Dynamics of an HIV Model with Multiple Infection Stages and Treatment with Different Drug Classes

Abstract

Highly active antiretroviral therapy can effectively control HIV replication in infected individuals. Some clinical and modeling studies suggested that viral decay dynamics may depend on the inhibited stages of the viral replication cycle. In this paper, we develop a general mathematical model incorporating multiple infection stages and various drug classes that can interfere with specific stages of the viral life cycle. We derive the basic reproductive number and obtain the global stability results of steady states. Using several simple cases of the general model, we study the effect of various drug classes on the dynamics of HIV decay. When drugs are assumed to be 100 % effective, drugs acting later in the viral life cycle lead to a faster or more rapid decay in viremia. This is consistent with some patient and experimental data, and also agrees with previous modeling results. When drugs are not 100 % effective, the viral decay dynamics are more complicated. Without a second population of long-lived infected cells, the viral load decline can have two phases if drugs act at an intermediate stage of the viral replication cycle. The slopes of viral load decline depend on the drug effectiveness, the death rate of infected cells at different stages, and the transition rate of infected cells from one to the next stage. With a second population of long-lived infected cells, the viral load decline can have three distinct phases, consistent with the observation in patients receiving antiretroviral therapy containing the integrase inhibitor raltegravir. We also fit modeling prediction to patient data under efavirenz (a nonnucleoside reverse-transcriptase inhibitor) and raltegravir treatment. The first-phase viral load decline under raltegravir therapy is longer than that under efavirenz, resulting in a lower viral load at initiation of the second-phase decline in patients taking raltegravir. This explains why patients taking a raltegravir-based therapy were faster to achieve viral suppression than those taking an efavirenz-based therapy. Taken together, this work provides a quantitative and systematic comparison of the effect of different drug classes on HIV decay dynamics and can explain the viral load decline in HIV patients treated with raltegravir-containing regimens.

Appendix

We prove the global stability of steady states of the general multi-stage model (Eq. 1). Inspired by the results in Huang et al. (2010), we define a Lyapunov function $L_0=L_0(x,y_1,y_2,\ldots ,y_n,v)$ as follows

$$\begin{aligned} L_0 \displaystyle= & {} x(t)-x_0-x_0\ln \frac{x(t)}{x_0}+y_1(t)+\frac{k_1}{\widetilde{k}_1}y_2(t)\\&\displaystyle \ \ +\,\frac{k_1k_2}{\widetilde{k}_1\widetilde{k}_2}y_3(t)+\cdots +\prod \limits _{i=1}^{n-1}\Big (\frac{k_i}{\widetilde{k}_i} \Big )y_n(t)+\prod \limits _{i=1}^n\Big (\frac{k_i}{\widetilde{k}_i}\Big )v(t). \end{aligned}$$

Calculating the derivative of $L_0$ along the positive solution of system (1) and using $\lambda =\mu x_0$, $R_0\le 1$, we have

$$\begin{aligned} \displaystyle \frac{{\hbox {d}}L_0}{{\hbox {d}}t}\Big |_{(1)} \displaystyle= & {} \lambda -\mu x-\frac{x_0}{x}(\lambda -\mu x)+(k x_0-\prod \limits _{i=1}^n \Big (\frac{k_i}{\widetilde{k}_i}\Big )c)v\nonumber \\ \displaystyle= & {} \mu (x_0-x)\left( 1-\frac{x_0}{x}\right) +\prod \limits _{i=1}^n \Big (\frac{k_i}{\widetilde{k}_i}\Big )c\left[ \frac{k\lambda }{c\mu }\prod \limits _{i=1}^n \Big (\frac{\widetilde{k}_i}{k_i}\Big )-1\right] v\nonumber \\ \displaystyle= & {} -\,\frac{\mu (x-x_0)^2}{x}+\prod \limits _{i=1}^n \Big (\frac{k_i}{\widetilde{k}_i}\Big )c(R_0-1)v\nonumber \\\le & {} 0. \end{aligned}$$

(15)

$\dot{L}_0\Big |_{(1)}=0$ if and only if $v(t)=0$ (or $R_0= 1$) and $x=x_0$. Therefore, the maximal compact invariant set in $\{\dot{L}_0\Big |_{(1)}=0\}$ is the singleton $\{E_0\}$. The global asymptotical stability of the infection-free steady state $E_0$ follows from LaSalle Invariance Principle (Hale et al. 1993).

