct drugs all can interact to critically affect the overall viral dynamics.

Appendices

A Extended Model with Reverse Transcription

We consider the following generalization of (3) with extra compartment explicitly tracking the process of reverse transcription (RT) during the eclipse phase of infected cell. Thus, the infected cells in the eclipse phase, $j(t,\tau )$, are separated into two classes $j_1(t,\tau _1)$ and $j_2(t,\tau _2)$ measuring infected cells $\tau _1$ units of time after cell infection, before RT, and $\tau _2$ units of time after RT, respectively. Then, the eclipse phase-infected cell equation in (3) is modified as follows:

$$\begin{aligned}&\left( {\partial \over \partial t}+{\partial \over \partial \tau _1}\right) j_1(t,\tau _1)=-(\nu _1(\tau _1)+\gamma _1(\tau _1))j(t,\tau _1), \quad j_1(t,0)=kS(t)V(t), \\&\left( {\partial \over \partial t}+{\partial \over \partial \tau }\right) j_2(t,\tau )=-(\nu _2(\tau )+\gamma _2(\tau ))j_2(t,\tau ),\\&j_2(t,0)=(1-\eta _1(t))\int \limits _0^\infty \gamma _1(\tau _1)j_1(t,\tau _1){\mathrm{d}}\tau _1, \\&\left( {\partial \over \partial t}+{\partial \over \partial a}\right) i(t,a)=-\mu (a)i(t,a), \quad i(t,0)=\int \limits _0^\infty \gamma _2(\tau )j_2(t,\tau ){\mathrm{d}}\tau . \end{aligned}$$

In the special case that $\gamma _1(\tau _1)=\delta (\tau _1-r)$, then we have

$$\begin{aligned} j_2(t,0)=(1-\eta _1(t)){\mathrm{e}}^{-\int _0^r\nu _1(s)}kS(t-r)V(t-r), \end{aligned}$$

which implies that

$$\begin{aligned} j_2(t,\tau _2)&=k(1-\eta _1(t-\tau _2)){\mathrm{e}}^{-\int _0^r\nu _1(s)}S(t-\tau _2-r)V(t-\tau _2-r)\\&\quad {\mathrm{e}}^{-\int _0^{\tau _2}(\nu _2(s)+\gamma _2(s)){\mathrm{d}}s} \\&=k(1-\eta _1(t-\tau +r)){\mathrm{e}}^{-\int _0^r\nu _1(s)}S(t-\tau )V(t-\tau ){\mathrm{e}}^{-\int _0^{\tau -r}(\nu _2(s)+\gamma _2(s)){\mathrm{d}}s}, \end{aligned}$$

where $\tau :=\tau _2+r$. Consequently, the differential equation for V(t) becomes

$$\begin{aligned} V'(t)&=p(t)\int \limits _0^\infty \int \limits _0^\infty q(a){\mathrm{e}}^{-\int \limits _0^a\mu (s){\mathrm{d}}s} \gamma _2(\tau ) {\mathrm{e}}^{-\int \limits _0^r\nu _1(s){\mathrm{d}}s-\int \limits _0^{\tau -r}(\nu _2(s)+\gamma _2(s)){\mathrm{d}}s}\beta (t-a-\tau +r) \\&\quad S(t-a-\tau )V(t-a-\tau ){\mathrm{d}}\tau {\mathrm{d}}\tau {\mathrm{d}}a - {\mathrm{d}}V, \end{aligned}$$

which is the same as Eq. (9) with the effective infection rate (affected by the RT inhibitor) shifted by r units of time, i.e., $\tilde{\beta }(t)=\beta (t+r)$. The corresponding relation between PDE and DDE is: $P(\tau )={\mathrm{e}}^{-\int \limits _0^r\nu _1(s){\mathrm{d}}s-\int \limits _0^{\tau -r}\nu _2(s){\mathrm{d}}s}$ and $\pi (\tau )=\gamma _2(\tau ){\mathrm{e}}^{-\int \limits _0^{\tau -r}\gamma _2(s){\mathrm{d}}s}$, $\theta :=\int \limits _0^\infty P(\tau )\pi (\tau ) {\mathrm{d}}\tau$, $f(\tau )=(P(\tau )\pi (\tau ))/ \theta$, along with $g(a)=(q(a)\sigma (a))/N$, $N=\int \limits _0^\infty q(a)\sigma (a) {\mathrm{d}}a$ where $\sigma (a)={\mathrm{e}}^{-\int \limits _0^a\mu (s){\mathrm{d}}s}$.

