Piecewise HIV virus dynamic model with CD4(+) T cell count-guided therapy: I

Transmission dynamics of symptom-dependent HIV/AIDS models

In this study, we proposed two, symptom-dependent, HIV/AIDS models to investigate the dynamical properties of HIV/AIDS in the Fujian Province. The basic reproduction number was obtained, and the local and global stabilities of the disease-free and endemic equilibrium points were verified to the deterministic HIV/AIDS model. Moreover, the indicators $ R_0^s $ and $ R_0^e $ were derived for the stochastic HIV/AIDS model, and the conditions for stationary distribution and stochastic extinction were investigated. By using the surveillance data from the Fujian Provincial Center for Disease Control and Prevention, some numerical simulations and future predictions on the scale of HIV/AIDS infections in the Fujian Province were conducted.

A modeling framework for adaptive collective defense: crisis response in social-insect colonies

Living systems, from cells to superorganismic insect colonies, have an organizational boundary between inside and outside and allocate resources to defend it. Whereas the micro-scale dynamics of cell walls can be difficult to study, the adaptive allocation of workers to defense in social-insect colonies is more conspicuous. This is particularly the case for Tetragonisca angustula stingless bees, which combine different defensive mechanisms found across other colonial animals: (1) morphological specialization (distinct soldiers (majors) are produced over weeks); (2) age-based polyethism (young majors transition to guarding tasks over days); and (3) task switching (small workers (minors) replace soldiers within minutes under crisis). To better understand how these timescales of reproduction, development, and behavior integrate to balance defensive demands with other colony needs, we developed a demographic Filippov ODE system to study the effect of these processes on task allocation and colony size. Our results show that colony size peaks at low proportions of majors, but colonies die if minors are too plastic or defensive demands are too high or if there is a high proportion of quickly developing majors. For fast maturation, increasing major production may decrease defenses. This model elucidates the demographic factors constraining collective defense regulation in social insects while also suggesting new explanations for variation in defensive allocation at smaller scales where the mechanisms underlying defensive processes are not easily observable. Moreover, our work helps to establish social insects as model organisms for understanding other systems where the transaction costs for component turnover are nontrivial, as in manufacturing systems and just-in-time supply chains.

Global Dynamics of a Predator–Prey Model with Fear Effect and Impulsive State Feedback Control

In this paper, a predator–prey model with fear effect and impulsive state control is proposed and analyzed. By constructing an appropriate Poincaré map, the dynamic properties of the system, including the existence, nonexistence, and stability of periodic solutions are studied. More specifically, based on the biological meaning, the pulse and the phase set are firstly defined in different regions as well as the corresponding Poincaré map. Subsequently, the properties of the Poincaré map are analyzed, and the existence of a periodic solution for the system is investigated according to the properties of the Poincaré map. We found that the existence of the periodic solution for the system completely depends on the property of the Poincaré map. Finally, several examples containing numerical simulations verify the obtained theoretical result.

Global dynamics for a Filippov system with media effects

In the process of spreading infectious diseases, the media accelerates the dissemination of information, and people have a deeper understanding of the disease, which will significantly change their behavior and reduce the disease transmission; it is very beneficial for people to prevent and control diseases effectively. We propose a Filippov epidemic model with nonlinear incidence to describe media's influence in the epidemic transmission process. Our proposed model extends existing models by introducing a threshold strategy to describe the effects of media coverage once the number of infected individuals exceeds a threshold. Meanwhile, we perform the stability of the equilibriua, boundary equilibrium bifurcation, and global dynamics. The system shows complex dynamical behaviors and eventually stabilizes at the equilibrium points of the subsystem or pseudo equilibrium. In addition, numerical simulation results show that choosing appropriate thresholds and control intensity can stop infectious disease outbreaks, and media coverage can reduce the burden of disease outbreaks and shorten the duration of disease eruptions.

Conflict and accord of optimal treatment strategies for HIV infection within and between hosts

Most of previous studies investigated the optimal control of HIV infection at either within-host or between-host level. However, the optimal treatment strategy for the individual may not be optimal for the population and vice versa. To determine when the two-level optimal controls are in accord or conflict, we develop a multi-scale model using various functions that link the viral load within host and the transmission rate between hosts, calibrated by cohort data. We obtain the within-host optimal treatment scheme that minimizes the viral load and maximizes the count of healthy cells at the individual level, and the coupled optimal scheme that minimizes the basic reproduction number at the population level. Mathematical analysis shows that whether the two-level optimal controls coincide depends on the sign of the product of their switching functions. Numerical results suggest that they are in accord for a high maximal drug efficacy but may conflict for a low drug efficacy. Using the multi-scale model, we also identify a threshold of the treatment effectiveness that determines how early treatment initiation can affect the disease dynamics among population. These results may help develop a synergistic treatment protocol beneficial to both HIV-infected individuals and the whole population.

