Stability in cyclic epidemic models

A general formulation for a family of cyclic epidemic models with density-dependent feedback mechanisms and removed classes is presented. A parameter, , related to the basic reproductive rate determines the asymptotic behaviour of solutions of the model. It is shown that if <1 the trivial solution is globally stable, and if >1 it is conditionally stable. The results are applied to a set of differential equations that has been used to model the life cycle of a parasite that has two hosts.     

Stability in a class of cyclic epidemic models with delay

A detailed analysis of a general class of SIRS epidemic models is given. Sufficient conditions are derived which guarantee the global stability of the endemic equilibrium solution. Further conditions are found which ensure instability for the equilibrium. Finally, the dependence of the stability on the contact number and the ratio of the mean length of infection to the mean removed time is considered.     

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).     

Population models with singular equilibrium

A class of models of biological population and communities with a singular equilibrium at the origin is analyzed; it is shown that these models can possess a dynamical regime of deterministic extinction, which is crucially important from the biological standpoint. This regime corresponds to the presence of a family of homoclinics to the origin, so-called elliptic sector. The complete analysis of possible topological structures in a neighborhood of the origin, as well as asymptotics to orbits tending to this point, is given. An algorithmic approach to analyze system behavior with parameter changes is presented. The developed methods and algorithm are applied to existing mathematical models of biological systems. In particular, we analyze a model of anticancer treatment with oncolytic viruses, a parasite-host interaction model, and a model of Chagas' disease.     

Oscillations in epidemic models with spread of awareness

Journal of Mathematical Biology - We study ODE models of epidemic spreading with a preventive behavioral response that is triggered by awareness of the infection. Previous studies of such models...     

Variational data assimilation with epidemic models

Mathematical modelling is playing an increasing role in developing an understanding of the dynamics of communicable disease and assisting the construction and implementation of intervention strategies. The threat of novel emergent pathogens in human and animal hosts implies the requirement for methods that can robustly estimate epidemiological parameters and provide forecasts. Here, a technique called variational data assimilation is introduced as a means of optimally melding dynamic epidemic models with epidemiological observations and data to provide forecasts and parameter estimates. Using data from a simulated epidemic process the method is used to estimate the start time of an epidemic, to provide a forecast of future epidemic behaviour and estimate the basic reproductive ratio. A feature of the method is that it uses a basic continuous-time SIR model, which is often the first point of departure for epidemiological modelling during the early stages of an outbreak. The method is illustrated by application to data gathered during an outbreak of influenza in a school environment.     

Susceptible-infected-removed epidemic models with dynamic partnerships

The author extends the classical, stochastic, Susceptible-Infected-Removed (SIR) epidemic model to allow for disease transmission through a dynamic network of partnerships. A new method of analysis allows for a fairly complete understanding of the dynamics of the system for small and large time. The key insight is to analyze the model by tracking the configurations of all possible dyads, rather than individuals. For large populations, the initial dynamics are approximated by a branching process whose threshold for growth determines the epidemic threshold, R ₀, and whose growth rate, , determines the rate at which the number of cases increases. The fraction of the population that is ever infected, , is shown to bear the same relationship to R ₀ as in models without partnerships. Explicit formulas for these three fundamental quantities are obtained for the simplest version of the model, in which the population is treated as homogeneous, and all transitions are Markov. The formulas allow a modeler to determine the error introduced by the usual assumption of instantaneous contacts for any particular set of biological and sociological parameters. The model and the formulas are then generalized to allow for non-Markov partnership dynamics, non-uniform contact rates within partnerships, and variable infectivity. The model and the method of analysis could also be further generalized to allow for demographic effects, recurrent susceptibility and heterogeneous populations, using the same strategies that have been developed for models without partnerships.     

SVIR epidemic models with vaccination strategies

Vaccination is important for the elimination of infectious diseases. To finish a vaccination process, doses usually should be taken several times and there must be some fixed time intervals between two doses. The vaccinees (susceptible individuals who have started the vaccination process) are different from both susceptible and recovered individuals. Considering the time for them to obtain immunity and the possibility for them to be infected before this, two SVIR models are established to describe continuous vaccination strategy and pulse vaccination strategy (PVS), respectively. It is shown that both systems exhibit strict threshold dynamics which depend on the basic reproduction number. If this number is below unity, the disease can be eradicated. And if it is above unity, the disease is endemic in the sense of global asymptotical stability of a positive equilibrium for continuous vaccination strategy and disease permanence for PVS. Mathematical results suggest that vaccination is helpful for disease control by decreasing the basic reproduction number. However, there is a necessary condition for successful elimination of disease. If the time for the vaccinees to obtain immunity or the possibility for them to be infected before this is neglected, this condition disappears and the disease can always be eradicated by some suitable vaccination strategies. This may lead to over-evaluating the effect of vaccination.     

Deterministic epidemic models with explicit household structure

For a wide range of airborne infectious diseases, transmission within the family or household is a key mechanism for the spread and persistence of infection. In general, household-based transmission is relatively strong but only involves a limited number of individuals in contact with each infectious person. In contrast, transmission outside the household can be characterised by many contacts but a lower probability of transmission. Here we develop a relatively simple dynamical model that captures these two transmission regimes. We compare the dynamics of such models for a range of household sizes, whilst constraining all models to have equal early growth rate so that all models fit to the same early incidence observations of an epidemic. Finally we consider the use of prophylactic vaccination, responsive vaccination, or antivirals to combat epidemic spread and focus on whether it is optimal to target controls at entire households or to treat individuals independently.     