B Proof of Theorem 4

We proceed in the following steps.

• 1. In Sect. 3.1, we have proved that $\Psi (t)$ is point dissipative and the trajectories of any given bounded set are uniformly bounded.

• 2. We show that $\Psi (t)$ is asymptotically smooth. Fix $C>\lambda /\min \{\delta ,d\}$. It follows from Burton and Hutson (1989), Lemma 3.2 that the set

  $$\begin{aligned} B_C:=\{u\in C_\alpha ^+: \sup _{\theta \le 0}u(\theta ){\mathrm{e}}^{\alpha \theta /2}\le C\} \end{aligned}$$

  is compact in $C_\alpha ^+$. We need to prove that $B_C\times B_C\times \mathbb R_T$ attracts all bounded invariant set $\Gamma$ in $X=C_\alpha ^+\times C_\alpha ^+\times \mathbb R_T$. Fix any $(S_r,V_r,r)$ in $\Gamma$, we denote $(S_t,V_t,t+r)=\Psi (t)(S_r,V_r,r)$ such that $(S(t),V(t))=(S_t(0),V_t(0))$ satisfies system (8), (9) for $t>r$ with the initial condition $(S(r+\theta ),V(r+\theta ))=(S_r(\theta ),V_r(\theta ))$ for $\theta \le 0$. Since the limit superior of S(t) is bounded above by $\lambda /\delta$, we have $S(t)<C$ for all large t. Let $t_0\ge 0$ be the largest $t\ge r$ such that $S(t)\ge C$. If $S(t)<C$ for all $t\ge r$, we set $t_0=r$. For $t>t_0$, define

  $$\begin{aligned} u_t(\theta ):={\left\{ \begin{array}{ll} S_t(\theta ),&{}\quad t_0-t\le \theta \le 0,\\ S(t_0){\mathrm{e}}^{-\alpha (\theta -t_0+t)/2},&{}\quad \theta \le t_0-t. \end{array}\right. } \end{aligned}$$

  It is readily seen that $u_t\in B_C$. Now, we intend to show that $\Vert u_t-S_t\Vert _\alpha \rightarrow 0$ as $t\rightarrow \infty$. For $\theta \in [t_0-t,0]$, we have $u_t(\theta )=S_t(\theta )$. As $t\rightarrow \infty$, we have

  $$\begin{aligned}&u_t(\theta ){\mathrm{e}}^{\alpha \theta }=S(t_0){\mathrm{e}}^{\alpha (\theta +t_0-t)/2}\le C{\mathrm{e}}^{\alpha (t_0-t)}\rightarrow 0,~\theta \le t_0-t;\\&S_t(\theta ){\mathrm{e}}^{\alpha \theta }\le S(t+\theta ){\mathrm{e}}^{\alpha (t_0-t)} \le \sup _{r\le s\le t_0}S(s){\mathrm{e}}^{\alpha (t_0-t)}\rightarrow 0,~\theta \in [r-t,t_0-t];\\&S_t(\theta ){\mathrm{e}}^{\alpha \theta }= S_r(t-r+\theta ){\mathrm{e}}^{\alpha (\theta +t-r)}{\mathrm{e}}^{-\alpha (t-r)} \le \Vert S_r\Vert _\alpha {\mathrm{e}}^{-\alpha (t-r)}\rightarrow 0,~\theta \le r-t. \end{aligned}$$

