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.     

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.     

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.     

A Contact-Network-Based Formulation of a Preferential Mixing Model

Heterogeneity in the number of potentially infectious contacts and connectivity correlations (“like attaches to like” i.e., assortatively mixed or “opposites attract” i.e., disassortatively mixed) have important implications for the value of the basic reproduction ratio R ₀ and final epidemic size. In this paper, we present a contact-network-based derivation of a simple differential equation model that accounts for preferential mixing based on the number of contacts. We show that results based on this model are in good qualitative agreement with results obtained from preferential mixing models used in the context of sexually transmitted diseases (STDs). This simple model can accommodate any mixing pattern ranging from completely disassortative to completely assortative and allows the derivation of a series of analytical results.     

Growth and reproduction in higher plants I. Theoretical analysis by mathematical models

To analyse the whole life of higher plants, an attempt was made to describe their growth and reproduction by mathematical models based on the elements determining matter production and economy of the matter. A plant body was regarded as a compound system of two parts; “productive part” and “reproductive part”. A parameter (reproductive index) was introduced to connect these two parts, and a set of the mathematical models describing the quantitative growth of these two parts were established. Two basic patterns of reproduction in higher plants were distinguished into “D-reproduction” and “I-reproduction”. The state of matter production of the mother plant determined an initial size of the daughter plant in theD-reproduction, while, in theI-reproduction, it did not determine the initial size of the daughter, but determined the number of propagules. The model of each reproduction pattern was also constructed. A formula determining the initial size of a plant in a given generation was constructed as the model of theD-reproduction. The model for theI-reproduction described the number of propagules produced in a given generation. Some aspects of the plant life, e.g. the optimum reproductive index, the switch-over time from the vegetative to the reproductive growth phase, the seed number, types of expansive reproduction, were theoretically analysed and discussed under these mathematical models.     

Modelling the effects of pre-exposure and post-exposure vaccines in tuberculosis control

Epidemic control strategies alter the spread of the disease in the host population. In this paper, we describe and discuss mathematical models that can be used to explore the potential of pre-exposure and post-exposure vaccines currently under development in the control of tuberculosis. A model with bacille Calmette-Guerin (BCG) vaccination for the susceptibles and treatment for the infectives is first presented. The epidemic thresholds known as the basic reproduction numbers and equilibria for the models are determined and stabilities are investigated. The reproduction numbers for the models are compared to assess the impact of the vaccines currently under development. The centre manifold theory is used to show the existence of backward bifurcation when the associated reproduction number is less than unity and that the unique endemic equilibrium is locally asymptotically stable when the associated reproduction number is greater than unity. From the study we conclude that the pre-exposure vaccine currently under development coupled with chemoprophylaxis for the latently infected and treatment of infectives is more effective when compared to the post-exposure vaccine currently under development for the latently infected coupled with treatment of the infectives.     

A theory of steady-state activity in nerve-fiber networks II: The simple circuit

It is found that for a simple circuit of neurons, if this contains an odd number of inhibitory fibers, or none at all, or if the product of the activity parameters is less than unity, then the stimulus pattern always determines uniquely the steady-state activity. For circuits not of one of these types, it is possible to classify exclusively and exhaustively all possible activity patterns into three types, here called “odd”, “even”, and “mixed”. For any pattern of odd type and any pattern of even type there always exists a stimulus pattern consistent with both, but in no other way can such an association of activity patterns be made.     

Global Stability of Equilibria in a Two-Sex HPV Vaccination Model

Human papillomavirus (HPV) is the primary cause of cervical carcinoma and its precursor lesions, and is associated with a variety of other cancers and diseases. A prophylactic quadrivalent vaccine against oncogenic HPV 16/18 and warts-causing genital HPV 6/11 types is currently available in several countries. Licensure of a bivalent vaccine against oncogenic HPV 16/18 is expected in the near future. This paper presents a two-sex, deterministic model for assessing the potential impact of a prophylactic HPV vaccine with several properties. The model is based on the susceptible-infective-removed (SIR) compartmental structure. Important epidemiological thresholds such as the basic and effective reproduction numbers and a measure of vaccine impact are derived. We find that if the effective reproduction number is greater than unity, there is a locally unstable infection-free equilibrium and a unique, globally asymptotically stable endemic equilibrium. If the effective reproduction number is less than unity, the infection-free equilibrium is globally asymptotically stable, and HPV will be eliminated.     