Household epidemic models with varying infection response

This paper is concerned with SIR (susceptible → infected → removed) household epidemic models in which the infection response may be either mild or severe, with the type of response also affecting the infectiousness of an individual. Two different models are analysed. In the first model, the infection status of an individual is predetermined, perhaps due to partial immunity, and in the second, the infection status of an individual depends on the infection status of its infector and on whether the individual was infected by a within- or between-household contact. The first scenario may be modelled using a multitype household epidemic model, and the second scenario by a model we denote by the infector-dependent-severity household epidemic model. Large population results of the two models are derived, with the focus being on the distribution of the total numbers of mild and severe cases in a typical household, of any given size, in the event that the epidemic becomes established. The aim of the paper is to investigate whether it is possible to determine which of the two underlying explanations is causing the varying response when given final size household outbreak data containing mild and severe cases. We conduct numerical studies which show that, given data on sufficiently many households, it is generally possible to discriminate between the two models by comparing the Kullback–Leibler divergence for the two fitted models to these data.     

Parameter identification in epidemic models

Starting from a recent paper of Pollicott, Wang and Weiss we try to obtain improved representation formulas for the estimation of the time-dependent transmission rate of an epidemic in terms of either incidence or prevalence data. Although the formulas are (trivially) mathematically equivalent to previous formulas, the new representations need no additional estimates and they should be more stable numerically.We review the discrete time and the stochastic continuous time approach. We replace the assumption that recovery follows an exponential distribution and get estimates for the transmission rate for constant duration of the infectious phase.     

Vaccination in density-dependent epidemic models

This paper examines the effect of vaccination for an epidemic model where the death rate depends on the number of individuals in the population. The basic model which is described is based on measles or other childhood diseases in developing countries or viral diseases such as rabies in animal populations. An equilibrium analysis of the model and the local stability of small perturbations about the equilibrium values are discussed. The biological implications of these results are examined and similar results presented for modifications of the basic model.     

Network epidemic models with two levels of mixing

The study of epidemics on social networks has attracted considerable attention recently. In this paper, we consider a stochastic SIR (susceptible-->infective-->removed) model for the spread of an epidemic on a finite network, having an arbitrary but specified degree distribution, in which individuals also make casual contacts, i.e. with people chosen uniformly from the population. The behaviour of the model as the network size tends to infinity is investigated. In particular, the basic reproduction number R(0), that governs whether or not an epidemic with few initial infectives can become established is determined, as are the probability that an epidemic becomes established and the proportion of the population who are ultimately infected by such an epidemic. For the case when the infectious period is constant and all individuals in the network have the same degree, the asymptotic variance and a central limit theorem for the size of an epidemic that becomes established are obtained. Letting the rate at which individuals make casual contacts decrease to zero yields, heuristically, corresponding results for the model without casual contacts, i.e. for the standard SIR network epidemic model. A deterministic model that approximates the spread of an epidemic that becomes established in a large population is also derived. The theory is illustrated by numerical studies, which demonstrate that the asymptotic approximations work well, even for only moderately sized networks, and that the degree distribution and the inclusion of casual contacts can each have a major impact on the outcome of an epidemic.     

Stochastic models of some endemic infections

Stochastic models are established and studied for several endemic infections with demography. Approximations of quasi-stationary distributions and of times to extinction are derived for stochastic versions of SI, SIS, SIR, and SIRS models. The approximations are valid for sufficiently large population sizes. Conditions for validity of the approximations are given for each of the models. These are also conditions for validity of the corresponding deterministic model. It is noted that some deterministic models are unacceptable approximations of the stochastic models for a large range of realistic parameter values.     

Bluff your way in epidemic models

The literature on the mathematical modelling of infectious diseases has grown enormously in recent years, both in quantity and quality. Here, we briefly point to the purposes of these modelling exercises and introduce the main ideas behind compartmental models, to act as a guide to the (bio)mathematical literature that is less directly accessible.     

Front dynamics in fractional-order epidemic models

A number of recent studies suggest that human and animal mobility patterns exhibit scale-free, Lévy-flight dynamics. However, current reaction-diffusion epidemics models do not account for the superdiffusive spread of modern epidemics due to Lévy flights. We have developed a SIR model to simulate the spatial spread of a hypothetical epidemic driven by long-range displacements in the infective and susceptible populations. The model has been obtained by replacing the second-order diffusion operator by a fractional-order operator. Theoretical developments and numerical simulations show that fractional-order diffusion leads to an exponential acceleration of the epidemic's front and a power-law decay of the front's leading tail. Our results indicate the potential of fractional-order reaction-diffusion models to represent modern epidemics.     

Serological differences in Neospora caninum-associated epidemic and endemic abortions.

A sandwich enzyme-linked immunosorbent assay (ELISA) for the sensitive and specific detection of bovine antibodies to Neospora caninum was developed and evaluated using sera from cattle experimentally infected with N. caninum, Toxoplasma gondii, Sarcocystis cruzi, Sarcocystis hominis, Sarcocystis hirsuta, Eimeria bovis, Cryptosporidium parvum, Babesia divergens, and field sera from naturally exposed animals. Field sera were classified using a gold standard that included the results from an indirect fluorescent antibody test (IFAT) and an immunoblot (IB). Based on these gold standard results, i.e., IFAT-IB results, an equal relative sensitivity and specificity of 94.2%(theta0) was reached when a cutoff of 0.034 (d0) was employed. The analysis of IFAT-IB-positive field sera showed that within groups of aborting and nonaborting dams, the animals from herds with endemic N. caninum-associated abortions had significantly higher ELISA indices than animals from herds with N. caninum-associated epidemic abortions. By contrast, IFAT-IB-positive aborting dams from herds with endemic N. caninum-associated abortions had significantly lower IFAT titers than IFAT-IB-positive aborting dams from herds with epidemic N. caninum-associated abortions. This is the first time that statistically significant serological differences between herds exhibiting epidemic and endemic N. caninum-associated abortions are described.