To prove the global stability of the infected steady state, we define the Lyapunov function $L_1=L_1(x,y_1,y_2,\ldots ,y_n,v)$ as follows

$$\begin{aligned} \begin{array}{llll} L_1&{}\displaystyle =x(t)-x^{*}-x^{*}\ln \frac{x(t)}{x^{*}}+y_1(t)-y_1^{*}-y_1^{*}\ln \frac{y_1(t)}{y_1^{*}}\\ &{}\displaystyle \ \ \ +\sum \limits _{i=2}^n\prod \limits _{j=1}^{i-1}\Big (\frac{k_j}{\widetilde{k}_j}\Big ) \left( y_i(t)-y_i^{*}-y_i^{*}\ln \frac{y_i(t)}{y_i^{*}}\right) \\ &{}\displaystyle \ \ \ +\prod \limits _{i=1}^n\Big (\frac{k_i}{\widetilde{k}_i}\Big ) \left( v(t)-v^{*}-v^{*}\ln \frac{v(t)}{v^{*}}\right) . \end{array} \end{aligned}$$

(16)

Calculating the derivative of $L_1$ along the positive solution of system (1), we obtain

$$\begin{aligned} \displaystyle \frac{{\hbox {d}}L_1}{{\hbox {d}}t}\Big |_{(1)} \displaystyle= & {} \dot{x}-\frac{x^{*}}{x}\dot{x}+\dot{y}_1-\frac{y_1^{*}}{y_1}\dot{y}_1+\prod \limits _{i=1}^n\frac{k_i}{\widetilde{k}_i}\left( \dot{v}-\frac{v^{*}}{v}\dot{v}\right) \nonumber \\&\displaystyle +\left\{ \frac{k_1}{\widetilde{k}_1}\left( \dot{y}_2-\frac{y_2^{*}}{y_2}\dot{y}_2\right) +\frac{k_1k_2}{\widetilde{k}_1\widetilde{k}_2}\left( \dot{y}_3-\frac{y_3^{*}}{y_3}\dot{y}_3\right) \right. \nonumber \\&\left. \quad +\cdots +\prod \limits _{i=1}^{n-1}\Big (\frac{k_i}{\widetilde{k}_i}\Big )\left( \dot{y}_n-\frac{y_n^{*}}{y_n}\dot{y}_n\right) \right\} \nonumber \\ \displaystyle= & {} \lambda -\mu x-k xv-\frac{x^{*}}{x}(\lambda -\mu x-k xv) +kxv-k_1y_1-\frac{y_1^{*}}{y_1}(k xv-k_1y_1)\nonumber \\&\displaystyle \ \ \ +\,\prod \limits _{i=1}^n\Big (\frac{k_i}{\widetilde{k}_i}\Big )(\widetilde{k}_ny_n-cv)-\prod \limits _{i=1}^n\Big (\frac{k_i}{\widetilde{k}_i}\Big )\frac{v^{*}}{v}(\widetilde{k}_ny_n-cv) \nonumber \\&\displaystyle +\frac{k_1}{\widetilde{k}_1}(\widetilde{k}_1y_1-k_2y_2)-\frac{k_1}{\widetilde{k}_1}\cdot \frac{y_2^{*}}{y_2}(\widetilde{k}_1y_1-k_2y_2)\nonumber \\&\displaystyle +\,\frac{k_1k_2}{\widetilde{k}_1\widetilde{k}_2}(\widetilde{k}_2y_2-k_3y_3)-\frac{k_1k_2}{\widetilde{k}_1\widetilde{k}_2}\cdot \frac{y_3^{*}}{y_3}(\widetilde{k}_2y_2-k_3y_3)+\cdots \nonumber \\&\displaystyle +\,\prod \limits _{i=1}^{n-1}\Big (\frac{k_i}{\widetilde{k}_i}\Big )(\widetilde{k}_{n-1}y_{n-1}-k_ny_n)-\prod \limits _{i=1}^{n-1}\Big (\frac{k_i}{\widetilde{k}_i}\Big )\frac{y_n^{*}}{y_n}(\widetilde{k}_{n-1}y_{n-1}-k_ny_n)\nonumber \\ \displaystyle= & {} \lambda -\mu x-\frac{x^{*}}{x}(\lambda -\mu x-k xv)-\frac{y_1^{*}}{y_1}(k xv-k_1y_1)\nonumber \\&\displaystyle -\,\prod \limits _{i=1}^n\Big (\frac{k_i}{\widetilde{k}_i}\Big )c v-\prod \limits _{i=1}^n\Big (\frac{k_i}{\widetilde{k}_i}\Big )\frac{v^{*}}{v}(\widetilde{k}_ny_n-cv) -\frac{k_1}{\widetilde{k}_1}\cdot \frac{y_2^{*}}{y_2}(\widetilde{k}_1y_1-k_2y_2)\nonumber \\&\displaystyle -\,\frac{k_1k_2}{\widetilde{k}_1\widetilde{k}_2}\cdot \frac{y_3^{*}}{y_3}(\widetilde{k}_2y_2-k_3y_3)-\cdots -\prod \limits _{i=1}^{n-1}\Big (\frac{k_i}{\widetilde{k}_i}\Big )\frac{y_n^{*}}{y_n}(\widetilde{k}_{n-1}y_{n-1}-k_ny_n). \end{aligned}$$

