To Cut or Not to Cut: A Modeling Approach for Assessing the Role of Male Circumcision in HIV Control

A recent randomized controlled trial shows a significant reduction in women-to-men transmission of HIV due to male circumcision. Such development calls for a rigorous mathematical study to ascertain the full impact of male circumcision in reducing HIV burden, especially in resource-poor nations where access to anti-retroviral drugs is limited. First of all, this paper presents a compartmental model for the transmission dynamics of HIV in a community where male circumcision is practiced. In addition to having a disease-free equilibrium, which is locally-asymptotically stable whenever a certain epidemiological threshold is less than unity, the model exhibits the phenomenon of backward bifurcation, where the disease-free equilibrium coexists with a stable endemic equilibrium when the threshold is less than unity. The implication of this result is that HIV may persist in the population even when the reproduction threshold is less than unity. Using partial data from South Africa, the study shows that male circumcision at 60% efficacy level can prevent up to 220,000 cases and 8,200 deaths in the country within a year. Further, it is shown that male circumcision can significantly reduce, but not eliminate, HIV burden in a community. However, disease elimination is feasible if male circumcision is combined with other interventions such as ARVs and condom use. It is shown that the combined use of male circumcision and ARVs is more effective in reducing disease burden than the combined use of male circumcision and condoms for a moderate condom compliance rate.     

The general mixing of addicts and needles in a variable-infectivity needle-sharing environment

 In this paper we develop and analyse a model for the spread of HIV/AIDS amongst a population of injecting drug users. The model we discuss focuses on the transmission of HIV through the sharing of contaminated drug injection equipment and in particular we examine the mixing of addicts and needles when the AIDS incubation period is divided into three distinct infectious stages. The impact of this assumption is to greatly increase the complexity of the HIV transmission mechanism. We begin the paper with a brief literature review followed by the derivation of a model which incorporates three classes of infectious addicts and three classes of infectious needles and where a general probability structure is used to represent the interaction of addicts and needles of varying levels of infectivity. We find that if the basic reproductive number is less than or equal to unity then there exists a globally stable disease free equilibrium. The model possesses an endemic equilibrium solution if the basic reproductive number exceeds unity. We then conduct a brief simulation study of our model. We find that the spread of disease is heavily influenced by the way addicts and needles of different levels of infectivity interact. Received: 20 September 2001 / Revised version: 21 December 2001 / Published online: 17 May 2002     

Mathematical Study of a Staged-Progression HIV Model with Imperfect Vaccine

A staged-progression HIV model is formulated and used to investigate the potential impact of an imperfect vaccine. The vaccine is assumed to have several desirable characteristics such as protecting against infection, causing bypass of the primary infection stage, and offering a disease-altering therapeutic effect (so that the vaccine induces reversal from the full blown AIDS stage to the asymptomatic stage). The model, which incorporates HIV transmission by individuals in the AIDS stage, is rigorously analyzed to gain insight into its qualitative features. Using a comparison theorem, the model with mass action incidence is shown to have a globally-asymptotically stable disease-free equilibrium whenever a certain threshold, known as the vaccination reproduction number, is less than unity. Furthermore, the model with mass action incidence has a unique endemic equilibrium whenever this threshold exceeds unity. Using the Li-Muldowney techniques for a reduced version of the mass action model, this endemic equilibrium is shown to be globally-asymptotically stable, under certain parameter restrictions. The epidemiological implications of these results are that an imperfect vaccine can eliminate HIV in a given community if it can reduce the reproduction number to a value less than unity, but the disease will persist otherwise. Furthermore, a future HIV vaccine that induces the bypass of primary infection amongst vaccinated individuals (who become infected) would decrease HIV prevalence, whereas a vaccine with therapeutic effect could have a positive or negative effect at the community level.     

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.     

Dynamical analysis of a multi-strain model of HIV in the presence of anti-retroviral drugs

One major drawback associated with the use of anti-retroviral drugs in curtailing HIV spread in a population is the emergence and transmission of HIV strains that are resistant to these drugs. This paper presents a deterministic HIV treatment model, which incorporates a wild (drug sensitive) and a drug-resistant strain, for gaining insights into the dynamical features of the two strains, and determining effective ways to control HIV spread under this situation. Rigorous qualitative analysis of the model reveals that it has a globally asymptotically stable disease-free equilibrium whenever a certain epidemiological threshold (R t 0) is less than unity and that the disease will persist in the population when this threshold exceeds unity. Further, for the case where R t 0 > 1, it is shown that the model can have two co-existing endemic equilibria, and competitive exclusion phenomenon occurs whenever the associated reproduction number of the resistant strain (R t r) is greater than that of the wild strain (R t w). Unlike in the treatment model, it is shown that the model without treatment can have a family of infinitely many endemic equilibria when its associated epidemiological threshold (R(0)) exceeds unity. For the case when [Formula in text], it is shown that the widespread use of treatment against the wild strain can lead to its elimination from the community if the associated reduction in infectiousness of infected individuals (treated for the wild strain) does not exceed a certain threshold value (in this case, the use of treatment is expected to make R t w < R t r.     

