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We consider a model for a disease with a progressing and a quiescent exposed class and variable susceptibility to super-infection. The model exhibits backward bifurcations under certain conditions, which allow for both stable and unstable endemic states when the basic reproduction number is smaller than one.
Cutaneous leishmaniasis is a vector-borne disease transmitted to humans by sandflies. In this paper, we develop a mathematical model which takes into account the seasonality of the vector population and the distribution of the latent period from infection to symptoms in humans. Parameters are fitted to real data from the province of Chichaoua, Morocco. We also introduce a generalization of the definition of the basic reproduction number R (0) which is adapted to periodic environments. This R (0) is estimated numerically for the epidemic in Chichaoua; approximately 1.94. The model suggests that the epidemic could be stopped if the vector population were reduced by a factor approximately 3.76.
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.
Empirical evidence shows that childhood diseases persist in large communities whereas in smaller communities the epidemic goes extinct (and is later reintroduced by immigration). The present paper treats a stochastic model describing the spread of an infectious disease giving life-long immunity, in a community where individuals die and new individuals are born. The time to extinction of the disease starting in quasi-stationarity (conditional on non-extinction) is exponentially distributed. As the population size grows the epidemic process converges to a diffusion process. Properties of the limiting diffusion are used to obtain an approximate expression for tau, the mean-parameter in the exponential distribution of the time to extinction for the finite population. The expression is used to study how tau depends on the community size but also on certain properties of the disease/community: the basic reproduction number and the means and variances of the latency period, infectious period and life-length. Effects of introducing a vaccination program are also discussed as is the notion of the critical community size, defined as the size which distinguishes between the two qualitatively different behaviours.
Drug treatment of patients with schistosomiasis may select for drug-resistant parasites. In this article, we formulate a deterministic model with multiple strains of schistosomes (helminth parasites with a two-host life cycles) in order to explore the role of drug treatment in the maintenance of a polymorphism of parasite strains that differ in their resistance levels. The basic reproductive numbers for all strains are computed, and are shown to determine the stabilities of equilibria of the model and consequently the distribution of parasite phenotypes with different levels of drug tolerance. Analysis of our model shows that the likelihood that resistant strains will increase in frequency depends on the interplay between their relative fitness, the cost of resistance, and the degree of selection pressure exerted by the drug treatments.
While there is a large literature on the form of epidemic waves in the time domain, models of their structure and shape in the spatial domain remain poorly developed. This paper concentrates on the changing spatial distribution of an epidemic wave over time and presents a simple method for identifying the leading and trailing edges of the spatial advance and retreat of such waves. Analysis of edge characteristics is used to (a) disaggregate waves into 'swash' and 'backwash' stages, (b) measure the phase transitions of areas from susceptible, S, through infective, I, to recovered, R, status (S --> I --> R) as dimensionless integrals and (c) estimate a spatial version of the basic reproduction number, R(0). The methods used are illustrated by application to measles waves in Iceland over a 60-year period from 1915 to 1974. Extensions of the methods for use with more complex waves are possible through modifying the threshold values used to define the start and end points of an event.
Nipah virus (NiV) is a highly pathogenic zoonotic RNA virus belonging to the genus Henipavirus within the family Paramyxoviridae, representing a continuing global health concern due to its high case fatality rate and potential for epidemic expansion in the era of increasing international connectivity. The virus demonstrates strong evolutionary adaptability driven by the absence of proofreading mechanisms during RNA replication, enabling genetic diversification that may influence host range, virulence, and transmission dynamics. Molecular pathogenesis of NiV is primarily mediated through interaction of viral glycoproteins with ephrin-B2 and ephrin-B3 receptors, facilitating host cell entry, endothelial damage, and neuroinvasion. Immune evasion facilitated by the action of accessory proteins encoded by the P gene (P, V, W, and C) acts to suppress innate antiviral immunity through the inhibition of interferon induction and JAK/STAT signaling. Human-to-human transmission of Nipah virus remains limited, with epidemiological evidence indicating basic reproduction numbers generally below unity; however, respiratory involvement and healthcare-associated exposure may enhance cluster outbreaks. Global travel, ecological disruption, and fragmented surveillance systems contribute to spillover risk, particularly in South and Southeast Asia where fruit bats of the genus Pteropus serve as natural reservoirs. Despite advances in vaccine technology, including subunit, viral vector, mRNA-based platforms, and monoclonal antibody therapies, no licensed prophylactic or therapeutic agent is currently available for human use. Global preparedness remains challenged by the scarcity of high-containment biosafety facilities, limited research funding, and absence of integrated One Health surveillance networks. Ethical considerations surrounding wildlife population control further complicate disease mitigation strategies. Emerging genomic surveillance, artificial intelligence-assisted predictive modeling, and regional data-sharing frameworks are essential for early detection and response. Strengthening molecular research on viral-host interactions and transmission determinants will be critical for preventing future Nipah virus outbreaks in an increasingly interconnected world.
