Global Dynamics of an In-host Viral Model with Intracellular Delay

The dynamics of a general in-host model with intracellular delay is studied. The model can describe in vivo infections of HIV-I, HCV, and HBV. It can also be considered as a model for HTLV-I infection. We derive the basic reproduction number R ₀ for the viral infection, and establish that the global dynamics are completely determined by the values of R ₀. If R ₀≤1, the infection-free equilibrium is globally asymptotically stable, and the virus are cleared. If R ₀>1, then the infection persists and the chronic-infection equilibrium is locally asymptotically stable. Furthermore, using the method of Lyapunov functional, we prove that the chronic-infection equilibrium is globally asymptotically stable when R ₀>1. Our results shows that for intercellular delays to generate sustained oscillations in in-host models it is necessary have a logistic mitosis term in target-cell compartments.     

A comparison of a deterministic and stochastic model for Hepatitis C with an isolation stage

We formulate a deterministic epidemic model for the spread of Hepatitis C containing an acute, chronic and isolation class and analyse the effects of the isolation class on the transmission dynamics of the disease. We calculate the basic reproduction number R₀ and show that for R₀≤1, the disease-free equilibrium is globally asymptotically stable. In addition, it is shown that for a special case when R₀>1, the endemic equilibrium is locally asymptotically stable. Furthermore, an analogous stochastic epidemic model for Hepatitis C is formulated using a continuous time Markov chain. Numerical simulations are used to estimate the mean, variance and probability distributions of the discrete random variables and these are compared to the steady-state solutions of the deterministic model. Finally, the expected time to disease extinction is estimated for the stochastic model and the impact of isolation on the time to extinction is explored.     

Age and Sex Structured Model for Assessing the Demographic Impact of Mother-to-Child Transmission of HIV/AIDS

Age and sex structured HIV/AIDS model with explicit incubation period is proposed as a system of delay differential equations. The model consists of two age groups that are children (0–14 years) and adults (15–49 years). Thus, the model considers both mother-to-child transmission (MTCT) and heterosexual transmission of HIV in a community. MTCT can occur prenatally, at labour and delivery or postnatally through breastfeeding. In the model, we consider the children age group as a one-sex formulation and divide the adult age group into a two-sex structure consisting of females and males. The important mathematical features of the model are analysed. The disease-free and endemic equilibria are found and their stabilities investigated. We use the Lyapunov functional approach to show the local stability of the endemic equilibrium. Qualitative analysis of the model including positivity and boundedness of solutions, and persistence are also presented. The basic reproductive number (ℛ₀) for the model shows that the adult population is responsible for the spread HIV/AIDS epidemic, thus up-to-date developed HIV/AIDS models to assess intervention strategies have focused much on heterosexual transmission by the adult population and the children population has received little attention. We numerically analyse the HIV/AIDS model to assess the community benefits of using antiretroviral drugs in reducing MTCT and the effects of breastfeeding in settings with high HIV/AIDS prevalence ratio using demographic and epidemiological parameters for Zimbabwe.     

Mathematical Analysis of a Two Strain HIV/AIDS Model with Antiretroviral Treatment

A two strain HIV/AIDS model with treatment which allows AIDS patients with sensitive HIV-strain to undergo amelioration is presented as a system of non-linear ordinary differential equations. The disease-free equilibrium is shown to be globally asymptotically stable when the associated epidemic threshold known as the basic reproduction number for the model is less than unity. The centre manifold theory is used to show that the sensitive HIV-strain only and resistant HIV-strain only endemic equilibria are locally asymptotically stable when the associated reproduction numbers are greater than unity. Qualitative analysis of the model including positivity, boundedness and persistence of solutions are presented. The model is numerically analysed to assess the effects of treatment with amelioration on the dynamics of a two strain HIV/AIDS model. Numerical simulations of the model show that the two strains co-exist whenever the reproduction numbers exceed unity. Further, treatment with amelioration may result in an increase in the total number of infective individuals (asymptomatic) but results in a decrease in the number of AIDS patients. Further, analysis of the reproduction numbers show that antiretroviral resistance increases with increase in antiretroviral use.     

