Modeling the role of macrophages in HIV persistence during antiretroviral therapy

Abstract

HIV preferentially infects activated CD4+ T cells. Current antiretroviral therapy cannot eradicate the virus. Viral infection of other cells such as macrophages may contribute to viral persistence during antiretroviral therapy. In addition to cell-free virus infection, macrophages can also get infected when engulfing infected CD4+ T cells as innate immune sentinels. How macrophages affect the dynamics of HIV infection remains unclear. In this paper, we develop an HIV model that includes the infection of CD4+ T cells and macrophages via cell-free virus infection and cell-to-cell viral transmission. We derive the basic reproduction number and obtain the local and global stability of the steady states. Sensitivity and viral dynamics simulations show that even when the infection of CD4+ T cells is completely blocked by therapy, virus can still persist and the steady-state viral load is not sensitive to the change of treatment efficacy. Analysis of the relative contributions to viral replication shows that cell-free virus infection leads to the majority of macrophage infection. Viral transmission from infected CD4+ T cells to macrophages during engulfment accounts for a small fraction of the macrophage infection and has a negligible effect on the total viral production. These results suggest that macrophage infection can be a source contributing to HIV persistence during suppressive therapy. Improving drug efficacies in heterogeneous target cells is crucial for achieving HIV eradication in infected individuals.

Appendices

Appendix A: Proof of Theorem 3.1

From (5), we obtain the characteristic polynomial for the infection-free steady state $E^{0}$

$$\begin{aligned} (\lambda +d_{1})(\lambda +d_{3})\bigg (\lambda +d_{2}(1-R_{01})\bigg )\bigg (\lambda ^{2}+(d_{4}+d_{5})\lambda +d_{4}d_{5}(1-R_{02})\bigg )=0, \end{aligned}$$

(22)

where $\lambda$ is the eigenvalue. It is easy to see that Eq. (22) has negative solutions $\lambda _{1}=-d_{1}$ and $\lambda _{2}=-d_{3}$. Thus, the stability of $E^{0}$ is determined by the roots of following equation

$$\begin{aligned} \bigg (\lambda +d_{2}(1-R_{01})\bigg )\bigg (\lambda ^{2}+(d_{4}+d_{5})\lambda +d_{4}d_{5}(1-R_{02})\bigg )=0. \end{aligned}$$

(23)

Let

$$\begin{aligned} F(\lambda )=\bigg (\lambda +d_{2}(1-R_{01})\bigg )\bigg (\lambda ^{2}+(d_{4}+d_{5})\lambda +d_{4}d_{5}(1-R_{02})\bigg ). \end{aligned}$$

When $R_{0}>1$, $F(\lambda )=0$ has at least one positive real root. When $R_{0}<1$, all roots of $F(\lambda )=0$ have negative real parts. Thus, the infection-free equilibrium $E^{0}$ is locally asymptotically stable when $R_{0}<1$ and unstable when $R_{0}>1$.

Appendix B: Proof of Theorem 3.2

From the Jacobian matrix (5), we obtain the characteristic equation of (2) at the infected steady state $\bar{E}$

$$\begin{aligned} (\lambda +d_{1})\bigg (\lambda +d_{2}(1-R_{01})\bigg )\bigg ((\lambda +d_{3}R_{02})[(\lambda +d_{4})(\lambda +d_{5})-d_{4}d_{5}]+d_{3}d_{4}d_{5}(R_{02}-1)\bigg )=0. \end{aligned}$$

(24)

It’s clear that $\lambda _{1}=-d_{1}<0$ and $\lambda _{2}=d_{2}(R_{01}-1)<0$ when $R_{01}<1$. The remaining eigenvalues are determined by the following equation

$$\begin{aligned} \lambda ^{3}+a_{1}\lambda ^{2}+a_{2}\lambda +a_{3}=0, \end{aligned}$$

(25)

where

$$\begin{aligned} a_{1}=d_{3}R_{02}+d_{4}+d_{5}>0, \quad a_{2}=d_{3}R_{02}(d_{4}+d_{5})>0,\quad a_{3}=d_{3}d_{4}d_{5}(R_{02}-1). \end{aligned}$$

(26)

When $R_{02}>1$, the inequality $a_{3}>0$ holds. Moreover, we have

$$\begin{aligned} a_{1}a_{2}-a_{3}&=d_{3}R_{02}(d_{4}+d_{5})(d_{3}R_{02}+d_{4}+d_{5})-d_{3}d_{4}d_{5}(R_{02}-1)\\&=d_{3}R_{02}(d_{4}+d_{5})(d_{3}R_{02}+d_{4})+d_{3}d_{5}(d_{5}R_{02}+d_{4})\\&>0. \end{aligned}$$

By the Routh-Hurwitz criterion, it follows that all roots of (24) have negative real parts. Thus, the equilibrium $\bar{E}$ is locally asymptotically stable when $R_{01}<1<R_{02}$.