A deterministic differential equation model for the population dynamics of the human malaria vector is derived and studied. Conditions for the existence and stability of a non-zero steady state vector population density are derived. These reveal that a threshold parameter, the vectorial basic reproduction number, exist and the vector can established itself in the community if and only if this parameter exceeds unity. When a non-zero steady state population density exists, it can be stable but it can also be driven to instability via a Hopf Bifurcation to periodic solutions, as a parameter is varied in parameter space. By considering a special case, an asymptotic perturbation analysis is used to derive the amplitude of the oscillating solutions for the full non-linear system. The present modelling exercise and results show that it is possible to study the population dynamics of disease vectors, and hence oscillatory behaviour as it is often observed in most indirectly transmitted infectious diseases of humans, without recourse to external seasonal forcing.
This paper discusses 2 prominent, contemporary issues in the epidemiological context of insect vector control: (i) the magnitude of the control problem, and (ii) non-linear processes influencing vector control. It concludes that we still cannot reliably measure the scale of some important control problems; e.g., there is considerable uncertainty about the basic reproduction number of malaria. The emergence of new concepts such as strain-specific immunity, and a growing emphasis on disease control as distinct from infection control, mean that some quantitative problems are being redefined more quickly than they are being solved. Population biologists have urged exploration of density-dependent processes which may help or hinder new methods of vector control. This brief review of non-linear phenomena such as facilitation and limitation finds little evidence that they will significantly influence, e.g., the introduction of some novel refractory mechanism into a vector population.
A model for the transmission dynamics of Chagas' disease is presented. The structure of the model is similar to that of the Ross-Macdonald model for malaria but includes an extra infectious compartment (chronically ill individuals) which is characteristic of Chagas' disease. In Chagas' disease there are two-main forms of transmission, by blood transfusion and by vector biting. The former is more common in urban environments and the latter is characteristic of rural settings. The characteristic long chronic (frequently asymptomatic) stage of Chagas' disease is potentially a risk factor that could enhance disease transmission by blood transfusion. The model evaluates the relative importance of both transmission modes in populations of constant size. The main results indicate that there is a strong tendency of the disease to reach an asymptotically stable endemic equilibrium point. Also, the magnitude of the basic reproductive number is very sensitive to the length of the chronic stage of the disease and hence it follows that early detection of cases (reducing the length of this stage) is important for the eventual eradication of the disease.
In this work we formulate and analyze a mathematical model for the transmission of West Nile Virus (WNV) infection between vector (mosquito) and avian population. We find the Basic Reproductive Number R0 in terms of measurable epidemiological and demographic parameters. R0 is the threshold condition that determines the dynamics of WNV infection: if R0< or =1 the disease fades out, and for R0 >1 the disease remains endemic. Using experimental and field data we estimate R0 for several species of birds. Numerical simulations of the temporal course of the infected bird proportion show damped oscillations approaching the endemic value.
The use of antibiotics for animal growth promotion has been controversial because of the potential transfer of antibiotic resistance from animals to humans. Such transfer could have severe public health implications in that treatment failures could result. We have followed a risk assessment approach to evaluate policy options for the streptogramin-class of antibiotics: virginiamycin, an animal growth promoter, and quinupristin/dalfopristin, a antibiotic used in humans. Under the assumption that resistance transfer is possible, models project a wide range of outcomes depending mainly on the basic reproductive number (R(0)) that determines the potential for person-to-person transmission. Counter-intuitively, the benefits of a ban on virginiamycin were highest for intermediate values of R(0), and lower for extremely high or low values of R(0).
