Global stability of an SEIS epidemic model with recruitment and a varying total population size

This paper considers an SEIS epidemic model that incorporates constant recruitment, disease-caused death and disease latency. The incidence term is of the bilinear mass-action form. It is shown that the global dynamics is completely determined by the basic reproduction number R(0). If R(0)1, a unique endemic equilibrium is globally stable in the interior of the feasible region and the disease persists at the endemic equilibrium.     

Susceptible-infected-recovered epidemics in dynamic contact networks

Contact patterns in populations fundamentally influence the spread of infectious diseases. Current mathematical methods for epidemiological forecasting on networks largely assume that contacts between individuals are fixed, at least for the duration of an outbreak. In reality, contact patterns may be quite fluid, with individuals frequently making and breaking social or sexual relationships. Here, we develop a mathematical approach to predicting disease transmission on dynamic networks in which each individual has a characteristic behaviour (typical contact number), but the identities of their contacts change in time. We show that dynamic contact patterns shape epidemiological dynamics in ways that cannot be adequately captured in static network models or mass-action models. Our new model interpolates smoothly between static network models and mass-action models using a mixing parameter, thereby providing a bridge between disparate classes of epidemiological models. Using epidemiological and sexual contact data from an Atlanta high school, we demonstrate the application of this method for forecasting and controlling sexually transmitted disease outbreaks.     

Sliding Mode Control of Outbreaks of Emerging Infectious Diseases

This paper proposes and analyzes a mathematical model of an infectious disease system with a piecewise control function concerning threshold policy for disease management strategy. The proposed models extend the classic models by including a piecewise incidence rate to represent control or precautionary measures being triggered once the number of infected individuals exceeds a threshold level. The long-term behaviour of the proposed non-smooth system under this strategy consists of the so-called sliding motion-a very rapid switching between application and interruption of the control action. Model solutions ultimately approach either one of two endemic states for two structures or the sliding equilibrium on the switching surface, depending on the threshold level. Our findings suggest that proper combinations of threshold densities and control intensities based on threshold policy can either preclude outbreaks or lead the number of infecteds to a previously chosen level.     

一个具有时滞的媒介传播流行病模型中的Hopt分支（英文）

文章研究的是一个具有时滞的媒介传播流行病模型.假定长期的发病率是双线性大规模行动的方式,确定了疾病是否流行的阈值R_0.当R_0≤1时,得到无病平衡点是全局稳定的,即疾病消失;当R_0〉1时,得到地方病平衡点.在具有时滞的微分模型中,时滞与载体转变成传染源的孵化期有关。我们研究了时滞对平衡点稳定性的影响,研究表明,在从寄生源到载体的传播过程中,时滞可以破坏动力系统并且得到了Hopt分支的周期解.     

Frequency-dependent incidence in models of sexually transmitted diseases: portrayal of pair-based transmission and effects of illness on contact behaviour

We explore the transmission process for sexually transmitted diseases (STDs). We derive the classical frequency-dependent incidence mechanistically from a pair-formation model, using an approximation that applies to populations with rapid pairing dynamics (such as core groups or non-pair-bonding animals). This mechanistic derivation provides a framework to assess how accurately frequency-dependent incidence portrays the pair-based transmission known to underlie STD dynamics. This accuracy depends strongly on the disease being studied: frequency-dependent formulations are more suitable for chronic less-transmissible infections than for transient highly transmissible infections. Our results thus support earlier proposals to divide STDs into these two functional classes, and we suggest guidelines to help assess under what conditions each class can be appropriately modelled using frequency-dependent incidence. We then extend the derivation to include situations where infected individuals exhibit altered pairing behaviour. For four cases of increasing behavioural complexity, analytic expressions are presented for the generalized frequency-dependent incidence rate, basic reproductive number (R0) and steady-state prevalence (i infinity) of an epidemic. The expression for R0 is identical for all cases, giving refined insights into determinants of invasibility of STDs. Potentially significant effects of infection-induced changes in contact behaviour are illustrated by simulating epidemics of bacterial and viral STDs. We discuss the application of our results to STDs (in humans and animals) and other infectious diseases.     

Effective degree network disease models

An effective degree approach to modeling the spread of infectious diseases on a network is introduced and applied to a disease that confers no immunity (a Susceptible-Infectious-Susceptible model, abbreviated as SIS) and to a disease that confers permanent immunity (a Susceptible-Infectious-Recovered model, abbreviated as SIR). Each model is formulated as a large system of ordinary differential equations that keeps track of the number of susceptible and infectious neighbors of an individual. From numerical simulations, these effective degree models are found to be in excellent agreement with the corresponding stochastic processes of the network on a random graph, in that they capture the initial exponential growth rates, the endemic equilibrium of an invading disease for the SIS model, and the epidemic peak for the SIR model. For each of these effective degree models, a formula for the disease threshold condition is derived. The threshold parameter for the SIS model is shown to be larger than that derived from percolation theory for a model with the same disease and network parameters, and consequently a disease may be able to invade with lower transmission than predicted by percolation theory. For the SIR model, the threshold condition is equal to that predicted by percolation theory. Thus unlike the classical homogeneous mixing disease models, the SIS and SIR effective degree models have different disease threshold conditions.     

