A mathematical model for Chagas disease with infection-age-dependent infectivity

In this paper we develop a mathematical model for Chagas disease with infection-age-dependent infectivity. The effects of vector and blood transfusion transmission are considered, and the infected population is structured by the infection age (the time elapsed from infection). The authors identify the basic reproduction ratio R0 and show that the disease can invade into the susceptible population and unique endemic steady state exists if R0 > 1, whereas the disease dies out if R0 is small enough. We show that depending on parameters, backward bifurcation of endemic steady state can occur, so even if R0 < 1, there could exist endemic steady states. We also discuss local and global stability of steady states.     

Mathematical analysis of an age-structured population model with space-limited recruitment

In this paper, we investigate structured population model of marine invertebrate whose life stage is composed of sessile adults and pelagic larvae, such as barnacles contained in a local habitat. First we formulate the basic model as an Cauchy problem on a Banach space to discuss the existence and uniqueness of non-negative solution. Next we define the basic reproduction number R0 to formulate the invasion condition under which the larvae can successfully settle down in the completely vacant habitat. Subsequently we examine existence and stability of steady states. We show that the trivial steady state is globally asymptotically stable if R0 < or = 1, whereas it is unstable if R0 > 1. Furthermore, we show that a positive (non-trivial) steady state uniquely exists if R0 > 1 and it is locally asymptotically stable as far as absolute value of R0 - 1 is small enough.     

Reproduction numbers and thresholds in stochastic epidemic models. I. Homogeneous populations.

We compare threshold results for the deterministic and stochastic versions of the homogeneous SI model with recruitment, death due to the disease, a background death rate, and transmission rate beta cXY/N. If an infective is introduced into a population of susceptibles, the basic reproduction number, R0, plays a fundamental role for both, though the threshold results differ somewhat. For the deterministic model, no epidemic can occur if R0 less than or equal to 1 and an epidemic occurs if R0 greater than 1. For the stochastic model we find that on average, no epidemic will occur if R0 less than or equal to 1. If R0 greater than 1, there is a finite probability, but less than 1, that an epidemic will develop and eventuate in an endemic quasi-equilibrium. However, there is also a finite probability of extinction of the infection, and the probability of extinction decreases as R0 increases above 1.     

A note on persistence about structured population models

In this paper, we report some results on persistence in two structured population models: a chronic- age-structured epidemic model and an age-duration-structured epidemic model. Regarding these models, we observe that the system is uniformly strongly persistent, which means, roughly speaking, that the proportion of infected subpopulation is bounded away from 0 and the bound does not depend on the initial data after a sufficient long time, if the basic reproduction ratio is larger than one. We derive this by adopting Thieme's technique, which requires some conditions about positivity and compactness. Although the compactness condition is rather difficult to show in general infinite-dimensional function spaces, we can apply Fréchet-Kolmogorov L(1)-compactness criteria to our models. The two examples that we study illuminate a useful method to show persistence in structured population models.     

A note on persistence about structured population models

In this paper, we report some results on persistence in two structured population models: a chronic- age-structured epidemic model and an age-duration-structured epidemic model. Regarding these models, we observe that the system is uniformly strongly persistent, which means, roughly speaking, that the proportion of infected subpopulation is bounded away from 0 and the bound does not depend on the initial data after a sufficient long time, if the basic reproduction ratio is larger than one. We derive this by adopting Thieme's technique, which requires some conditions about positivity and compactness. Although the compactness condition is rather difficult to show in general infinite-dimensional function spaces, we can apply Fréchet–Kolmogorov L ¹-compactness criteria to our models. The two examples that we study illuminate a useful method to show persistence in structured population models.     

一类带有肝炎B病毒感染的数学模型的全局稳定性分析（英文）

本文主要分析了一类具有肝炎B病毒感染且带有治愈率的典型的数学模型（HBV）.通过稳定性分析,得到了该模型的无病平衡点与地方病平衡点全局稳定的充分条件,并且证明了当基本再生数R0〈1, HBV感染消失;当R0〉1,HBV感染持续.     