(17)

From the infected steady state, we have

$$\begin{aligned} \displaystyle k x^{*}v^{*}= & {} k_1y_1^{*},\ \lambda =\mu x^{*}+k x^{*}v^{*}=\mu x^{*}+k_1y_1^{*}, \nonumber \\ \displaystyle \widetilde{k}_1y_1^{*}= & {} k_2y_2^{*},\ \widetilde{k}_2y_2^{*}=k_3y_3^{*},\ \cdots ,\ \widetilde{k}_{n-1}y_{n-1}^{*}=k_ny_n^{*},\nonumber \\ \widetilde{k}_ny_n^{*}= & {} cv^{*},\ cv^{*}=k_1y_1^{*}\prod \limits _{i=1}^n \Big (\frac{\widetilde{k}_i}{k_i}\Big ),\ k_1y_1^{*}=\prod \limits _{i=1}^{n-1}\Big ( \frac{k_i}{\widetilde{k}_i}\Big )k_ny_n^{*},\nonumber \\ \displaystyle \frac{k_1k_2}{\widetilde{k}_1}y_2^{*}= & {} \frac{k_1}{\widetilde{k}_1}\cdot k_2y_2^{*}=\frac{k_1}{\widetilde{k}_1}\cdot \widetilde{k}_1y_1^{*}=k_1y_1^{*}, \nonumber \\ \displaystyle \frac{k_1k_2k_3}{\widetilde{k}_1\widetilde{k}_2}y_3^{*}= & {} \frac{k_1k_2}{\widetilde{k}_1\widetilde{k}_2}\cdot \widetilde{k}_2y_2^{*}=\frac{k_1}{\widetilde{k}_1}\cdot k_2y_2^{*}=\frac{k_1}{\widetilde{k}_1}\cdot \widetilde{k}_1y_1^{*}=k_1y_1^{*},\nonumber \\&\cdots \nonumber \\ \displaystyle \prod \limits _{i=1}^{n-1}\Big ( \frac{k_i}{\widetilde{k}_i}\Big )k_ny_n^{*}= & {} \prod \limits _{i=1}^{n-1}\Big ( \frac{k_i}{\widetilde{k}_i}\Big )\widetilde{k}_{n-1}y_{n-1}^{*}=\prod \limits _{i=1}^{n-2}\Big ( \frac{k_i}{\widetilde{k}_i}\Big )k_{n-1}y_{n-1}^{*}=\cdots =k_1y_1^{*}. \end{aligned}$$

(18)