Modelling the Transmission Dynamics and Control of the Novel 2009 Swine Influenza (H1N1) Pandemic

The paper presents a deterministic compartmental model for the transmission dynamics of swine influenza (H1N1) pandemic in a population in the presence of an imperfect vaccine and use of drug therapy for confirmed cases. Rigorous analysis of the model, which stratifies the infected population in terms of their risk of developing severe illness, reveals that it exhibits a vaccine-induced backward bifurcation when the associated reproduction number is less than unity. The epidemiological consequence of this result is that the effective control of H1N1, when the reproduction number is less than unity, in the population would then be dependent on the initial sizes of the subpopulations of the model. For the case where the vaccine is perfect, it is shown that having the reproduction number less than unity is necessary and sufficient for effective control of H1N1 in the population (in such a case, the associated disease-free equilibrium is globally asymptotically stable). The model has a unique endemic equilibrium when the reproduction number exceeds unity. Numerical simulations of the model, using data relevant to the province of Manitoba, Canada, show that it reasonably mimics the observed H1N1 pandemic data for Manitoba during the first (Spring) wave of the pandemic. Further, it is shown that the timely implementation of a mass vaccination program together with the size of the Manitoban population that have preexisting infection-acquired immunity (from the first wave) are crucial to the magnitude of the expected burden of disease associated with the second wave of the H1N1 pandemic. With an estimated vaccine efficacy of approximately 80%, it is projected that at least 60% of Manitobans need to be vaccinated in order for the effective control or elimination of the H1N1 pandemic in the province to be feasible. Finally, it is shown that the burden of the second wave of H1N1 is expected to be at least three times that of the first wave, and that the second wave would last until the end of January or early February, 2010.     

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.     

Dynamics Analysis of a Multi-strain Cholera Model with an Imperfect Vaccine

A new two-strain model, for assessing the impact of basic control measures, treatment and dose-structured mass vaccination on cholera transmission dynamics in a population, is designed. The model has a globally-asymptotically stable disease-free equilibrium whenever its associated reproduction number is less than unity. The model has a unique, and locally-asymptotically stable, endemic equilibrium when the threshold quantity exceeds unity and another condition holds. Numerical simulations of the model show that, with the expected 50 % minimum efficacy of the first vaccine dose, vaccinating 55 % of the susceptible population with the first vaccine dose will be sufficient to effectively control the spread of cholera in the community. Such effective control can also be achieved if 50 % of the first vaccine dose recipients take the second dose. It is shown that a control strategy that emphasizes the use of antibiotic treatment is more effective than one that emphasizes the use of basic (non-pharmaceutical) anti-cholera control measures only. Numerical simulations show that, while the universal strategy (involving all three control measures) gives the best outcome in minimizing cholera burden in the community, the combined basic anti-cholera control measures and treatment strategy also has very effective community-wide impact.     

Analysis of Risk-Structured Vaccination Model for the Dynamics of Oncogenic and Warts-Causing HPV Types

A new deterministic model is designed and used to assess the community-wide impact of mass vaccination of new sexually active individuals on the dynamics of the oncogenic and warts-causing HPV types. Rigorous qualitative analyses of the model, which incorporates the two currently available anti-HPV vaccines, reveal that it undergoes competitive exclusion when the reproduction of one HPV risk type (low/high) exceeds unity, while that of the other HPV risk type is less than unity. For the case when the reproduction numbers of the two HPV risk types (low/high) exceed unity, the two risk types co-exist. It is shown that the sub-model with the low-risk HPV types only has at least one endemic equilibrium whenever the associated reproduction threshold exceeds unity. Furthermore, this sub-model undergoes a re-infection-induced backward bifurcation under certain conditions. In the absence of the re-infection of recovered individuals and cancer-induced mortality in males, the associated disease-free equilibrium of the full (risk-structured) model is shown to be globally asymptotically stable whenever the reproduction number of the model is less than unity (that is, the full model does not undergo backward bifurcation under this setting). It is shown, via numerical simulations, that the use of the Gardasil vaccine could lead to the effective control of HPV in the community if the coverage rate is in the range of 73–95 % (84 %). If 70 % of the new sexually active susceptible females are vaccinated with the Gardasil vaccine, additionally vaccinating 34–56 % (45 %) of the new sexually active susceptible males can lead to the effective community-wide control (or elimination) of the HPV types.     

Role of incidence function in vaccine-induced backward bifurcation in some HIV models

The phenomenon of backward bifurcation in disease models, where a stable endemic equilibrium co-exists with a stable disease-free equilibrium when the associated reproduction number is less than unity, has important implications for disease control. In such a scenario, the classical requirement of the reproduction number being less than unity becomes only a necessary, but not sufficient, condition for disease elimination. This paper addresses the role of the choice of incidence function in a vaccine-induced backward bifurcation in HIV models. Several examples are given where backward bifurcations occur using standard incidence, but not with their equivalents that employ mass action incidence. Furthermore, this result is independent of the type of vaccination program adopted. These results emphasize the need for further work on the incidence functions used in HIV models.     

