Analysis of an HIV model with post-treatment control

Recent investigation indicated that latent reservoir and immune impairment are responsible for the post-treatment control of HIV infection. In this paper, we simplify the disease model with latent reservoir and immune impairment and perform a series of mathematical analysis. We obtain the basic infection reproductive number R0 to characterize the viral dynamics. We prove that when R0 < 1, the uninfected equilibrium of the proposed model is globally asymptotically stable. When R0 > 1, we obtain two thresholds, the post-treatment immune control threshold and the elite control threshold. The model has bistable behaviors in the interval between the two thresholds. If the proliferation rate of CTLs is less than the post-treatment immune control threshold, the model does not have positive equilibria. In this case, the immune free equilibrium is stable and the system will have virus rebound. On the other hand, when the proliferation rate of CTLs is greater than the elite control threshold, the system has stable positive immune equilibrium and unstable immune free equilibrium. Thus, the system is under elite control. Author summary In this article, we use mathematical model to investigate the combined effect of latent reservoir and immune impairment on the post-treatment control of HIV infection. By simplifying an HIV model with latent reservoir and immune impairment, and performing mathematical analysis, we obtain the post-treatment immune control threshold and the elite control threshold for the HIV dynamics when R0 > 1. The HIV model displays bistable behaviors in the interval between the two thresholds. We illustrate our results using both mathematical analysis and numerical simulation. Our result is consistent with recent medical experiment. We show that patient with low proliferation rate of CTLs may undergo virus rebound, and patient with high proliferation rate of CTLs may obtain elite control of HIV infection. We perform bifurcation analysis to illustrate the infection status of patient with the variation of proliferation rate of CTLs, which potentially explain the reason behind different outcomes among HIV patients.


Introduction
In 2010, an HIV-infected mother gave birth to a baby prematurely in a Mississippi 14 clinic. The infant was known as the 'Mississippi baby'. Before delivery, the mother was 15 not diagnosed with HIV infection did not receive antiretroviral treatment [26]. At the 16 age of 30 hours, the baby received liquid, triple-drug antiretroviral treatment. Such 17 treatment was terminated at the age of 18 months and since then, the virus level in the 18 baby remains undetectable. Though it was thought that the baby was cured of HIV, a 19 routine clinical test on July 10, 2014 showed that the level of virus in the 'Mississippi 20 baby' became detectable (16,750 copies/ml) [26]. 21 Antiretroviral therapy (ART) is effective in inhibiting the HIV infection and 22 prolongs the life of infected individuals. However, due to the existence of latent 23 reservoirs, it is unable to totally eliminate the virus infection [7,8,12,13,48]. The time 24 it takes the virus to rebound varies. For example, the virus level of the Mississippi baby 25 remains undetectable for years before the virus rebound [26,30]. Sometimes, a host may 26 have low virus load after antiretroviral therapy. Investigations have been carried out to 27 reveal the causes of low virus level and virus rebound [9,30,38]. 28 Conway and Perelson constructed a mathematical model to investigate the dynamics 29 of virus rebound [9]. Their investigation reveals the interplay between immune response 30 and latent reservoir, and shows that post-treatment control may appear. Recent 31 investigations indicated that early antiretroviral therapy may be responsible for the 32 development of post-treatment control with plasma virus remaining undetectable after 33 the cessation of treatment. However, only a small proportion of patients receiving early 34 antiretroviral therapy developed post-treatment control. Further investigations are to be 35 carried out to reveal the reasons behind this. 36 Treasure et al investigated the HIV rebound in patients who terminated the 37 antiretroviral therapy. They showed that a patient who discontinued the antiretroviral 38 therapy may or may not undergo immediate HIV rebound [38]. 39 As an important approach to investigate disease transmission, mathematical 40 modeling provides insights into interactions between viral and host factors. Evaluating 41 the behaviors of the viral models yields a better understanding of the disease and is 42 beneficial to the development of appropriate therapy strategies. In the literature, 43 mathematical models of within-host viral dynamics have been designed 44 [1,3,10,11,15,[27][28][29][44][45][46]. Immune response has also been integrated into within 45 host models to investigate the combined effects of viral dynamics and immune process 46 of the host [6,16,23,36,40,41,43,49]. 47 Regoes et al. [32] incorporated immune impairment into viral models to consider the 48 effects that target cell limitation and immune responses have on the evolution of virus. 49 Their investigations indicated that the immune system of the host may collapse when 50 the impairment rate of HIV surpasses its threshold value. Iwami et al. [17,18] 51 investigated the HIV dynamics with immune impairment using mathematical models.