Many infectious diseases spread through populations via the networks formed by physical contacts among individuals. The patterns of these contacts tend to be highly heterogeneous. Traditional "compartmental" modeling in epidemiology, however, assumes that population groups are fully mixed, that is, every individual has an equal chance of spreading the disease to every other. Applications of compartmental models to Severe Acute Respiratory Syndrome (SARS) resulted in estimates of the fundamental quantity called the basic reproductive number R0--the number of new cases of SARS resulting from a single initial case--above one, implying that, without public health intervention, most outbreaks should spark large-scale epidemics. Here we compare these predictions to the early epidemiology of SARS. We apply the methods of contact network epidemiology to illustrate that for a single value of R0, any two outbreaks, even in the same setting, may have very different epidemiological outcomes. We offer quantitative insight into the heterogeneity of SARS outbreaks worldwide, and illustrate the utility of this approach for assessing public health strategies.
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.
Genetic studies report the existence of a mutant allele Delta32 of CCR5 chemokine receptor gene at high allele frequencies (approximately 10%) in Caucasian populations. The presence of this allele is believed to provide partial or full resistance to HIV. In this study, we look at the impact of education, temporarily effective vaccines and therapies on the dynamics of HIV in homosexually active populations. In our model, it is assumed that some individuals possess one or two mutant alleles (like Delta32 of CCR5) that prevent the successful invasion or replication of HIV. Our model therefore differentiates by genetic and epidemiological status and naturally ignores the reproduction process. Furthermore, HIV infected individuals are classified as rapid, normal or slow progressors. In this complex setting, the basic reproductive number R0 is derived in various situations. The separate or combined effects of therapies, education, vaccines, and genetic resistance are analyzed. Our results support the conclusions of Hsu Schmitz that some integrated intervention strategies are far superior to those based on a single approach. However, treatment programs may have effects which counteract each other, as may genetic resistance.
We develop and analyze a simple SIV epidemic model including susceptible, infected and perfectly vaccinated classes, with a generalized non-linear incidence rate subject only to a few general conditions. These conditions are satisfied by many models appearing in the literature. The detailed dynamics analysis of the model, using the Poincaré index theory, shows that non-linearity of the incidence rate leads to vital dynamics, such as bistability and periodicity, without seasonal forcing or being cyclic. Furthermore, it is shown that the basic reproductive number is independent of the functional form of the non-linear incidence rate. Under certain, well-defined conditions, the model undergoes a Hopf bifurcation. Using the normal form of the model, the first Lyapunov coefficient is computed to determine the various types of Hopf bifurcation the model undergoes. These general results are applied to two examples: unbounded and saturated contact rates; in both cases, forward or backward Hopf bifurcations occur for two distinct values of the contact parameter. It is also shown that the model may undergo a subcritical Hopf bifurcation leading to the appearance of two concentric limit cycles. The results are illustrated by numerical simulations with realistic model parameters estimated for some infectious diseases of childhood.
An integral equation model of a smallpox epidemic is proposed. The model structures the incidence of infection among the household, the workplace, the wider community and a health-care facility; and incorporates a finite incubation period and plausible infectivity functions. Linearisation of the model is appropriate for small epidemics, and enables analytic expressions to be derived for the basic reproduction number and the size of the epidemic. The effects of control interventions (vaccination, isolation, quarantine and public education) are explored for a smallpox epidemic following an imported case. It is found that the rapid identification and isolation of cases, the quarantine of affected households and a public education campaign to reduce contact would be capable of bringing an epidemic under control. This could be used in conjunction with the vaccination of healthcare workers and contacts. Our results suggest that prior mass vaccination would be an inefficient method of containing an outbreak.
The invasion of new species and the spread of emergent infectious diseases in spatially structured populations has stimulated the study of explicit spatial models such as cellular automata, network models and lattice models. However, the analytic intractability of these models calls for the development of tractable mathematical approximations that can capture the dynamics of discrete, spatially-structured populations. Here we explore moment closure approximations for the invasion of an SIS epidemic on a regular lattice. We use moment closure methods to derive an expression for the basic reproductive number, R(0), in a lattice population. On lattices, R(0) should be bounded above by the number of neighbors per individual. However, we show that conventional pair approximations actually predict unbounded growth in R(0) with increasing transmission rates. To correct this problem, we propose an 'invasory' pair approximation which yields a relatively simple expression for R(0) that remains bounded above, and also predicts R(0) values from lattice model simulations more accurately than conventional pair and triple approximations. The invasory pair approximation is applicable to any spatial model, since it takes into account characteristics of invasions that are common to all spatially structured populations.