The Clinical and Economic Impact of Point-of-Care CD4 Testing in Mozambique and Other Resource-Limited Settings: A Cost-Effectiveness Analysis

Background

Point-of-care CD4 tests at HIV diagnosis could improve linkage to care in resource-limited settings. Our objective is to evaluate the clinical and economic impact of point-of-care CD4 tests compared to laboratory-based tests in Mozambique.

Methods and Findings

We use a validated model of HIV testing, linkage, and treatment (CEPAC-International) to examine two strategies of immunological staging in Mozambique: (1) laboratory-based CD4 testing (LAB-CD4) and (2) point-of-care CD4 testing (POC-CD4). Model outcomes include 5-y survival, life expectancy, lifetime costs, and incremental cost-effectiveness ratios (ICERs). Input parameters include linkage to care (LAB-CD4, 34%; POC-CD4, 61%), probability of correctly detecting antiretroviral therapy (ART) eligibility (sensitivity: LAB-CD4, 100%; POC-CD4, 90%) or ART ineligibility (specificity: LAB-CD4, 100%; POC-CD4, 85%), and test cost (LAB-CD4, US$10; POC-CD4, US$24). In sensitivity analyses, we vary POC-CD4-specific parameters, as well as cohort and setting parameters to reflect a range of scenarios in sub-Saharan Africa. We consider ICERs less than three times the per capita gross domestic product in Mozambique (US$570) to be cost-effective, and ICERs less than one times the per capita gross domestic product in Mozambique to be very cost-effective. Projected 5-y survival in HIV-infected persons with LAB-CD4 is 60.9% (95% CI, 60.9%–61.0%), increasing to 65.0% (95% CI, 64.9%–65.1%) with POC-CD4. Discounted life expectancy and per person lifetime costs with LAB-CD4 are 9.6 y (95% CI, 9.6–9.6 y) and US$2,440 (95% CI, US$2,440–US$2,450) and increase with POC-CD4 to 10.3 y (95% CI, 10.3–10.3 y) and US$2,800 (95% CI, US$2,790–US$2,800); the ICER of POC-CD4 compared to LAB-CD4 is US$500/year of life saved (YLS) (95% CI, US$480–US$520/YLS). POC-CD4 improves clinical outcomes and remains near the very cost-effective threshold in sensitivity analyses, even if point-of-care CD4 tests have lower sensitivity/specificity and higher cost than published values. In other resource-limited settings with fewer opportunities to access care, POC-CD4 has a greater impact on clinical outcomes and remains cost-effective compared to LAB-CD4. Limitations of the analysis include the uncertainty around input parameters, which is examined in sensitivity analyses. The potential added benefits due to decreased transmission are excluded; their inclusion would likely further increase the value of POC-CD4 compared to LAB-CD4.

Conclusions

POC-CD4 at the time of HIV diagnosis could improve survival and be cost-effective compared to LAB-CD4 in Mozambique, if it improves linkage to care. POC-CD4 could have the greatest impact on mortality in settings where resources for HIV testing and linkage are most limited. Please see later in the article for the Editors'' Summary     

Mathematical Study of the Role of Gametocytes and an Imperfect Vaccine on Malaria Transmission Dynamics

A mathematical model is developed to assess the role of gametocytes (the infectious sexual stage of the malaria parasite) in malaria transmission dynamics in a community. The model is rigorously analysed to gain insights into its dynamical features. It is shown that, in the absence of disease-induced mortality, the model has a globally-asymptotically stable disease-free equilibrium whenever a certain epidemiological threshold, known as the basic reproduction number (denoted by ℛ₀), is less than unity. Further, it has a unique endemic equilibrium if ℛ₀>1. The model is extended to incorporate an imperfect vaccine with some assumed therapeutic characteristics. Theoretical analyses of the model with vaccination show that an imperfect malaria vaccine could have negative or positive impact (in reducing disease burden) depending on whether or not a certain threshold (denoted by ∇) is less than unity. Numerical simulations of the vaccination model show that such an imperfect anti-malaria vaccine (with a modest efficacy and coverage rate) can lead to effective disease control if the reproduction threshold (denoted by ℛ_vac) of the disease is reasonably small. On the other hand, the disease cannot be effectively controlled using such a vaccine if ℛ_vac is high. Finally, it is shown that the average number of days spent in the class of infectious individuals with higher level of gametocyte is critically important to the malaria burden in the community.     