Appendix C: Uniform persistence of system (2)

In this Appendix, we will show that system (2) is uniformly persistent when $R_{01}>1$.

We define the following sets

$$\begin{aligned} X&=\big \{(H_{1},I_{1},H_{2},I_{2},V)\in \mathbb {R}^{5}|H_{1}>0,I_{1}\ge 0,H_{2}>0,I_{2}\ge 0,V\ge 0\big \},\\ X_{0}&=\big \{(H_{1},I_{1},H_{2},I_{2},V)\in X|H_{1}>0,I_{1}>0,H_{2}>0,I_{2}>0,V>0\big \},\\ \partial X_{0}&=X{\setminus } X_{0}. \end{aligned}$$

It suffices to show that model (2) is uniformly persistent with respect to $(X_{0},\partial X_{0})$. From model (2), we have that X and $X_{0}$ are positively invariant. Obviously, $\partial X_{0}$ is relatively closed in X and (2) is point dissipative.

Let $\psi (t)$ be the solution semiflow of system (2) for all $t>0$. Denote the global attractor of $\psi (t)$ in $\partial X_{0}$ by $A_{\partial }$, we have

$$\begin{aligned} \bigcup _{x\in A_{\partial }}\omega (x)=\big \{E_{0},\bar{E}\big \}. \end{aligned}$$

By Theorems 4.1 and 4.2, we conclude that $\big \{E_{0},\bar{E}\big \}$ is isolated and is an acyclic covering in $\partial X_{0}$. According to Theorem 4.6 of Thieme (1993), we only need to show that $W^{s}(E_{0})\cap X_{0}=\varnothing$ and $W^{s}(\bar{E})\cap X_{0}=\varnothing$ if $R_{01}>1$.

We first prove that $W^{s}(E_{0})\cap X_{0}=\varnothing$. Assume the contrary, that is, $W^{s}(E_{0})\cap X_{0}\ne \varnothing$. Then, there exists a positive solution $(\tilde{H_{1}}(t),\tilde{I_{1}}(t),\tilde{H_{2}}(t),\tilde{I_{2}}(t),\tilde{V}(t))$ with $(\tilde{H_{1}}(0),\tilde{I_{1}}(0),\tilde{H_{2}}(0),\tilde{I_{2}}(0),\tilde{V}(0))\in X_{0}$ such that $(\tilde{H_{1}}(t),\tilde{I_{1}}(t),\tilde{H_{2}}(t),\tilde{I_{2}}(t),\tilde{V}(t))\rightarrow E_{0}=(H_{1}^{0},0,H_{2}^{0},0,0)$ as $t\rightarrow +\infty$. Because $R_{01}>1$, we can choose a small $\delta >0$ such that $(1-g\epsilon )\beta _{1}(H_{1}^{0}-\delta )>d_{2}$. Thus, for sufficiently large t, we have

$$\begin{aligned} H_{1}^{0}-\delta&<\tilde{H_{1}}\le H_{1}^{0}+\delta , \quad 0\le \tilde{I_{1}}\le \delta , \end{aligned}$$

and

$$\begin{aligned} \tilde{I_{1}}^{'}\ge (1-g\epsilon )\beta _{1}(H_{1}^{0}-\delta )\tilde{I_{1}}-d_{2}\tilde{I_{1}}. \end{aligned}$$

Consider the following perturbed system

$$\begin{aligned} I_{1}^{'}=(1-g\epsilon )\beta _{1}(H_{1}^{0}-\delta )I_{1}-d_{2}I_{1}. \end{aligned}$$

Direct calculation shows that when $(1-g\epsilon )\beta _{1}(H_{1}^{0}-\delta )>d_{2}$, the above system has a unique positive equilibrium $I_{1}^{*}(\delta )=\frac{(1-g\epsilon )\beta _{1}(H_{1}^{0}-\delta )}{d_{2}}$, and it is globally asymptotically stable. This, together with the comparison principle, implies that $\tilde{I_{1}}(t)>0$ for all sufficiently large t. It is a contradiction with $\tilde{I_{1}}(t)\rightarrow 0$ as $t\rightarrow +\infty$. Thus, $W^{s}(E_{0})\cap X_{0}=\varnothing$. Similarly, we can prove $W^{s}(\bar{E})\cap X_{0}=\varnothing$. Thus, the system (2) is uniformly persistent when $R_{01}>1$.