An age-dependent epidemic model with application to measles

A combined epidemic-demographic model is developed which models the spread of an infectious disease throughout a population of constant size. The model allows for births, deaths, temporary or permanent immunity, and immunization. The relationship of this model to previously studied epidemic and demographic models is illustrated. An advantage of this model is that all epidemic and demographic parameters may be estimated. The stability of the equilibrium point corresponding to the elimination of the disease is studied and a threshold value is found which indicates whether the disease will die out or remain endemic in the population. The application of the model to measles indicates that immunization levels needed to reduce the incidence to near zero may not be as high as previously predicted.     

Global dynamics of an SIRS epidemic model with saturation incidence

In this paper, the dynamical behavior of an SIRS epidemic model with birth pulse, pulse vaccination, and saturation incidence is studied. By using a discrete map, the existence and stability of the infection-free periodic solution and the endemic periodic solution are investigated. The conditions required for the existence of supercritical bifurcation are derived. A threshold for a disease to be extinct or endemic is established. The Poincaré map and center manifold theorem are used to discuss flip bifurcation of the endemic periodic solution. Moreover, numerical simulations for bifurcation diagrams, phase portraits and periodic solutions, which are illustrated with an example, are in good agreement with the theoretical analysis.     

Effects of predation on host-pathogen dynamics in SIR models

The integration of infectious disease epidemiology with community ecology is an active area of research. Recent studies using SI models without acquired immunity have demonstrated that predation can suppress infectious disease levels. The authors recently showed that incorporating immunity (SIR models) can produce a “hump”-shaped relationship between disease prevalence and predation pressure; thus, low to moderate levels of predation can boost prevalence in hosts with acquired immunity. Here we examine the robustness of this pattern to realistic extensions of a basic SIR model, including density-dependent host regulation, predator saturation, interference, frequency-dependent transmission, predator numerical responses, and explicit resource dynamics. A non-monotonic relationship between disease prevalence and predation pressure holds across all these scenarios. With saturation, there can also be complex responses of mean host abundance to increasing predation, as well as bifurcations leading to unstable cycles (epidemics) and pathogen extinction at larger predator numbers. Firm predictions about the relationship between prevalence and predation thus require one to consider the complex interplay of acquired immunity, host regulation, and foraging behavior of the predator.     

A two-component model for counts of infectious diseases

We propose a stochastic model for the analysis of time series of disease counts as collected in typical surveillance systems on notifiable infectious diseases. The model is based on a Poisson or negative binomial observation model with two components: a parameter-driven component relates the disease incidence to latent parameters describing endemic seasonal patterns, which are typical for infectious disease surveillance data. An observation-driven or epidemic component is modeled with an autoregression on the number of cases at the previous time points. The autoregressive parameter is allowed to change over time according to a Bayesian changepoint model with unknown number of changepoints. Parameter estimates are obtained through the Bayesian model averaging using Markov chain Monte Carlo techniques. We illustrate our approach through analysis of simulated data and real notification data obtained from the German infectious disease surveillance system, administered by the Robert Koch Institute in Berlin. Software to fit the proposed model can be obtained from http://www.statistik.lmu.de/ approximately mhofmann/twins.     

Optimal Timing of Disease Transmission in an Age-Structured Population

It is a common medical folk-practice for parents to encourage their children to contract certain infectious diseases while they are young. This folk-practice is controversial, in part, because it contradicts the long-term public health goal of minimizing disease incidence. We study an epidemiological model of infectious disease in an age-structured population where virulence is age-dependent and show that, in some cases, the optimal behavior will increase disease transmission. This provides a rigorous justification of the concept of “endemic stability,” and demonstrates that folk-practices may have been historically justified.     

The control of vector-borne disease epidemics

The theoretical underpinning of our struggle with vector-borne disease, and still our strongest tool, remains the basic reproduction number, R₀, the measure of long term endemicity. Despite its widespread application, R₀ does not address the dynamics of epidemics in a model that has an endemic equilibrium. We use the concept of reactivity to derive a threshold index for epidemicity, E₀, which gives the maximum number of new infections produced by an infective individual at a disease free equilibrium. This index describes the transitory behavior of disease following a temporary perturbation in prevalence. We demonstrate that if the threshold for epidemicity is surpassed, then an epidemic peak can occur, that is, prevalence can increase further, even when the disease is not endemic and so dies out. The relative influence of parameters on E₀ and R₀ may differ and lead to different strategies for control. We apply this new threshold index for epidemicity to models of vector-borne disease because these models have a long history of mathematical analysis and application. We find that both the transmission efficiency from hosts to vectors and the vector-host ratio may have a stronger effect on epidemicity than endemicity. The duration of the extrinsic incubation period required by the pathogen to transform an infected vector to an infectious vector, however, may have a stronger effect on endemicity than epidemicity. We use the index E₀ to examine how vector behavior affects epidemicity. We find that parasite modified behavior, feeding bias by vectors for infected hosts, and heterogeneous host attractiveness contribute significantly to transitory epidemics. We anticipate that the epidemicity index will lead to a reevaluation of control strategies for vector-borne disease and be applicable to other disease transmission models.     