Theoretical Assessment of Public Health Impact of Imperfect Prophylactic HIV-1 Vaccines with Therapeutic Benefits

This paper presents a number of deterministic models for theoretically assessing the potential impact of an imperfect prophylactic HIV-1 vaccine that has five biological modes of action, namely “take,” “degree,” “duration,” “infectiousness,” and “progression,” and can lead to increased risky behavior. The models, which are of the form of systems of nonlinear differential equations, are constructed via a progressive refinement of a basic model to incorporate more realistic features of HIV pathogenesis and epidemiology such as staged progression, differential infectivity, and HIV transmission by AIDS patients. The models are analyzed to gain insights into the qualitative features of the associated equilibria. This allows the determination of important epidemiological thresholds such as the basic reproduction numbers and a measure for vaccine impact or efficacy. The key findings of the study include the following (i) if the vaccinated reproduction number is greater than unity, each of the models considered has a locally unstable disease-free equilibrium and a unique endemic equilibrium; (ii) owing to the vaccine-induced backward bifurcation in these models, the classical epidemiological requirement of vaccinated reproduction number being less than unity does not guarantee disease elimination in these models; (iii) an imperfect vaccine will reduce HIV prevalence and mortality if the reproduction number for a wholly vaccinated population is less than the corresponding reproduction number in the absence of vaccination; (iv) the expressions for the vaccine characteristics of the refined models take the same general structure as those of the basic model.     

Cost-effectiveness of HIV screening in STD clinics, emergency departments, and inpatient units: a model-based analysis

Background

Identifying and treating persons with human immunodeficiency virus (HIV) infection early in their disease stage is considered an effective means of reducing the impact of the disease. We compared the cost-effectiveness of HIV screening in three settings, sexually transmitted disease (STD) clinics serving men who have sex with men, hospital emergency departments (EDs), settings where patients are likely to be diagnosed early, and inpatient diagnosis based on clinical manifestations.

Methods and Findings

We developed the Progression and Transmission of HIV/AIDS model, a health state transition model that tracks index patients and their infected partners from HIV infection to death. We used program characteristics for each setting to compare the incremental cost per quality-adjusted life year gained from early versus late diagnosis and treatment. We ran the model for 10,000 index patients for each setting, examining alternative scenarios, excluding and including transmission to partners, and assuming HAART was initiated at a CD4 count of either 350 or 500 cells/µL. Screening in STD clinics and EDs was cost-effective compared with diagnosing inpatients, even when including only the benefits to the index patients. Screening patients in STD clinics, who have less-advanced disease, was cost-effective compared with ED screening when treatment with HAART was initiated at a CD4 count of 500 cells/µL. When the benefits of reduced transmission to partners from early diagnosis were included, screening in settings with less-advanced disease stages was cost-saving compared with screening later in the course of infection. The study was limited by a small number of observations on CD4 count at diagnosis and by including transmission only to first generation partners of the index patients.

Conclusions

HIV prevention efforts can be advanced by screening in settings where patients present with less-advanced stages of HIV infection and by initiating treatment with HAART earlier in the course of infection.     

On the dynamics of SEIRS epidemic model with transport-related infection

Transportation amongst cities is found as one of the main factors which affect the outbreak of diseases. To understand the effect of transport-related infection on disease spread, an SEIRS (Susceptible, Exposed, Infectious, Recovered) epidemic model for two cities is formulated and analyzed. The epidemiological threshold, known as the basic reproduction number, of the model is derived. If the basic reproduction number is below unity, the disease-free equilibrium is locally asymptotically stable. Thus, the disease can be eradicated from the community. There exists an endemic equilibrium which is locally asymptotically stable if the reproduction number is larger than unity. This means that the disease will persist within the community. The results show that transportation among regions will change the disease dynamics and break infection out even if infectious diseases will go to extinction in each isolated region without transport-related infection. In addition, the result shows that transport-related infection intensifies the disease spread if infectious diseases break out to cause an endemic situation in each region, in the sense of that both the absolute and relative size of patients increase. Further, the formulated model is applied to the real data of SARS outbreak in 2003 to study the transmission of disease during the movement between two regions. The results show that the transport-related infection is effected to the number of infected individuals and the duration of outbreak in such the way that the disease becomes more endemic due to the movement between two cities. This study can be helpful in providing the information to public health authorities and policy maker to reduce spreading disease when its occurs.     