A two-thresholds policy to interrupt transmission of West Nile Virus to birds

This paper proposes a model of West Nile Virus (WNV) including threshold control policies concerning the culling of mosquitoes and birds under different conditions. Two thresholds are introduced to estimate whether and which control strategy should be implemented. For each mosquito threshold level [Formula: see text] the dynamical behaviour of the proposed non-smooth system is investigated as the bird threshold level [Formula: see text] varies, focusing on the existence of sliding domains, the existence of pseudo-equilibria, real or virtual of the endemic equilibria, global stability of these steady states, and the most interesting case of the occurrence of a novel globally asymptotically stable pseudo-attractor. The model solutions ultimately converge to a real equilibrium or a pseudo-equilibrium (if it exists), or a pseudo-attractor if no equilibrium is real and no pseudo-equilibrium exists. Here within, we show that the free system has a single stable endemic equilibrium under biologically reasonable assumptions, and show that when the control system has: (1) a bird-culling threshold that is above the bird equilibrium, culling has no advantage; (2) a bird-culling threshold that is below the bird equilibrium, but a mosquito-culling threshold that lies above the mosquito equilibrium, the infected bird population can be reduced but the infected mosquito population will remain the same; (3) a bird-culling threshold and a mosquito-culling threshold that both lie below their respective equilibrium values of the free system, then both the infected bird and mosquito populations can be reduced to lower levels. The results suggest that preset levels of the number of infected birds and infected mosquitoes can be maintained simultaneously when threshold values are chosen properly, which provides a possible control strategy when an emergent infectious disease cannot be eradicated immediately.

Personalized life expectancy and treatment benefit index of antiretroviral therapy

BACKGROUND

The progression of Human Immunodeficiency Virus (HIV) within host includes typical stages and the Antiretroviral Therapy (ART) is shown to be effective in slowing down this progression. There are great challenges in describing the entire HIV disease progression and evaluating comprehensive effects of ART on life expectancy for HIV infected individuals on ART.

METHODS

We develop a novel summative treatment benefit index (TBI), based on an HIV viral dynamics model and linking the infection and viral production rates to the Weibull function. This index summarizes the integrated effect of ART on the life expectancy (LE) of a patient, and more importantly, can be reconstructed from the individual clinic data.

RESULTS

The proposed model, faithfully mimicking the entire HIV disease progression, enables us to predict life expectancy and trace back the timing of infection. We fit the model to the longitudinal data in a cohort study in China to reconstruct the treatment benefit index, and we describe the dependence of individual life expectancy on key ART treatment specifics including the timing of ART initiation, timing of emergence of drug resistant virus variants and ART adherence.

CONCLUSIONS

We show that combining model predictions with monitored CD4 counts and viral loads can provide critical information about the disease progression, to assist the design of ART regimen for maximizing the treatment benefits.

A piecewise model of virus-immune system with two thresholds

The combined antiretroviral therapy with interleukin (IL)-2 treatment may not be enough to preclude exceptionally high growth of HIV virus nor rebuilt the HIV-specific CD4 or CD8 T-cell proliferative immune response for management of HIV infected patients. Whether extra inclusion of immune therapy can induce the HIV-specific immune response and control HIV replication remains challenging. Here a piecewise virus-immune model with two thresholds is proposed to represent the HIV-1 RNA and effector cell-guided therapy strategies. We first analyze the dynamics of the virus-immune system with effector cell-guided immune therapy only and prove that there exists a critical level of the intensity of immune therapy determining whether the HIV-1 RAN virus loads can be controlled below a relative low level. Our analysis of the global dynamics of the proposed model shows that the pseudo-equilibrium can be globally stable or locally bistable with order 1 periodic solution or bistable with the virus-free periodic solution under various appropriate conditions. This indicates that HIV viral loads can either be eradicated or stabilize at a previously given level or go to infinity (corresponding to the effector cells oscillating), depending on the threshold levels and the initial HIV virus loads and effector cell counts. Comparing with the single threshold therapy strategy we obtain that with two thresholds therapy strategies either virus can be eradicated or the controllable region, where HIV viral loads can be maintained below a certain value, can be enlarged.