A baseline model for the co-evolution of hosts and pathogens

The basic reproduction ratio (R (0)) is the expected number of secondary cases per primary in a totally susceptible population. In a baseline model, faced with an individual host strain pathogen virulence evolves to maximise R (0) which yields monomorphism. The basic depression ratio (D (0)) is the amount by which the total population is decreased, per infected individual, due to the presence of infection. Again, in a baseline model, faced with an individual pathogen strain host resistance evolves to minimise D (0) which yields monomorphism. With this in mind we analyse the community dynamics of the interaction between R (0) and D (0) and show that multi-strain co-existence (polymorphism) is possible and we discuss the possibility of stable cycles occuring within the co-existence states. We show for co-existence, the number of host and pathogen strains present need to be identical in order to achieve stable equilibria. For polymorphic states we observe contingencies (outcome dependent on initial conditions) between both point equilibrium and sustained oscillations. Invasion criteria for host and pathogen strains are identified.     

Theory versus data: how to calculate R0?

To predict the potential severity of outbreaks of infectious diseases such as SARS, HIV, TB and smallpox, a summary parameter, the basic reproduction number R(0), is generally calculated from a population-level model. R(0) specifies the average number of secondary infections caused by one infected individual during his/her entire infectious period at the start of an outbreak. R(0) is used to assess the severity of the outbreak, as well as the strength of the medical and/or behavioral interventions necessary for control. Conventionally, it is assumed that if R(0)>1 the outbreak generates an epidemic, and if R(0)<1 the outbreak becomes extinct. Here, we use computational and analytical methods to calculate the average number of secondary infections and to show that it does not necessarily represent an epidemic threshold parameter (as it has been generally assumed). Previously we have constructed a new type of individual-level model (ILM) and linked it with a population-level model. Our ILM generates the same temporal incidence and prevalence patterns as the population-level model; we use our ILM to directly calculate the average number of secondary infections (i.e., R(0)). Surprisingly, we find that this value of R(0) calculated from the ILM is very different from the epidemic threshold calculated from the population-level model. This occurs because many different individual-level processes can generate the same incidence and prevalence patterns. We show that obtaining R(0) from empirical contact tracing data collected by epidemiologists and using this R(0) as a threshold parameter for a population-level model could produce extremely misleading estimates of the infectiousness of the pathogen, the severity of an outbreak, and the strength of the medical and/or behavioral interventions necessary for control.     

Stability analysis of within-host parasite models with delays

We provide a global analysis of systems of within-host parasitic infections. The systems studied have parallel classes of different length of latently infected target cells. These systems can also be thought as systems arising from within-host parasitic systems with distributed continuous delays. We compute the basic reproduction ratio R0 for the systems under consideration. If R0< or =1 the parasite is cleared, if R0>1 and if a sufficient condition is satisfied we conclude to the global asymptotic stability (GAS) of the endemic equilibrium. For some generic class of models this condition reduces to R0>1. These results make possible to revisit some parasitic models including intracellular delays and to study their global stability.     

Mathematical analysis of the global dynamics of a model for HIV infection of CD4+ T cells

A mathematical model that describes HIV infection of CD4(+) T cells is analyzed. Global dynamics of the model is rigorously established. We prove that, if the basic reproduction number R(0) < or = 1, the HIV infection is cleared from the T-cell population; if R(0) > 1, the HIV infection persists. For an open set of parameter values, the chronic-infection equilibrium P* can be unstable and periodic solutions may exist. We establish parameter regions for which P* is globally stable.     

Stochastic and deterministic models for SIS epidemics among a population partitioned into households

Models for the spread of an SIS epidemic among a population consisting of m households, each containing n individuals, are considered and their behaviour is analysed under the practically relevant situation when m is large and n small. A threshold parameter R* is determined. For the stochastic model it is shown that the epidemic has a non-zero probability of taking off if and only if R* > 1, and the extension to unequal household sizes is also considered. For the deterministic model, with households of size 2, it is shown that if R* < or = 1 then the epidemic dies out, whilst if R* > 1 the epidemic settles down to an endemic equilibrium. The usual basic reproductive ratio R0 does not provide a good indicator for the behaviour of these household epidemic models unless the household size n is large.     

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.     