Global behaviour of an age-infection-structured HIV model with impulsive drug-treatment strategy

In this paper, we develop an HIV model that considers the dependence of HIV/AIDS progress on infection age (duration since infection), chronological age and impulsive antiretroviral treatment. The analytical results show that there exists a globally asymptotically stable infection-free state when the impulsive period T and drug-treatment proportion p satisfy R₀(p,T)<1. The numerical simulation results indicate that there exist T and p such that R₀(p,T)<1. Due to pulses, it would be difficult to obtain the optimal pulse interval in the age distribution of population. However, our results demonstrate the effect of the impulsive drug-treatment strategy on the dynamics of HIV/AIDS.     

A model of bi-mode transmission dynamics of hepatitis C with optimal control

In this paper, we present a rigorous mathematical analysis of a deterministic model for the transmission dynamics of hepatitis C. The model is suitable for populations where two frequent modes of transmission of hepatitis C virus, namely unsafe blood transfusions and intravenous drug use, are dominant. The susceptible population is divided into two distinct compartments, the intravenous drug users and individuals undergoing unsafe blood transfusions. Individuals belonging to each compartment may develop acute and then possibly chronic infections. Chronically infected individuals may be quarantined. The analysis indicates that the eradication and persistence of the disease is completely determined by the magnitude of basic reproduction number R _c. It is shown that for the basic reproduction number R _c < 1, the disease-free equilibrium is locally and globally asymptotically stable. For R _c > 1, an endemic equilibrium exists and the disease is uniformly persistent. In addition, we present the uncertainty and sensitivity analyses to investigate the influence of different important model parameters on the disease prevalence. When the infected population persists, we have designed a time-dependent optimal quarantine strategy to minimize it. The Pontryagin’s Maximum Principle is used to characterize the optimal control in terms of an optimality system which is solved numerically. Numerical results for the optimal control are compared against the constant controls and their efficiency is discussed.     

Modelling HIV/AIDS in the presence of an HIV testing and screening campaign

Preventing and managing the HIV/AIDS epidemic in South Africa will dominate the next decade and beyond. Reduction of new HIV infections by implementing a comprehensive national HIV prevention programme at a sufficient scale to have real impact remains a priority. In this paper, a deterministic HIV/AIDS model that incorporates condom use, screening through HIV counseling and testing (HCT), regular testing and treatment as control strategies is proposed with the objective of quantifying the effectiveness of HCT in preventing new infections and predicting the long-term dynamics of the epidemic. It is found that a backward bifurcation occurs if the rate of screening is below a certain threshold, suggesting that the classical requirement for the basic reproduction number to be below unity though necessary, is not sufficient for disease control in this case. The global stabilities of the equilibria under certain conditions are determined in terms of the model reproduction number R₀. Numerical simulations are performed and the model is fitted to data on HIV prevalence in South Africa. The effects of changes in some key epidemiological parameters are investigated. Projections are made to predict the long-term dynamics of the disease. The epidemiological implications of such projections on public health planning and management are discussed.     

A Tuberculosis Model with Seasonality

The statistical data of tuberculosis (TB) cases show seasonal fluctuations in many countries. A TB model incorporating seasonality is developed and the basic reproduction ratio R ₀ is defined. It is shown that the disease-free equilibrium is globally asymptotically stable and the disease eventually disappears if R ₀<1, and there exists at least one positive periodic solution and the disease is uniformly persistent if R ₀>1. Numerical simulations indicate that there may be a unique positive periodic solution which is globally asymptotically stable if R ₀>1. Parameter values of the model are estimated according to demographic and epidemiological data in China. The simulation results are in good accordance with the seasonal variation of the reported cases of active TB in China.     

Modeling the Population Level Effects of an HIV-1 Vaccine in an Era of Highly Active Antiretroviral Therapy

First generation HIV vaccines may have limited ability to prevent infection. Instead, they may delay the onset of AIDS or reduce the infectiousness of vaccinated individuals who become infected. To assess the population level effects of such a vaccine, we formulate a deterministic model for the spread of HIV in a homosexual population in which the use of highly active antiretroviral therapy (HAART) to treat HIV infection is incorporated. The basic reproduction number R ₀ is obtained under this model. We then expand the model to include the potential effects of a prophylactic HIV vaccine. The reproduction number R _f is derived for a population in which a fraction f of susceptible individuals is vaccinated and continues to benefit from vaccination. We define f ^* as the minimum vaccination fraction for which R _f ≤1 and describe situations in which it equals the critical vaccination fraction necessary to eliminate disease. When R ₀ is large or an HIV vaccine is only partially effective, the critical vaccination fraction may exceed one. HIV vaccination, however, may still reduce the prevalence of disease if the reduction in infectiousness is at least as great as the reduction in the rate of disease progression. In particular, a vaccine that reduces infectiousness during acute infection may have an important public health impact especially if coupled with counseling to reduce risky behavior.     