Early HAART Initiation May Not Reduce Actual Reproduction Number and Prevalence of MSM Infection: Perspectives from Coupled within- and between-Host Modelling Studies of Chinese MSM Populations

Having a thorough understanding of the infectivity of HIV, time of initiating treatment and emergence of drug resistant virus variants is crucial in mitigating HIV infection. There are many challenges to evaluating the long-term effect of the Highly Active Antiretroviral Therapy (HAART) on disease transmission at the population level. We proposed an individual based model by coupling within-host dynamics and between-host dynamics and conduct stochastic simulation in the group of men who have sex with men (MSM). The mean actual reproduction number is estimated to be 3.6320 (95% confidence interval: [3.46, 3.80]) for MSM group without treatment. Stochastic simulations show that given relatively high (low) level of drug efficacy after emergence of drug resistant variants, early initiation of treatment leads to a less (greater) actual reproduction number, lower (higher) prevalence and less (more) incidences, compared to late initiation of treatment. This implies early initiation of HAART may not always lower the actual reproduction number and prevalence of infection, depending on the level of treatment efficacy after emergence of drug resistant virus variants, frequency of high-risk behaviors and etc. This finding strongly suggests early initiation of HAART should be implemented with great care especially in the settings where the effective drugs are limited. Coupling within-host dynamics with between-host dynamics can provide critical information about impact of HAART on disease transmission and thus help to assist treatment strategy design and HIV/AIDS prevention and control.

An avian-only Filippov model incorporating culling of both susceptible and infected birds in combating avian influenza

Depopulation of birds has always been an effective method not only to control the transmission of avian influenza in bird populations but also to eliminate influenza viruses. We introduce a Filippov avian-only model with culling of susceptible and/or infected birds. For each susceptible threshold level [Formula: see text], we derive the phase portrait for the dynamical system as we vary the infected threshold level [Formula: see text], focusing on the existence of endemic states; the endemic states are represented by real equilibria, pseudoequilibria and pseudo-attractors. We show generically that all solutions of this model will approach one of the endemic states. Our results suggest that the spread of avian influenza in bird populations is tolerable if the trajectories converge to the equilibrium point that lies in the region below the threshold level [Formula: see text] or if they converge to one of the pseudoequilibria or a pseudo-attractor on the surface of discontinuity. However, we have to cull birds whenever the solution of this model converges to an equilibrium point that lies in the region above the threshold level [Formula: see text] in order to control the outbreak. Hence a good threshold policy is required to combat bird flu successfully and to prevent overkilling birds.

Piecewise virus-immune dynamic model with HIV-1 RNA-guided therapy

Clinical studies have used CD4 T cell counts to evaluate the safety or risk of plasma HIV-1 RNA-guided structured treatment interruptions (STIs), aimed at maintaining CD4 T cell counts above a safe level and plasma HIV-1 RNA below a certain level. However, quantifying and evaluating the impact of STIs on the control of HIV replication and on activation of the immune response remains challenging. Here we extend the virus-immune dynamic system by including a piecewise smooth function to describe the elimination of HIV viral loads and the activation of effector cells under plasma HIV-1 RNA-guided therapy, in order to quantitatively explore the STI strategies. We theoretically investigate the global dynamics of the proposed Filippov system. Our main results indicate that HIV viral loads could either go to infinity or be maintained below a certain level or stabilize at a previously given level, depending on the threshold level and initial HIV virus loads and effector cell counts. This suggests that proper combinations of threshold and initial HIV virus loads and effector cell counts, based on threshold policy, can successfully preclude exceptionally high growth of HIV virus and, in particular, maximize the controllable region.

Global stability of an infection-age structured HIV-1 model linking within-host and between-host dynamics

Although much evidence shows the inseparable interaction between the within-host progression of HIV-1 infection and the transmission of the disease at the population level, few models coupling the within-host and between-host dynamics have been developed. In this paper, we adopt the nested approach, viewing the transmission rate at each stage (primary, chronic, and AIDS stage) of HIV-1 infection as a saturated function of the viral load, to formulate an infection-age structured epidemic model. We explicitly link the individual and the host population scale, and derive the basic reproduction number R0 for the coupled system. To analyze the model and perform a detailed global dynamics analysis, two Lyapunov functionals are constructed to prove the global asymptotical stability of the disease-free and endemic equilibria. Theoretical results indicate that R0 provides a threshold value determining whether or not the disease dies out. Numerical simulations are presented to quantitatively investigate the influence of the within-host viral dynamics on between-host transmission dynamics. The results suggest that increasing the effectiveness of inhibitors can decrease the basic reproduction number, but can also increase the overall infected population because of a lower disease-induced mortality rate and a longer lifespan of HIV infected individuals.

Modeling antiretroviral drug responses for HIV-1 infected patients using differential equation models

We review mathematical modeling and related statistical issues of HIV dynamics primarily in response to antiretroviral drug therapy in this article. We start from a basic model of virus infection and then review a number of more advanced models with consideration of pharmacokinetic factors, adherence and drug resistance. Specifically, we illustrate how mathematical models can be developed and parameterized to understand the effects of long-term treatment and different treatment strategies on disease progression. In addition, we discuss a variety of parameter estimation methods for differential equation models that are applicable to either within- or between-host viral dynamics.