Likely health outcomes for untreated acute febrile illness in the tropics in decision and economic models; a Delphi survey

Background

Modelling is widely used to inform decisions about management of malaria and acute febrile illnesses. Most models depend on estimates of the probability that untreated patients with malaria or bacterial illnesses will progress to severe disease or death. However, data on these key parameters are lacking and assumptions are frequently made based on expert opinion. Widely diverse opinions can lead to conflicting outcomes in models they inform.

Methods and Findings

A Delphi survey was conducted with malaria experts aiming to reach consensus on key parameters for public health and economic models, relating to the outcome of untreated febrile illnesses. Survey questions were stratified by malaria transmission intensity, patient age, and HIV prevalence. The impact of the variability in opinion on decision models is illustrated with a model previously used to assess the cost-effectiveness of malaria rapid diagnostic tests. Some consensus was reached around the probability that patients from higher transmission settings with untreated malaria would progress to severe disease (median 3%, inter-quartile range (IQR) 1–5%), and the probability that a non-malaria illness required antibiotics in areas of low HIV prevalence (median 20%). Children living in low transmission areas were considered to be at higher risk of progressing to severe malaria (median 30%, IQR 10–58%) than those from higher transmission areas (median 13%, IQR 7–30%). Estimates of the probability of dying from severe malaria were high in all settings (medians 60–73%). However, opinions varied widely for most parameters, and did not converge on resurveying.

Conclusions

This study highlights the uncertainty around potential consequences of untreated malaria and bacterial illnesses. The lack of consensus on most parameters, the wide range of estimates, and the impact of variability in estimates on model outputs, demonstrate the importance of sensitivity analysis for decision models employing expert opinion. Results of such models should be interpreted cautiously. The diversity of expert opinion should be recognised when policy options are debated.     

Endemic threshold results in an age-duration-structured population model for HIV infection

In this paper we consider an age-duration-structured population model for HIV infection in a homosexual community. First we investigate the invasion problem to establish the basic reproduction ratio R(0) for the HIV/AIDS epidemic by which we can state the threshold criteria: The disease can invade into the completely susceptible population if R(0)>1, whereas it cannot if R(0)<1. Subsequently, we examine existence and uniqueness of endemic steady states. We will show sufficient conditions for a backward or a forward bifurcation to occur when the basic reproduction ratio crosses unity. That is, in contrast with classical epidemic models, for our HIV model there could exist multiple endemic steady states even if R(0) is less than one. Finally, we show sufficient conditions for the local stability of the endemic steady states.     

Modelling to infer the role of animals in gambiense human African trypanosomiasis transmission and elimination in the DRC

Gambiense human African trypanosomiasis (gHAT) has been targeted for elimination of transmission (EoT) to humans by 2030. Whilst this ambitious goal is rapidly approaching, there remain fundamental questions about the presence of non-human animal transmission cycles and their potential role in slowing progress towards, or even preventing, EoT. In this study we focus on the country with the most gHAT disease burden, the Democratic Republic of Congo (DRC), and use mathematical modelling to assess whether animals may contribute to transmission in specific regions, and if so, how their presence could impact the likelihood and timing of EoT.By fitting two model variants—one with, and one without animal transmission—to the human case data from 2000–2016 we estimate model parameters for 158 endemic health zones of the DRC. We evaluate the statistical support for each model variant in each health zone and infer the contribution of animals to overall transmission and how this could impact predicted time to EoT.We conclude that there are 24/158 health zones where there is substantial to decisive statistical support for some animal transmission. However—even in these regions—we estimate that animals would be extremely unlikely to maintain transmission on their own. Animal transmission could hamper progress towards EoT in some settings, with projections under continuing interventions indicating that the number of health zones expected to achieve EoT by 2030 reduces from 68/158 to 61/158 if animal transmission is included in the model. With supplementary vector control (at a modest 60% tsetse reduction) added to medical screening and treatment interventions, the predicted number of health zones meeting the goal increases to 147/158 for the model including animal transmission. This is due to the impact of vector reduction on transmission to and from all hosts.     