The Ross-Macdonald model in a patchy environment

We generalize to n patches the Ross-Macdonald model which describes the dynamics of malaria. We incorporate in our model the fact that some patches can be vector free. We assume that the hosts can migrate between patches, but not the vectors. The susceptible and infectious individuals have the same dispersal rate. We compute the basic reproduction ratio R(0). We prove that if R(0)1, then the disease-free equilibrium is globally asymptotically stable. When R(0)>1, we prove that there exists a unique endemic equilibrium, which is globally asymptotically stable on the biological domain minus the disease-free equilibrium.     

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.     

Competition in the presence of a virus in an aquatic system: an SIS model in the chemostat

Recent research indicates that viruses are much more prevalent in aquatic environments than previously imagined. We derive a model of competition between two populations of bacteria for a single limiting nutrient in a chemostat where a virus is present. It is assumed that the virus can only infect one of the populations, the population that would be a more efficient consumer of the resource in a virus free environment, in order to determine whether introduction of a virus can result in coexistence of the competing populations. We also analyze the subsystem that results when the resistant competitor is absent. The model takes the form of an SIS epidemic model. Criteria for the global stability of the disease free and endemic steady states are obtained for both the subsystem as well as for the full competition model. However, for certain parameter ranges, bi-stability, and/or multiple periodic orbits is possible and both disease induced oscillations and competition induced oscillations are possible. It is proved that persistence of the vulnerable and resistant populations can occur, but only when the disease is endemic in the population. It is also shown that it is possible to have multiple attracting endemic steady states, oscillatory behavior involving Hopf, saddle-node, and homoclinic bifurcations, and a hysteresis effect. An explicit expression for the basic reproduction number for the epidemic is given in terms of biologically meaningful parameters. Mathematical tools that are used include Lyapunov functions, persistence theory, and bifurcation analysis.     

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.     

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.     

一类CD4＋T细胞感染HIV病毒模型的全局稳定性

研究了一类具有标准发生率的CD4＋T细胞感染HIV病毒模型的动力学性质．通过分析,得到了病毒消除与否的阚值一基本再生数．证明了当基本再生数小于1时,未感染病毒平衡点全局渐近稳定,病毒将在宿主体内被清除．当基本再生数大于1时,病毒将在宿主体内持续生存,进一步给出了病毒感染平衡点全局渐近稳定的条件．最后对所得结论进行了数值模拟．     

Modelling sexually transmitted infections: the effect of partnership activity and number of partners on R0

We model a sexually transmitted infection in a network population where individuals have different numbers of partners, separated into steady and casual partnerships, where the risk of transmission is higher in steady partnerships. An individual's number of partners of the two types defines its degree, and the degrees in the community specify the degree distribution. For this structured network population a simple model for disease transmission is defined and the basic reproduction number R0 is derived, R0 being a size-biased (i.e. biasing individuals with many partners) average number of new infections caused by individuals during the early stages of the epidemic. First a homosexual population is considered and then a heterosexual population. The heterosexual model is fitted to data from a census survey on sexual activity from the Swedish island of Gotland. The main empirical finding is that, for relevant transmission rates, the effect that so-called superspreaders have on R0 is over-estimated when not admitting for different types of partnerships.     

On a temporal model for the Chikungunya disease: modeling, theory and numerics

Reunion Island faced two episodes of Chikungunya, a vector-borne disease, in 2005 and in 2006. The latter was of unprecedented magnitude: one third of the population was infected. Until the severe episode of 2006, our knowledge of Chikungunya was very limited. The principal aim of our study is to propose a model, including human and mosquito compartments, that is associated to the time course of the first epidemic of Chikungunya. By computing the basic reproduction number R(0), we show there exists a disease-free equilibrium that is locally asymptotically stable if the basic reproduction number is less than 1. Moreover, we give a necessary condition for global asymptotic stability of the disease-free equilibrium. Then, we propose a numerical scheme that is qualitatively stable and present several simulations as well as numerical estimates of the basic reproduction number for some cities of Reunion Island. For the episode of 2005, R(0) was less than one, which partly explains why no outbreak appeared. Using recent entomological results, we investigate links between the episode of 2005 and the outbreak of 2006. Finally, our work shows that R(0) varied from place to place on the island, indicating that quick and focused interventions, like the destruction of breeding sites, may be effective for controlling the disease.