  Therefore,

  $$\begin{aligned} \Vert u_t-S_t\Vert _\alpha&=\sup _{\theta \le t_0-t}|u_t(\theta )-S_t(\theta )|{\mathrm{e}}^{\alpha \theta } \\&\le C{\mathrm{e}}^{\alpha (t_0-t)}+\max \{\sup _{r\le s\le t_0}S(s){\mathrm{e}}^{\alpha (t_0-t)},\Vert S_r\Vert _\alpha {\mathrm{e}}^{-\alpha (t-r)}\} \rightarrow 0, \end{aligned}$$

  as $t\rightarrow \infty$. Similarly, we define

  $$\begin{aligned} v_t(\theta ):={\left\{ \begin{array}{ll} V_t(\theta ),&{}~t_1-t\le \theta \le 0,\\ V(t_1){\mathrm{e}}^{-\alpha (\theta -t_1+t)/2},&{}~\theta \le t_1-t, \end{array}\right. } \end{aligned}$$

  where $t_1$ is the largest $t\ge r$ such that $V(t)\ge C$; if $V(t)<C$ for all $t\ge r$, then we set $t_1=r$. It can be shown that $v_t\in B_C$ and $\Vert v_t-V_t\Vert _\alpha \rightarrow 0$ as $t\rightarrow \infty$. Therefore, the compact set $B_C\times B_C\times \mathbb R_T$ attracts all bounded invariant set $\Gamma \in X$, which proves asymptotic smoothness of system (8), (9).

• 3. By Hale and Waltman (1989), Theorem 2.1, $\Psi (t)$ possesses a nonempty global attractor in X. Denote $X_0=\{(u,v,r)\in X: v(0)>0\}$ and $\partial X_0=X{\setminus } X_0=\{(u,v,r)\in X: v(0)=0\}$. Introduce a generalized distance function $p:X\rightarrow \mathbb R_+$ as $p(u,v,r)=v(0)$. It is readily seen that $p^{-1}(0)=\partial X_0$ and $p^{-1}(0,\infty )=X_0$. Furthermore, by comparison principle, $p(\Psi (t)x)>0$ for all $x\in X_0$. Hence, the condition (P) in Smith and Zhao (2001), Section 3 is verified; see also Zhao (2017b), Definition 1.3.1. We now prove that the basin of attraction for $E_0\times \mathbb R_T$ does not intersect $p^{-1}(0,\infty )=X_0$. Assume to the contrary that there exists $(S_0,V_0,t_0)\in X_0$ such that $(S(t),V(t))\rightarrow (\bar{S},0)$ as $t\rightarrow \infty$, where $(S(t),V(t))=(S_t(0),V_t(0))$ with $(S_t,V_t)=U(t,t_0)(S_0,V_0)$. Since $V(0)>0$, comparison principle shows that $V(t)>0$ for all $t\ge 0$. For any $\mu ,\nu >0$, we introduce a parametrized operator on $\mathbb P_T$:

  $$\begin{aligned} (L_{\mu ,\nu }\phi )(t)= & {} \theta N\bar{S}\int \limits _0^\infty \int \limits _0^\infty \int \limits _0^\infty {\mathrm{e}}^{-\mu (s+a+\tau )-(d+\nu )s}p(t-s) g(a) f(\tau ) \beta (t-s-a-\tau )\\&\phi (t-s-a-\tau ){\mathrm{d}}\tau {\mathrm{d}}a{\mathrm{d}}s. \end{aligned}$$

  Clearly, $\rho (L_{0,0})=R_0>1$. It follows from continuity (Degla 2008, Theorem 2.1) and monotonicity (Burlando 1991, Theorem 1.1) of $L_{\mu ,\nu }$ on both $\mu$ and $\nu$ that $\rho (L_{\delta ,\delta })>1$ for some small $\delta >0$. Krein–Rutman theorem guarantees that the principal eigenfunction $\phi$ of $L_{\delta ,\delta }$ is positive. Set $\varepsilon =\bar{S}-\bar{S}/\rho (L_{\delta ,\delta })>0$ and $v(t)={\mathrm{e}}^{\delta t}\phi (t)$. It is easily seen that

  $$\begin{aligned} v(t)= & {} \theta N(\bar{S}-\varepsilon )\int \limits _0^\infty \int \limits _0^\infty \int \limits _0^\infty {\mathrm{e}}^{-(d+\delta )s}p(t-s) g(a) f(\tau ) \beta (t-s-a-\tau ) \\&v(t-s-a-\tau ){\mathrm{d}}\tau {\mathrm{d}}a{\mathrm{d}}s. \end{aligned}$$