Effects of public health educational campaigns and the role of sex workers on the spread of HIV/AIDS among heterosexuals

This paper presents a sex-structured model for heterosexual transmission of HIV/AIDS in which the population is divided into three subgroups: susceptibles, infectives and AIDS cases. The subgroups are further divided into two classes, consisting of individuals involved in high-risk sexual activities and individuals involved in low-risk sexual activities. The model considers the movement of individuals from high to low sexual activity groups as a result of public health educational campaigns. Thus, in this case public health educational campaigns are resulting in the split of the population into risk groups. The equilibrium and epidemic threshold, which is known as the basic reproductive number (R0), are obtained, and stability (local and global) of the disease-free equilibrium is investigated. The model is extended to incorporate sex workers, and their role in the spread of HIV/AIDS in settings with heterosexual transmission is explored. Comprehensive analytic and numerical techniques are employed in assessing the possible community benefits of public health educational campaigns in controlling HIV/AIDS. From the study, we conclude that the presence of sex workers enlarges the epidemic threshold R0, thus fuels the epidemic among the heterosexuals, and that public health educational campaigns among the high-risk heterosexual population reduces R0, thus can help slow or eradicate the epidemic.     

Investigating alcohol consumption as a risk factor for HIV transmission in heterosexual settings in sub-Saharan African communities

Alcohol consumption and abuse is widespread in sub-Saharan Africa where most HIV infections occur and has been associated with risky sexual behaviors. It may therefore be one of the most common, potentially modifiable HIV risk factors in this region. A deterministic system of ordinary differential equations incorporating heterogeneity and biased sexual preferences is formulated to assess the effects of alcohol consumption on the transmission dynamics of the disease in heterosexual settings. Extensive qualitative analysis of the model is carried out and epidemic threshold such as the alcohol-induced reproductive number $({\mathcal{R}}_{A})$ , and equilibria are derived and their stabilities examined. The disease-free equilibrium is found to be globally attracting whenever the reproductive number is less than unity. In the model, heterosexuality is the source of transmissions, and therefore, targeting a reduction of the basic reproductive number $({\mathcal{R}}_{0})$ should be primary objective for any intervention programme. We show that the preference to form partnerships amongst the heterogeneous groups influences the severity of disease and its evolution, and consequently the rate of partnership formation between females and alcohol consumers and their relative infectiousness over nondrinkers has a huge positive correlation with the alcohol-induced reproductive number and hence the epidemic. The proportion or absolute number of drinkers is shown to have minimal influence on the disease dynamics, and in a community with alcohol consumers, it is more prudent to reduce their risk sexual behavior rather than to fight the spread of alcohol consumption. Thus, intervention measures targeted at reducing heterogeneous group interactions and behavior change are the key to disease control in these settings.     

Reducing Plasmodium falciparum Malaria Transmission in Africa: A Model-Based Evaluation of Intervention Strategies

Background

Over the past decade malaria intervention coverage has been scaled up across Africa. However, it remains unclear what overall reduction in transmission is achievable using currently available tools.

Methods and Findings

We developed an individual-based simulation model for Plasmodium falciparum transmission in an African context incorporating the three major vector species (Anopheles gambiae s.s., An. arabiensis, and An. funestus) with parameters obtained by fitting to parasite prevalence data from 34 transmission settings across Africa. We incorporated the effect of the switch to artemisinin-combination therapy (ACT) and increasing coverage of long-lasting insecticide treated nets (LLINs) from the year 2000 onwards. We then explored the impact on transmission of continued roll-out of LLINs, additional rounds of indoor residual spraying (IRS), mass screening and treatment (MSAT), and a future RTS,S/AS01 vaccine in six representative settings with varying transmission intensity (as summarized by the annual entomological inoculation rate, EIR: 1 setting with low, 3 with moderate, and 2 with high EIRs), vector–species combinations, and patterns of seasonality. In all settings we considered a realistic target of 80% coverage of interventions. In the low-transmission setting (EIR∼3 ibppy [infectious bites per person per year]), LLINs have the potential to reduce malaria transmission to low levels (<1% parasite prevalence in all age-groups) provided usage levels are high and sustained. In two of the moderate-transmission settings (EIR∼43 and 81 ibppy), additional rounds of IRS with DDT coupled with MSAT could drive parasite prevalence below a 1% threshold. However, in the third (EIR = 46) with An. arabiensis prevailing, these interventions are insufficient to reach this threshold. In both high-transmission settings (EIR∼586 and 675 ibppy), either unrealistically high coverage levels (>90%) or novel tools and/or substantial social improvements will be required, although considerable reductions in prevalence can be achieved with existing tools and realistic coverage levels.