Stochastic models for virus and immune system dynamics

New stochastic models are developed for the dynamics of a viral infection and an immune response during the early stages of infection. The stochastic models are derived based on the dynamics of deterministic models. The simplest deterministic model is a well-known system of ordinary differential equations which consists of three populations: uninfected cells, actively infected cells, and virus particles. This basic model is extended to include some factors of the immune response related to Human Immunodeficiency Virus-1 (HIV-1) infection. For the deterministic models, the basic reproduction number, R₀, is calculated and it is shown that if R₀<1, the disease-free equilibrium is locally asymptotically stable and is globally asymptotically stable in some special cases. The new stochastic models are systems of stochastic differential equations (SDEs) and continuous-time Markov chain (CTMC) models that account for the variability in cellular reproduction and death, the infection process, the immune system activation, and viral reproduction. Two viral release strategies are considered: budding and bursting. The CTMC model is used to estimate the probability of virus extinction during the early stages of infection. Numerical simulations are carried out using parameter values applicable to HIV-1 dynamics. The stochastic models provide new insights, distinct from the basic deterministic models. For the case R₀>1, the deterministic models predict the viral infection persists in the host. But for the stochastic models, there is a positive probability of viral extinction. It is shown that the probability of a successful invasion depends on the initial viral dose, whether the immune system is activated, and whether the release strategy is bursting or budding.     

The role of screening and treatment in the transmission dynamics of HIV/AIDS and tuberculosis co-infection: a mathematical study

In this paper, we present a deterministic non-linear mathematical model for the transmission dynamics of HIV and TB co-infection and analyze it in the presence of screening and treatment. The equilibria of the model are computed and stability of these equilibria is discussed. The basic reproduction numbers corresponding to both HIV and TB are found and we show that the disease-free equilibrium is stable only when the basic reproduction numbers for both the diseases are less than one. When both the reproduction numbers are greater than one, the co-infection equilibrium point may exist. The co-infection equilibrium is found to be locally stable whenever it exists. The TB-only and HIV-only equilibria are locally asymptotically stable under some restriction on parameters. We present numerical simulation results to support the analytical findings. We observe that screening with proper counseling of HIV infectives results in a significant reduction of the number of individuals progressing to HIV. Additionally, the screening of TB reduces the infection prevalence of TB disease. The results reported in this paper clearly indicate that proper screening and counseling can check the spread of HIV and TB diseases and effective control strategies can be formulated around ‘screening with proper counseling’.     

Spatial Heterogeneity,Host Movement and Mosquito-Borne Disease Transmission

Mosquito-borne diseases are a global health priority disproportionately affecting low-income populations in tropical and sub-tropical countries. These pathogens live in mosquitoes and hosts that interact in spatially heterogeneous environments where hosts move between regions of varying transmission intensity. Although there is increasing interest in the implications of spatial processes for mosquito-borne disease dynamics, most of our understanding derives from models that assume spatially homogeneous transmission. Spatial variation in contact rates can influence transmission and the risk of epidemics, yet the interaction between spatial heterogeneity and movement of hosts remains relatively unexplored. Here we explore, analytically and through numerical simulations, how human mobility connects spatially heterogeneous mosquito populations, thereby influencing disease persistence (determined by the basic reproduction number R ₀), prevalence and their relationship. We show that, when local transmission rates are highly heterogeneous, R ₀ declines asymptotically as human mobility increases, but infection prevalence peaks at low to intermediate rates of movement and decreases asymptotically after this peak. Movement can reduce heterogeneity in exposure to mosquito biting. As a result, if biting intensity is high but uneven, infection prevalence increases with mobility despite reductions in R ₀. This increase in prevalence decreases with further increase in mobility because individuals do not spend enough time in high transmission patches, hence decreasing the number of new infections and overall prevalence. These results provide a better basis for understanding the interplay between spatial transmission heterogeneity and human mobility, and their combined influence on prevalence and R ₀.     