Therefore, we have

$$\begin{aligned} \displaystyle k x^{*}v= & {} k_1y_1^{*}\cdot \frac{x^{*}v}{x^{*}v^{*}}=k_1y_1^{*}\cdot \frac{v}{v^{*}},\nonumber \\ \displaystyle \frac{y_1^{*}}{y_1}\cdot kxv= & {} \frac{y_1^{*}}{y_1}\cdot \frac{k_1y_1^{*}}{x^{*}v^{*}}xv=k_1y_1^{*}\cdot \frac{y_1^{*}xv}{y_1x^{*}v^{*}},\nonumber \\ \displaystyle \frac{k_1k_2}{\widetilde{k}_1}y_2^{*}= & {} \frac{k_1k_2k_3}{\widetilde{k}_1\widetilde{k}_2}y_3^{*}=\cdots =\prod \limits _{i=1}^{n-1}\Big (\frac{k_i}{\widetilde{k}_i}\Big )\cdot k_ny_n^{*}=k_1y_1^{*},\nonumber \\ \displaystyle \prod \limits _{i=1}^n\Big (\frac{k_i}{\widetilde{k}_i}\Big )cv= & {} k_1y_1^{*}\cdot \frac{v}{v^{*}},\ \prod \limits _{i=1}^n \Big (\frac{k_i}{\widetilde{k}_i}\Big )cv^{*}=k_1y_1^{*},\nonumber \\ \displaystyle \prod \limits _{i=1}^n\Big (\frac{k_i}{\widetilde{k}_i}\Big )\cdot \frac{v^{*}}{v}\widetilde{k}_{n}y_{n}= & {} \prod \limits _{i=1}^{n-1} \Big (\frac{k_i}{\widetilde{k}_i}\Big )k_ny_n\cdot \frac{v^{*}}{v} =k_1y_1^{*}\cdot \frac{y_nv^{*}}{y_n^{*}v},\nonumber \\ \displaystyle k_1\frac{y_1y_2^{*}}{y_2}= & {} k_1y_1\cdot \frac{y_2^{*}}{y_2}=k_1y_1^{*}\cdot \frac{y_1y_2^{*}}{y_1^{*}y_2},\nonumber \\ \displaystyle \frac{k_1k_2}{\widetilde{k}_1}\frac{y_2y_3^{*}}{y_3}= & {} \frac{k_1}{\widetilde{k}_1}k_2y_2\cdot \frac{y_3^{*}}{y_3}=k_1y_1^{*}\cdot \frac{y_2y_3^{*}}{y_2^{*}y_3}, \nonumber \\&\vdots&\nonumber \\ \displaystyle \prod \limits _{i=1}^{n-2}\Big (\frac{k_i}{\widetilde{k}_i}\Big )k_{n-1}\frac{y_{n-1}y_n^{*}}{y_n}= & {} k_1y_1^{*}\cdot \frac{y_{n-1}y_n^{*}}{y_{n-1}^{*}y_n}. \end{aligned}$$

(19)