  Differentiating both sides gives a periodic renewal equation

  $$\begin{aligned} v'(t)&=-(d+\delta )v(t)+\theta N(\bar{S}-\varepsilon )p(t)\int ^{\infty }_0\int ^{\infty }_0\beta (t-a-\tau )g(a)f(\tau )\\&\quad v(t-a-\tau ){\mathrm{d}}a{\mathrm{d}}\tau . \end{aligned}$$

  Since $S(t)\rightarrow \bar{S}$ as $t\rightarrow \infty$, there exists $t_0>0$ such that $S(t)>\bar{S}-\varepsilon$ for all $t>t_0$. Define

  $$\begin{aligned} F(t)=\theta N(\bar{S}-\varepsilon )p(t)\iint _{\tau +a\ge t-t_0}\beta (t-a-\tau )g(a)f(\tau )v(t-a-\tau ){\mathrm{d}}a{\mathrm{d}}\tau . \end{aligned}$$

  It is easy to show that $F(t)\rightarrow 0$ as $t\rightarrow \infty$. On the other hand, $v(t)={\mathrm{e}}^{\delta t}\phi (t)\rightarrow \infty$ as $t\rightarrow \infty$. There exists $t_1>t_0$, such that $F(t)<\delta v(t)$ for all $t>t_1$. Consequently, we obtain

  $$\begin{aligned} v'(t)\le & {} -d v(t)+\theta N(\bar{S}-\varepsilon )p(t)\iint _{\tau +a\le t-t_0}\beta (t-a-\tau )g(a)f(\tau )\\&v(t-a-\tau ){\mathrm{d}}a{\mathrm{d}}\tau \end{aligned}$$

  for all $t\ge t_1$. On the other hand,

  $$\begin{aligned} V'(t)\ge & {} -d V(t)+\theta N(\bar{S}-\varepsilon )p(t)\iint _{\tau +a\le t-t_0}\beta (t-a-\tau )g(a)f(\tau )\\&V(t-a-\tau ){\mathrm{d}}a{\mathrm{d}}\tau \end{aligned}$$

  for all $t\ge t_1$. Let $C=\max _{t\in [t_0,t_1]}[v(t)/V(t)]$. It follows from comparison principle that $CV(t)\ge v(t)$ for all $t\ge t_0$. This leads to a contradiction because v(t) is unbounded but V(t) vanishes as $t\rightarrow \infty$.

• 4. We demonstrate that $E_0\times \mathbb R_T$ is isolated and acyclic. Obviously, $E_0\times \mathbb R_T$ is isolated. If to the contrary $E_0\times \mathbb R_T$ is cyclic, namely, there exists a homoclinic orbit $\{S(t),V(t)\}$ that connects $E_0$ as $t\rightarrow \pm \infty$. We claim that $V(t)=0$ for all t. Otherwise, if $V(t_0)>0$ for some $t_0\in \mathbb R$, then by (9), $V(t)>0$ for all $t\ge t_0$. A similar argument as in the previous step shows that V(t) cannot converge to 0 at infinity. Hence, $V(t)=0$ for all t, which reduces (8) to a single ordinary equation and contradicts to the existence of homoclinic orbit.

• 5. All the conditions in Smith and Zhao (2001), Theorem 4.7 (see also Zhao 2017b, Theorem 1.3.2) have been verified. Therefore, there exists $\delta _0>0$ such that $\liminf _{t\rightarrow \infty }p(\Psi (t)x)>\delta _0$ for any $x\in X_0$. Let (S, V) be the solution of (8), (9) with the initial condition $(u_0,v_0)\in C_\alpha \times C_\alpha$ such that $v_0(0)>0$. Denote $S_t(\theta )=S(t+\theta )$ and $V_t(\theta )=V(t+\theta )$ for all $t\ge 0$ and $\theta \le 0$. We then have $(u_0,v_0,0)\in X_0$ and $(S_t,V_t,t)=\Psi (t)(u_0,v_0,0)$. The persistent of $\Psi (t)$ with respect to the distance function p implies that $\liminf _{t\rightarrow \infty }V(t)>\delta _0$. By choosing $\delta _0>0$ sufficiently small (and still independent of initial condition), we also obtain from (8) that $\liminf _{t\rightarrow \infty }S(t)>\delta _0$. This completes the proof.