Threshold virus dynamics with impulsive antiretroviral drug effects

Ahmadini A. Analysis and simulation study of the HIV/AIDS model using the real cases

We construct a model to investigate HIV/AIDS dynamics in real cases and study its mathematical analysis. The study examines the qualitative outcomes and confirms the local and global asymptotic stability of both the endemic equilibrium and the disease-free equilibrium. The model's criteria for exhibiting both local and global asymptotically stable behavior are examined. We compute the endemic equilibria and obtain the existence of a unique positive endemic equilibrium. The data is fitted to the model using the idea of nonlinear least-squares fitting. Accurate parameter values are achieved by fitting the data to the model using a 95% confidence interval. The basic reproduction number is computed using parameters that have been fitted or estimated. Sensitivity analysis is performed to discover the influential parameters that impact the reproduction number and the eradication of the disease. The results show that implementing preventive measures can reduce HIV/AIDS cases.

Sliding mode dynamics and optimal control for HIV model

Considering the drug treatment strategy in both virus-to-cell and cell-to-cell transmissions, this paper presents an HIV model with Filippov control. Given the threshold level $ N_t $, when the total number of infected cells is less or greater than threshold level $ N_t $, the threshold dynamics of the model are studied by using the Routh-Hurwitz Criterion. When the total number of infected cells is equal to $ N_t $, the sliding mode equations are obtained by Utkin equivalent control method, and the dynamics are studied. In addition, the optimal control strategy is introduced for the case that the number of infected cells is greater than $ N_t $. By dynamic programming, the Hamilton-Jacobi-Bellman (HJB) equation is constructed, and the optimal control is obtained. Numerical simulations are presented to illustrate the validity of our results.

Towards a new combination therapy with vectored immunoprophylaxis for HIV: Modeling "shock and kill" strategy

Latently infected cells are considered as a major barrier to curing Human Immunodeficiency Virus (HIV) infection. Reactivation of latently infected cells followed by killing the actively infected cells may be a potential strategy ("shock and kill") to purge the latent reservoir. Based on vectored immunoprophylaxis (VIP) experiment that can elicit bNAbs, in this paper a mathematical model is formulated to explore the efficacy of "shock and kill" strategy with VIP. We derive the basic reproduction number R₀ of the model and show that R₀ completely determines the dynamics of the model: if R₀<1, the disease-free equilibrium is globally asymptotically stable; if R₀>1, the system is uniformly persistent. Numerical simulations suggest that the "shock and kill" strategy with VIP can effectively control HIV infection while this strategy cannot eradicate the reservoir without VIP although it can alleviate the HIV infection. To model the administration of drugs and vaccine more realistically, pharmacokinetics and pulse vaccination are incorporated into the model of ordinary differential equations. The resultants are described by impulsive differential equations. The thresholds are obtained for the frequency and strength of the vaccination to eliminate the viruses. Furthermore, the most appropriate times are numerically investigated for starting a short-term latency-reversing agents (LRAs) treatment relative to ART considering the toxicity of LRAs. The results show that LRAs treatment at the beginning of ART might be a better option. These results have important implications for the design of HIV cure-related clinical trials.

Impulsive Reaction-Diffusion Delayed Models in Biology: Integral Manifolds Approach

In this paper we study an impulsive delayed reaction-diffusion model applied in biology. The introduced model generalizes existing reaction-diffusion delayed epidemic models to the impulsive case. The integral manifolds notion has been introduced to the model under consideration. This notion extends the single state notion and has important applications in the study of multi-stable systems. By means of an extension of the Lyapunov method integral manifolds' existence, results are established. Based on the Lyapunov functions technique combined with a Poincarè-type inequality qualitative criteria related to boundedness, permanence, and stability of the integral manifolds are also presented. The application of the proposed impulsive control model is closely related to a most important problems in the mathematical biology-the problem of optimal control of epidemic models. The considered impulsive effects can be used by epidemiologists as a very effective therapy control strategy. In addition, since the integral manifolds approach is relevant in various contexts, our results can be applied in the qualitative investigations of many problems in the epidemiology of diverse interest.

Effect of DAA therapy in hepatitis C treatment - an impulsive control approach

In this article, we have presented a mathematical model to study the dynamics of hepatitis C virus (HCV) disease considering three populations namely the uninfected liver cells, infected liver cells, and HCV with the aim to control the disease. The model possesses two equilibria namely the disease-free steady state and the endemically infected state. There exists a threshold condition (basic reproduction number) that determines the stability of the disease-free equilibrium and the number of the endemic states. We have further introduced impulsive periodic therapy using DAA into the system and studied the efficacy of the DAA therapy for hepatitis C infected patients in terms of a threshold condition. Finally, impulse periodic dosing with varied rate and time interval is adopted for cost effective disease control for finding the proper dose and dosing interval for the control of HCV disease.

Impulsive Control of Nonlinear Systems With Time-Varying Delay and Applications

Impulsive control of nonlinear delay systems is studied in this paper, where the time delays addressed may be the constant delay, bounded time-varying delay, or unbounded time-varying delay. Based on the impulsive control theory and some analysis techniques, a new theoretical result for global exponential stability is derived from the impulsive control point of view. The significance of the presented result is that the stability can be achieved via the impulsive control at certain impulse points despite the existence of impulsive perturbations which causes negative effect to the control. That is, the impulsive control provides a super performance to allow the existence of impulsive perturbations. In addition, we apply the theoretical result to the problem of impulsive control of delayed neural networks. Some results for global exponential stability and synchronization control of neural networks with time delays are derived via impulsive control. Three illustrated examples are given to show the effectiveness and distinctiveness of the proposed impulsive control schemes.