Conclusions

Interventions using current tools can result in major reductions in P. falciparum malaria transmission and the associated disease burden in Africa. Reduction to the 1% parasite prevalence threshold is possible in low- to moderate-transmission settings when vectors are primarily endophilic (indoor-resting), provided a comprehensive and sustained intervention program is achieved through roll-out of interventions. In high-transmission settings and those in which vectors are mainly exophilic (outdoor-resting), additional new tools that target exophagic (outdoor-biting), exophilic, and partly zoophagic mosquitoes will be required. Please see later in the article for the Editors'' Summary     

HIV model parameter estimates from interruption trial data including drug efficacy and reservoir dynamics

Mathematical models based on ordinary differential equations (ODE) have had significant impact on understanding HIV disease dynamics and optimizing patient treatment. A model that characterizes the essential disease dynamics can be used for prediction only if the model parameters are identifiable from clinical data. Most previous parameter identification studies for HIV have used sparsely sampled data from the decay phase following the introduction of therapy. In this paper, model parameters are identified from frequently sampled viral-load data taken from ten patients enrolled in the previously published AutoVac HAART interruption study, providing between 69 and 114 viral load measurements from 3-5 phases of viral decay and rebound for each patient. This dataset is considerably larger than those used in previously published parameter estimation studies. Furthermore, the measurements come from two separate experimental conditions, which allows for the direct estimation of drug efficacy and reservoir contribution rates, two parameters that cannot be identified from decay-phase data alone. A Markov-Chain Monte-Carlo method is used to estimate the model parameter values, with initial estimates obtained using nonlinear least-squares methods. The posterior distributions of the parameter estimates are reported and compared for all patients.     

Threshold dynamics of a non-autonomous SEIRS model with quarantine and isolation

A model for assessing the effect of periodic fluctuations on the transmission dynamics of a communicable disease, subject to quarantine (of asymptomatic cases) and isolation (of individuals with clinical symptoms of the disease), is considered. The model, which is of a form of a non-autonomous system of non-linear differential equations, is analysed qualitatively and numerically. It is shown that the disease-free solution is globally-asymptotically stable whenever the associated basic reproduction ratio of the model is less than unity, and the disease persists in the population when the reproduction ratio exceeds unity. This study shows that adding periodicity to the autonomous quarantine/isolation model developed in Safi and Gumel (Discret Contin Dyn Syst Ser B 14:209–231, 2010) does not alter the threshold dynamics of the autonomous system with respect to the elimination or persistence of the disease in the population.     

The dilution effect of the domestic animal population on the transmission of P. vivax malaria

The diversion of disease carrying insect from humans to animals may reduce transmission of diseases such as malaria. The use of animals to mitigate mosquito bites on human is called ‘zooprophylaxis’. We introduce a mathematical model for Plasmodium vivax malaria transmission with two bloodmeal hosts (humans and domestic animals) to study the effect of zooprophylaxis. After computing the basic reproduction number from the proposed model, we explore how perturbations in the parameters, sensitive to the effects of control measures, affect its value. Zooprophylaxis is shown to determine whether a basic reproduction becomes bigger than an outbreak threshold value or not. Sensitivity analysis shows that increasing the relative animal population size works better in P. vivax malaria control than decreasing the mosquito population when the relative animal population size is larger than a threshold value.     

An age-structured within-host HIV-1 infection model with virus-to-cell and cell-to-cell transmissions

In this paper, a within-host HIV-1 infection model with virus-to-cell and direct cell-to-cell transmission and explicit age-since-infection structure for infected cells is investigated. It is shown that the model demonstrates a global threshold dynamics, fully described by the basic reproduction number. By analysing the corresponding characteristic equations, the local stability of an infection-free steady state and a chronic-infection steady state of the model is established. By using the persistence theory in infinite dimensional system, the uniform persistence of the system is established when the basic reproduction number is greater than unity. By means of suitable Lyapunov functionals and LaSalle's invariance principle, it is shown that if the basic reproduction number is less than unity, the infection-free steady state is globally asymptotically stable; if the basic reproduction number is greater than unity, the chronic-infection steady state is globally asymptotically stable. Numerical simulations are carried out to illustrate the feasibility of the theoretical results.