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The authors got the 'risky threshold' and 'immunodeficiency threshold' by performing 53 analysis. The results implied that the immune system always collapses when the 54 impairment rate is greater than the threshold value. Immune impairment in within-host 55 virus models have received much attention in the literature [2,37,39].

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HIV latent reservoir is responsible for the rebound in HIV viral load. As a major 57 barrier to the eradication of HIV-1 virus, latent reservoir poses persistent risks to the 58 hosts. The infected cells in the latent reservoir remain undetectable to the immune 59 system and can be reactivated to produce virions with the termination of drug therapy 60 [19,20,33,34,42]. Investigations showed that the size of the virus reservoir is relatively 61 stable [42]. For a patient under sufficient antiretroviral therapy (ART), ongoing viral 62 replication rate in the reservoir remains low [19] Perelson integrated the post treatment into an HIV model and performed analysis [9]. 69 Here, we simplify the model proposed in [9] to obtain effective, we have = 1 [9,33].

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In the literature, the immune and immune impairment function cyz 1+ηy − bz − myz 76 has been applied to the viral models to characterize the interaction between the immune 77 cells and the productively infected CD4 + T cells [11,31,39]. Wang and Liu [39] 78 constructed a within-host viral dynamics models to consider HIV infection with immune 79 impairment. In this article, we consider the post-treatment immune control, the 80 biological implication behind the 'Mississippi baby'. By mathematical analysis, we 81 obtain the threshold of proliferation rate of CTLs, which determines the HIV infection 82 status. We also perform bifurcation analysis and demonstrate the bistable behavior of 83 the model, which is consistence with results from recent medical trial.  In the following, we show that system (1.1) is well-posed.
Furthermore, the solution is bounded.

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Proof. It follows from the fundamental theory of ordinary differential equations [14] 94 that there exists a unique solution to system (1.1) with nonnegative initial conditions.

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In this section, we consider the existence of the equilibria to system (1.1).
is the basic infection reproductive number. Here, R 0 is the expected number of newly 119 infected cells generated from an infected cell at the beginning of the infectious process. 120 bmη and c 2 = m + bη + 2 √ bmη. We then get the existence conditions 122 for the positive equilibria.
(iii) If R * − > 1 and c > c 2 , system (1.1) has an immune equilibrium Here Denote We then have the following results. 124 If c < c * and one of the conditions c < c 1 or c > c 2 holds, then R * − is always greater  Table 1. The existence of the positive equilibria when 1 < R 0 < R c . Table 2. The existence of the positive equilibria when R 0 > R c > 1.
In this section, we consider the stability of the equilibria of system (1.1).

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LetẼ be any arbitrary equilibrium of system (1.1). Its corresponding Jacobian 150 matrix is obtained as The characteristic equation of the linearized system of (1.1) atẼ is given by It is easy to see that equation (3.2) has two negative roots, obtained as The other eigenvalues are determined by If R 0 < 1, we have a 1 > 0 and a 2 > 0, and as such equation (3.4) has two negative 161 roots. Thus, E 0 is locally stable for R 0 < 1.

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If R 0 > 1, from (3.5) we know that E 0 is a saddle, and hence unstable. The proof of 163 Theorem 3.1 is complete.

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Biologically, the global asymptotic stability of the uninfected equilibrium E 0 of 175 system (1.1) implies that the virus will die out in the host if the treatment is strong 176 enough to ensure R 0 < 1.

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Proof. The characteristic equation of the linearized system of (1.1) at E 1 is given by Clearly, We then consider the sign of the eigenvalue Let ∆ = 0, we have c = c 1 or c = c 2 .

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(i) If ∆ = 0, then c = c 1 or c = c 2 , which is a critical situation.