Modeling HIV Dynamics Under Combination Therapy with Inducers and Antibodies

A mathematical model is proposed to simulate the "shock-kill" strategy where broadly neutralizing antibodies (bNAbs) are injected with a combination of HIV latency activators to reduce persistent HIV reservoirs. The basic reproductive ratio of virus is computed to extrapolate how the combinational therapy of inducers and antibodies affects the persistence of HIV infection. Numerical simulations demonstrate that a proper combination of inducers and bNAbs can drive the basic reproductive ratio below unity. Interestingly, it is found that a longer dosage interval leads to the higher HIV survival opportunity and a smaller dosage interval is preferred, which is fundamental to design an optimal therapeutic scheme. Further simulations reveal the conditions under which the joint therapy of inducer and antibodies induces a large extension of viral rebound time, which highlights the mechanism of delayed viral rebound from the experiment (Halper-Stromberg et al. in Cell 158:989-999, 2014). Optimal time for cessation of treatment is also analyzed to aid practical applications.

Using non-smooth models to determine thresholds for microbial pest management

Releasing infectious pests could successfully control and eventually maintain the number of pests below a threshold level. To address this from a mathematical point of view, two non-smooth microbial pest-management models with threshold policy are proposed and investigated in the present paper. First, we establish an impulsive model with state-dependent control to describe the cultural control strategies, including releasing infectious pests and spraying chemical pesticide. We examine the existence and stability of an order-1 periodic solution, the existence of order-k periodic solutions and chaotic phenomena of this model by analyzing the properties of the Poincaré map. Secondly, we establish and analyze a Filippov model. By examining the sliding dynamics, we investigate the global stability of both the pseudo-equilibria and regular equilibria. The findings suggest that we can choose appropriate threshold levels and control intensity to maintain the number of pests at or below the economic threshold. The modelling and control outcomes presented here extend the results for the system with impulsive interventions at fixed moments.

Modelling the HIV persistence through the network of lymphocyte recirculation in vivo

Human Immunodeficiency Virus (HIV) is able to persist in cellular and/or anatomical viral reservoirs, despite the effective inhibition of virus replication by the antiretroviral therapy (ART). Here we develop a mathematical model to gain some insights of HIV persistence relevant to the lymphocyte recirculation network of immune system and the central nervous system (CNS). Our simulations and analyses illustrate the role of the CNS as a virus reservoir to prevent antiretroviral drugs from penetrating the blood-brain (or blood-testis) barrier, and we examine the long-term impact of this reservoir on the transmissibility of an infected individual. We observe numerically that level of HIV in peripheral blood may not accurately reflect the true mechanisms occurring within other organs.

Threshold dynamics of a periodic SIR model with delay in an infected compartment

Threshold dynamics of epidemic models in periodic environments attract more attention. But there are few papers which are concerned with the case where the infected compartments satisfy a delay differential equation. For this reason, we investigate the dynamical behavior of a periodic SIR model with delay in an infected compartment. We first introduce the basic reproduction number R0 for the model, and then show that it can act as a threshold parameter that determines the uniform persistence or extinction of the disease. Numerical simulations are performed to confirm the analytical results and illustrate the dependence of R0 on the seasonality and the latent period.

A PERIODIC ROSS-MACDONALD MODEL IN A PATCHY ENVIRONMENT

Based on the classical Ross-Macdonald model, in this paper we propose a periodic malaria model to incorporate the effects of temporal and spatial heterogeneity on disease transmission. The temporal heterogeneity is described by assuming that some model coefficients are time-periodic, while the spatial heterogeneity is modeled by using a multi-patch structure and assuming that individuals travel among patches. We calculate the basic reproduction number [Formula: see text] and show that either the disease-free periodic solution is globally asymptotically stable if [Formula: see text] or the positive periodic solution is globally asymptotically stable if [Formula: see text]. Numerical simulations are conducted to confirm the analytical results and explore the effect of travel control on the disease prevalence.

Resistance to protease inhibitors in a model of HIV-1 infection with impulsive drug effects

BACKGROUND

The emergence of drug resistance is one of the most prevalent reasons for treatment failure in HIV therapy. This has severe implications for the cost of treatment, survival and quality of life.

METHODS

We use mathematical modelling to describe the interaction between T cells, HIV-1 and protease inhibitors. We use impulsive differential equations to examine the effects of different levels of protease inhibitors in a T cell. We classify three different regimes according to whether the drug efficacy is low, intermediate or high. The model includes two strains: the wild-type strain, which initially dominates in the absence of drugs, and the mutant strain, which is the less efficient competitor, but has more resistance to the drugs.

RESULTS

Drug regimes may take trajectories through one, two or all three regimes, depending on the dosage and the dosing schedule. Stability analysis shows that resistance does not emerge at low drug levels. At intermediate drug levels, drug resistance is guaranteed to emerge. At high drug levels, either the drug-resistant strain will dominate or, in the absence of longer-lived reservoirs of infected cells, a region exists where viral elimination could theoretically occur. We provide estimates of a range of dosages and dosing schedules where the trajectories lie either solely within a region or cross multiple regions.

CONCLUSION

Under specific circumstances, if the drug level is physiologically tolerable, elimination of free virus is theoretically possible. This forms the basis for theoretical control using combination therapy and for understanding the effects of partial adherence.