From (17) and (19), we get

$$\begin{aligned} \displaystyle \frac{{\hbox {d}}L_1}{{\hbox {d}}t}\Big |_{(1)} \displaystyle= & {} \lambda -\mu x-\frac{x^{*}}{x}(\lambda -\mu x)+k x^{*}v-\frac{y_1^{*}}{y_1}k xv+k_1y_1^{*}\nonumber \\&\displaystyle -\,k_1\cdot \frac{y_1y_2^{*}}{y_2}-\frac{k_1k_2}{\widetilde{k}_1}\cdot \frac{y_2y_3^{*}}{y_3}-\cdots - \prod \limits _{i=1}^{n-2}\Big (\frac{k_i}{\widetilde{k}_i}\Big )k_{n-1}\frac{y_{n-1}y_n^{*}}{y_n}\nonumber \\&\displaystyle +\,\frac{k_1k_2}{\widetilde{k}_1}y_2^{*}+\frac{k_1k_2k_3}{\widetilde{k}_1\widetilde{k}_2}y_3^{*}+\cdots + \prod \limits _{i=1}^{n-1}\Big (\frac{k_i}{\widetilde{k}_i}\Big )k_ny_n^{*}\nonumber \\&\displaystyle -\,\prod \limits _{i=1}^{n}\Big (\frac{k_i}{\widetilde{k}_i}\Big )cv+\prod \limits _{i=1}^{n}\Big (\frac{k_i}{\widetilde{k}_i}\Big )c v^{*}-\prod \limits _{i=1}^{n}\Big (\frac{k_i}{\widetilde{k}_i}\Big )\frac{v^{*}}{v}\widetilde{k}_ny_n\nonumber \\ \displaystyle= & {} \mu x^{*}+k_1y_1^{*}-\mu x-\frac{x^{*}}{x}(\mu x^{*}+k_1y_1^{*}-\mu x)+k_1y_1^{*}\frac{v}{v^{*}}-k_1y_1^{*}\frac{y_1^{*}xv}{y_1x^{*}v^{*}}\nonumber \\&\displaystyle +\,k_1y_1^{*} -k_1y_1^{*}\frac{v}{v^{*}}+k_1y_1^{*}-k_1y_1^{*}\frac{y_nv^{*}}{y_n^{*}v}-k_1y_1^{*}\frac{y_1y_2^{*}}{y_1^{*}y_2}\nonumber \\&\displaystyle -\,\frac{k_1}{\widetilde{k}_1}k_2y_2\cdot \frac{y_3^{*}}{y_3} -\cdots -\prod \limits _{i=1}^{n-2}\Big (\frac{k_i}{\widetilde{k}_i}\Big )\cdot k_{n-1}y_{n-1}\cdot \frac{y_n^{*}}{y_n}+(n-1)k_1y_1^{*}\nonumber \\ \displaystyle= & {} \mu x^{*}(1-\frac{x^{*}}{x})-\mu x(1-\frac{x^{*}}{x})+(n+2)k_1y_1^{*}\nonumber \\&\displaystyle +\,k_1y_1^{*}\left( -\frac{y_1^{*}xv}{y_1x^{*}v^{*}}-\frac{y_nv^{*}}{y_n^{*}v}-\frac{x^{*}}{x}\right) \nonumber \\&\displaystyle -\,k_1y_1^{*}\left( \frac{y_1y_2^{*}}{y_1^{*}y_2}+\frac{y_2y_3^{*}}{y_2^{*}y_3}+\cdots +\frac{y_{n-1}y_n^{*}}{y_{n-1}^{*}y_n}\right) \nonumber \\ \displaystyle= & {} -\,\frac{\mu (x-x^{*})^2}{x}+k_1y_1^{*}\left( n+2-\frac{y_1y_2^{*}}{y_1^{*}y_2}-\frac{y_2y_3^{*}}{y_2^{*}y_3}-\cdots - \frac{y_{n-1}y_n^{*}}{y_{n-1}^{*}y_n}\right. \nonumber \\&\displaystyle \left. -\,\frac{y_nv^{*}}{y_n^{*}v}-\frac{x^{*}}{x}-\frac{y_1^{*}xv}{y_1x^{*}v^{*}}\right) \nonumber \\ \displaystyle\le & {} 0. \end{aligned}$$

(20)

The last inequality follows from the fact that the arithmetic mean is always greater than or equal to the geometric mean, i.e.,

$$\begin{aligned} \frac{y_1y_2^{*}}{y_1^{*}y_2}+\frac{y_2y_3^{*}}{y_2^{*}y_3}+\cdots + \frac{y_{n-1}y_n^{*}}{y_{n-1}^{*}y_n}+\frac{y_nv^{*}}{y_n^{*}v}+\frac{x^{*}}{x}+\frac{y_1^{*}xv}{y_1x^{*}v^{*}}\ge n+2. \end{aligned}$$

$\dot{L}_1\Big |_{(1)}=0$ if and only if $x=x^*$ and the left side of the above inequality is equal to $n+2$, i.e.,

$$\begin{aligned} \frac{y_1}{y_1^{*}}=\frac{y_2}{y_2^{*}}=\cdots =\frac{y_n}{y_n^{*}} =\frac{v}{v^{*}}. \end{aligned}$$

(21)

From Eq. (21), we have $v=v^*$ and $y_i=y_i^*$, $i=1,2,\cdots ,n$. Therefore, we conclude that $\dot{L}_1\Big |_{(1)}\le 0$ holds for all $x,y_i,v>0$. The maximal compact invariant set in $\{\dot{L}_1\Big |_{(1)}=0\}$ is the singleton $\{E^{*}\}$. We finish the proof of the global asymptotical attractivity of the infected steady state $E^{*}$.