On the role of long incubation periods in the dynamics of acquired immunodeficiency syndrome (AIDS)

In this study, we investigate systematically the role played by the reproductive number (the number of secondary infections generated by an infectious individual in a population of susceptibles) on single group populations models of the spread of HIV/AIDS. Our results for a single group model show that if R 1, the disease will die out, and strongly suggest that if R > 1 the disease will persist regardless of initial conditions. Our extensive (but incomplete) mathematical analysis and the numerical simulations of various research groups support the conclusion that the reproductive number R is a global bifurcation parameter. The bifurcation that takes place as R is varied is a transcritical bifurcation; in other words, when R crosses 1 there is a global transfer of stability from the infection-free state to the endemic equilibrium, and vice versa. These results do not depend on the distribution of times spent in the infectious categories (the survivorship functions). Furthermore, by keeping all the key statistics fixed, we can compare two extremes: exponential survivorship versus piecewise constant survivorship (individuals remain infectious for a fixed length of time). By choosing some realistic parameters we can see (at least in these cases) that the reproductive numbers corresponding to these two extreme cases do not differ significantly whenever the two distributions have the same mean. At any rate a formula is provided that allows us to estimate the role played by the survivorship function (and hence the incubation period) in the global dynamics of HIV. These results support the conclusion that single population models of this type are robust and hence are good building blocks for the construction of multiple group models. Our understanding of the dynamics of HIV in the context of mathematical models for multiple groups is critical to our understanding of the dynamics of HIV in a highly heterogeneous population.     

Dynamics of an epidemic model with non-local infections for diseases with latency over a patchy environment

In this paper, with the assumptions that an infectious disease in a population has a fixed latent period and the latent individuals of the population may disperse, we formulate an SIR model with a simple demographic structure for the population living in an n-patch environment (cities, towns, or countries, etc.). The model is given by a system of delay differential equations with a fixed delay accounting for the latency and a non-local term caused by the mobility of the individuals during the latent period. Assuming irreducibility of the travel matrices of the infection related classes, an expression for the basic reproduction number R₀{\mathcal{R}_0} is derived, and it is shown that the disease free equilibrium is globally asymptotically stable if R₀ < 1{\mathcal{R}_0 < 1} , and becomes unstable if ${\mathcal{R}_0 > 1}${\mathcal{R}_0 > 1} . In the latter case, there is at least one endemic equilibrium and the disease will be uniformly persistent. When n = 2, two special cases allowing reducible travel matrices are considered to illustrate joint impact of the disease latency and population mobility on the disease dynamics. In addition to the existence of the disease free equilibrium and interior endemic equilibrium, the existence of a boundary equilibrium and its stability are discussed for these two special cases.     

Thresholds for macroparasite infections

We analyse here the equilibria of an infinite system of partial differential equations modelling the dynamics of a population infected by macroparasites. We find that it is possible to define a reproduction number R₀ that satisfies the intuitive definition, and yields a sharp threshold in the behaviour of the system: if R₀ < 1, the parasite-free equilibrium (PFE) is asymptotically stable and there are no endemic equilibria; if R₀ > 1, the PFE is unstable and there exists a unique endemic equilibrium. The results mainly confirm what had been obtained in simplified models, except for the fact that no backward bifurcation occurs in this model. The stability of equilibria is established by extending an abstract linearization principle and by analysing the spectra of appropriate operators.Revised version: 14 November 2003Supported in part by CNR under Grant n. 00.0142.ST74 Metodi e modelli matematici nello studio dei fenomeni biologici     

An SIS patch model with variable transmission coefficients

In this paper, an SIS patch model with non-constant transmission coefficients is formulated to investigate the effect of media coverage and human movement on the spread of infectious diseases among patches. The basic reproduction number R₀ is determined. It is shown that the disease-free equilibrium is globally asymptotically stable if R₀?1, and the disease is uniformly persistent and there exists at least one endemic equilibrium if R₀>1. In particular, when the disease is non-fatal and the travel rates of susceptible and infectious individuals in each patch are the same, the endemic equilibrium is unique and is globally asymptotically stable as R₀>1. Numerical calculations are performed to illustrate some results for the case with two patches.     