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Biologically, if the proliferation rate of CTLs is less than the critical value c * * , the 196 viral load can be at high level.  system (1.1) has an immune equilibrium E * − , which is a stable node.

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Proof. The characteristic equation of the linearized system of (1.1) at an arbitrary 208 positive equilibrium E * is given by . It thus follows that 212 Routh-Hurartz Criterion, we know that the positive equilibrium E * + is a stable node in 215 this case.

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(ii) For equilibrium E * + , if R 0 > R c > 1 and c 2 < c < c * * , then c (1+ηy * + ) 2 − m < 0 217 and A 4 < 0. Thus, equilibrium E * + is an unstable saddle for R 0 > R c and c 2 < c < c * * . 218 By Theorem 3.3 and Theorem 3.4, we have the following result.  Tables 3 and 4. Table 3. The stabilities of the equilibria and the behaviors of system (1.1) in the case 1 < R 0 < R c . Here, c * * is the critical value, and we assume Table 4. The stabilities of the equilibria and the behaviors of system (1.1) in the case R 0 > R c > 1. Here, c 2 , c * and c * * are critical values, and c 2 is a saddle-node bifurcation point. Here we assume Sensitive analysis provides insights into the basic infection reproductive number R 0 with 226 respect to system parameters [47]. In this section, we use latin hypercube sampling 227 (LHS) and partial rank correlation coefficients (PRCCs) [4,24] to reveal the dependence 228 of the basic infection reproduction number R 0 on a variety of system parameters. As a 229 statistical sampling method, LHS provides efficient analysis of parameter variations 230 across simultaneous uncertainty ranges in each parameter [4]. PRCC, which is obtained 231 from the rank transformed LHS matrix and output matrix [24], indicates the 232 parameters that have the most significant influences on the behaviors of the model. In 233 this work, we perform 4000 simulations per run. We use a uniform distribution function 234 to test the PRCCs for a variety of system parameters.

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The PRCC results of the model, Fig. 1, illustrate the dependence of R 0 on different 236 system parameters. The estimations of the distributions for R 0 is approximately a 237 normal distribution. We use |PRCC| as an index to test if the parameter has important 238 correlation with the infection reproduction number R 0 . If |PRCC| > 0.4, we say that 239 the correlation is strong. If 0.4 ≥ |PRCC| > 0.2, we say that the correlation is moderate. 240 For 0.2 ≥ |PRCC| > 0, there correlation is weak. As is shown in Fig. 1, the general rate 241 of CD4 + T cells s, the decay rate of CD4 + T cells d, the infection rate of CD4 + T cells 242 β, the drug efficacy and the latently infected cell death rate d L have significant 243 influence on the infection reproduction number R 0 . In this section, we carry out numerical simulations to consider the HIV dynamics of our 246 model. The parameter values are listed in Table 5. We then calculate the thresholds 247 R 0 ≈ 3.0030 > 1, R c ≈ 1.4243, c 2 ≈ 0.2914 and c * * ≈ 0.4988. Notice that We then get the bistable interval 249 (0.2914, 0.4988). In this case, when c < c 2 , the immune-free equilibrium E 1 is stable.

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When c 2 < c < c * * , the immune-free equilibrium E 1 and the positive equilibrium E * + 251 are stable. When c > c * * , only the positive equilibrium E * + is stable.
252 Fig.2 shows that there is no positive equilibrium if c < 0.2914 and a saddle-node 253 bifurcation appear when c passes through 0.2914. The system display bistable behavior 254 for 0.2914 < c < 0.4988. As an example, we simulate the time history of the system for 255 c = 0.45 ∈ (0.2914, 0.4988) with different initial conditions (see Fig. 3). We find that, 256 with the same parameter values and different initial conditions, the system may 257 converge to different equilibriums. Such simulation result is consistent with recent clinic 258 trial performed by Treasure et al [38].
We also consider the influence of system parameters on the elite control threshold The proliferation rate of latently infected cells ρ plays an important role in the elite 284 control. It is worth carrying out further investigation to reveal the viral dynamics of the 285 within host model with logistic proliferation rate of latently infected cells, given by 286 system (5.1).
Using the same method of analyzing system (1.1), we can get theoretical results.