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     

Backward bifurcations in dengue transmission dynamics

A deterministic model for the transmission dynamics of a strain of dengue disease, which allows transmission by exposed humans and mosquitoes, is developed and rigorously analysed. The model, consisting of seven mutually-exclusive compartments representing the human and vector dynamics, has a locally-asymptotically stable disease-free equilibrium (DFE) whenever a certain epidemiological threshold, known as the basic reproduction number(R(0)) is less than unity. Further, the model exhibits the phenomenon of backward bifurcation, where the stable DFE coexists with a stable endemic equilibrium. The epidemiological consequence of this phenomenon is that the classical epidemiological requirement of making R(0) less than unity is no longer sufficient, although necessary, for effectively controlling the spread of dengue in a community. The model is extended to incorporate an imperfect vaccine against the strain of dengue. Using the theory of centre manifold, the extended model is also shown to undergo backward bifurcation. In both the original and the extended models, it is shown, using Lyapunov function theory and LaSalle Invariance Principle, that the backward bifurcation phenomenon can be removed by substituting the associated standard incidence function with a mass action incidence. In other words, in addition to establishing the presence of backward bifurcation in models of dengue transmission, this study shows that the use of standard incidence in modelling dengue disease causes the backward bifurcation phenomenon of dengue disease.     

“Ignition” phenomena in random nets

The spread of excitation in a “random net” is investigated. It is shown that if the thresholds of individual neurons in the net are equal to unity, a positive steady state of excitation will be reached equal to γ, which previously had been computed as the weak connectivity of the net. If, however, the individual thresholds are greater than unity, either no positive steady state exists, or two such states depending on the magnitude of the axone density. In the latter case the smaller of the two steady states is unstable and hence resembles an “ignition point” of the net. If the initial stimulation (assumed instantaneous) exceeds the “ignition point,” the excitation of the net eventually assumes the greater steady state. Possible connections between this model and the phenomenon of the “preset” response are discussed.     

Stability of differential susceptibility and infectivity epidemic models

We introduce classes of differential susceptibility and infectivity epidemic models. These models address the problem of flows between the different susceptible, infectious and infected compartments and differential death rates as well. We prove the global stability of the disease free equilibrium when the basic reproduction ratio R₀ ￡ 1{\mathcal{R}_0 \leq 1} and the existence and uniqueness of an endemic equilibrium when ${\mathcal{R}_0 >1 }${\mathcal{R}_0 >1 } . We also prove the global asymptotic stability of the endemic equilibrium for a differential susceptibility and staged progression infectivity model, when ${\mathcal{R}_0 >1 }${\mathcal{R}_0 >1 } . Our results encompass and generalize those of Hyman and Li (J Math Biol 50:626–644, 2005; Math Biosci Eng 3:89–100, 2006).     

Spread of disease with transport-related infection and entry screening

An SIQS model is proposed to study the effect of transport-related infection and entry screening. If the basic reproduction number is below unity, the disease free equilibrium is locally asymptotically stable. There exists an endemic equilibrium which is locally asymptotically stable if the reproduction number is larger than unity. It is shown that the disease is endemic in the sense of permanence if and only if the endemic equilibrium exists. Entry screening is shown to be helpful for disease eradication since it can always have the possibility to eradicate the disease led by transport-related infection and furthermore have the possibility to eradicate disease even when the disease is endemic in both isolated cities.     

Opportunistic infection as a cause of transient viremia in chronically infected HIV patients under treatment with HAART

When highly active antiretroviral therapy is administered for long periods of time to HIV-1 infected patients, most patients achieve viral loads that are “undetectable” by standard assay (i.e., HIV-1 RNA < 50 copies/ml). Yet despite exhibiting sustained viral loads below the level of detection, a number of these patients experience unexplained episodes of transient viremia or viral “blips”. We propose here that transient activation of the immune system by opportunistic infection may explain these episodes of viremia. Indeed, immune activation by opportunistic infection may spur HIV replication, replenish viral reservoirs and contribute to accelerated disease progression. In order to investigate the effects of intercurrent infection on chronically infected HIV patients under treatment with highly active antiretroviral therapy (HAART), we extend a simple dynamic model of the effects of vaccination on HIV infection [Jones, L.E., Perelson, A.S., 2002. Modeling the effects of vaccination on chronically infected HIV-positive patients. JAIDS 31, 369–377] to include growing pathogens. We then propose a more realistic model for immune cell expansion in the presence of pathogen, and include this in a set of competing models that allow low baseline viral loads in the presence of drug treatment. Programmed expansion of immune cells upon exposure to antigen is a feature not previously included in HIV models, and one that is especially important to consider when simulating an immune response to opportunistic infection. Using these models we show that viral blips with realistic duration and amplitude can be generated by intercurrent infections in HAART treated patients.     

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.