Sex-structured HIV/AIDS model to analyse the effects of condom use with application to Zimbabwe

We present a sex-structured model for heterosexual transmission of HIV/AIDS in a community. The model is formulated using integro-differential equations, which are shown to be equivalent to delay differential equations with a time delay due to incubation period. The sex-structured HIV/AIDS model divides the population into a two sex-structure consisting of females and males. The threshold and equilibria for the model are determined and stabilities are examined. We extend the model to focus on the effects of condom use as a single-strategy approach in HIV prevention in the absence of any treatment. Initially we model the use of male condoms and further extend the model to incorporate the use of both female and male condoms. The model includes two primary factors in condom use to control HIV that are condom efficacy and compliance. The exposure risk of infection after each intervention is obtained. Basic reproductive numbers for these models are computed and compared to assess the effectiveness of male and female condom use in a community. The models are numerically analysed to assess the effectiveness of condom use on the transmission dynamics of HIV/AIDS using demographic and epidemiological parameters for Zimbabwe. The study demonstrates the use of sex-structured HIV/AIDS models in assessing the effectiveness of female and male condom use as a preventative strategy in a heterosexually active population. Z. Mukandavire would like to acknowledge financial support given by the National University of Science and Technology through a Staff Development Scholarship. The authors are grateful to Eagle Insurance Company of Zimbabwe for financial support.     

When Did HIV Incidence Peak in Harare,Zimbabwe? Back-Calculation from Mortality Statistics

HIV prevalence has recently begun to decline in Zimbabwe, a result of both high levels of AIDS mortality and a reduction in incident infections. An important component in understanding the dynamics in HIV prevalence is knowledge of past trends in incidence, such as when incidence peaked and at what level. However, empirical measurements of incidence over an extended time period are not available from Zimbabwe or elsewhere in sub-Saharan Africa. Using mortality data, we use a back-calculation technique to reconstruct historic trends in incidence. From AIDS mortality data, extracted from death registration in Harare, together with an estimate of survival post-infection, HIV incidence trends were reconstructed that would give rise to the observed patterns of AIDS mortality. Models were fitted assuming three parametric forms of the incidence curve and under nine different assumptions regarding combinations of trends in non-AIDS mortality and patterns of survival post-infection with HIV. HIV prevalence was forward-projected from the fitted incidence and mortality curves. Models that constrained the incidence pattern to a cubic spline function were flexible and produced well-fitting, realistic patterns of incidence. In models assuming constant levels of non-AIDS mortality, annual incidence peaked between 4 and 5% between 1988 and 1990. Under other assumptions the peak level ranged from 3 to 8% per annum. However, scenarios assuming increasing levels of non-AIDS mortality resulted in implausibly low estimates of peak prevalence (11%), whereas models with decreasing underlying crude mortality could be consistent with the prevalence and mortality data. HIV incidence is most likely to have peaked in Harare between 1988 and 1990, which may have preceded the peak elsewhere in Zimbabwe. This finding, considered alongside the timing and location of HIV prevention activities, will give insight into the decline of HIV prevalence in Zimbabwe.     

Interaction of maturation delay and nonlinear birth in population and epidemic models

 A population with birth rate function B(N) N and linear death rate for the adult stage is assumed to have a maturation delay T>0. Thus the growth equation N′(t)=B(N(t−T)) N(t−T) e⁻ d ₁ T−dN(t) governs the adult population, with the death rate in previous life stages d ₁≧0. Standard assumptions are made on B(N) so that a unique equilibrium N _e exists. When B(N) N is not monotone, the delay T can qualitatively change the dynamics. For some fixed values of the parameters with d ₁>0, as T increases the equilibrium N _e can switch from being stable to unstable (with numerically observed periodic solutions) and then back to stable. When disease that does not cause death is introduced into the population, a threshold parameter R ₀ is identified. When R ₀<1, the disease dies out; when R ₀>1, the disease remains endemic, either tending to an equilibrium value or oscillating about this value. Numerical simulations indicate that oscillations can also be induced by disease related death in a model with maturation delay. Received: 2 November 1998 / Revised version: 26 February 1999     

Global dynamics of a staged-progression model with amelioration for infectious diseases

We analyze the global dynamics of a mathematical model for infectious diseases that progress through distinct stages within infected hosts with possibility of amelioration. An example of such diseases is HIV/AIDS that progresses through several stages with varying degrees of infectivity; amelioration can result from a host's immune action or more commonly from antiretroviral therapies, such as highly active antiretroviral therapy. For a general n-stage model with constant recruitment and bilinear incidence that incorporates amelioration, we prove that the global dynamics are completely determined by the basic reproduction number R(0). If R(0)≤1, then the disease-free equilibrium P(0) is globally asymptotically stable, and the disease always dies out. If R(0)>1, P(0) is unstable, a unique endemic equilibrium P* is globally asymptotically stable, and the disease persists at the endemic equilibrium. Impacts of amelioration on the basic reproduction number are also investigated.