Do fatal infectious diseases eradicate host species?

In simple SI epidemic and endemic models, three classes of incidence functions are identified for their potential to be associated with host extinction: weakly upper density-dependent incidences are never associated with host extinction. Power incidences that depend on the number of susceptibles and infectives by powers strictly between 0 and 1 are associated with initial-constellation-dependent host extinction for all parameter values. Homogeneous incidences, of which frequency-dependent incidence is a very particular case, and power incidences are associated with global host extinction for certain parameter constellations and with host survival for others. Laboratory infection experiments with salamander larvae are equally well fitted by power incidences and certain upper density-dependent incidences such as the negative binomial incidence and do not rule out homogeneous incidences such as an asymmetric frequency-dependent incidence either.

     

Disease-induced stabilization of predator-prey oscillations

Parasites are an integral part of virtually all food webs and species communities. Here we consider the invasion of a resident predator-prey system by an infectious disease with frequency-dependent transmission spreading within the predator population. We derive biologically plausible and insightful quantities (demographic and epizootiological reproduction numbers) that allow us to completely determine community composition. Successful disease invasion can have two contrary effects in driving its host population to extinction or in stabilizing predator-prey cycles. Our findings contradict predictions from previous models suggesting a destabilizing effect of parasites. We show that predator infection counteracts the paradox of enrichment. In turn, parasite removal from food webs can have catastrophic effects. We discuss the implications for biological control and resource management on more than one trophic level.     

Why genetic selection to reduce the prevalence of infectious diseases is way more promising than currently believed

Genetic selection for improved disease resistance is an important part of strategies to combat infectious diseases in agriculture. Quantitative genetic analyses of binary disease status, however, indicate low heritability for most diseases, which restricts the rate of genetic reduction in disease prevalence. Moreover, the common liability threshold model suggests that eradication of an infectious disease via genetic selection is impossible because the observed-scale heritability goes to zero when the prevalence approaches zero. From infectious disease epidemiology, however, we know that eradication of infectious diseases is possible, both in theory and practice, because of positive feedback mechanisms leading to the phenomenon known as herd immunity. The common quantitative genetic models, however, ignore these feedback mechanisms. Here, we integrate quantitative genetic analysis of binary disease status with epidemiological models of transmission, aiming to identify the potential response to selection for reducing the prevalence of endemic infectious diseases. The results show that typical heritability values of binary disease status correspond to a very substantial genetic variation in disease susceptibility among individuals. Moreover, our results show that eradication of infectious diseases by genetic selection is possible in principle. These findings strongly disagree with predictions based on common quantitative genetic models, which ignore the positive feedback effects that occur when reducing the transmission of infectious diseases. Those feedback effects are a specific kind of Indirect Genetic Effects; they contribute substantially to the response to selection and the development of herd immunity (i.e., an effective reproduction ratio less than one).     

Oscillatory phenomena in a model of infectious diseases

Oscillations of the number of cases around an average endemic level are common in several infectious diseases. In this paper we study simple deterministic models, where the oscillations arise either solely from periodically varying contact rates or from the combined effect of large initial perturbation, small periodic variation of the contact rate, and the destabilizing nature of infectious and latent periods when described as time delays. The main results are: (a) For a model with a periodically varying contact rate and a recovery rate, a threshold amplitude of variation is found by numerical and analytic methods at which 2-year subharmonic resonance appears. (b) Approximate analytic relationships are derived for the amplitude and phase of the forced 1-year oscillations below this threshold and for the 2-year oscillations above it—in terms of the reproduction rate of the infection. (c) Similar calculations are performed when the recovery rate is replaced by a fixed infectious period represented by a pure time delay. The threshold amplitude of variation in the contact rate is found here to be smaller than in the recovery rate model. (d) A model with a fixed infectious period and a constant contact rate is considered. The nontrivial steady state is shown to be locally stable for the parameter range of interest. However, the ratio of the imaginary to real parts of the eigenvalues in the characteristic equation is increased as compared to the corresponding model with a recovery rate. (e) For the model with a fixed infectious period and a constant contact rate an approximation method indicates consistency in a certain range of contact rates with the existence of an unstable periodic solution about the locally stable steady state. The actual existence of such a solution is not verified. The interpretation is that the destabilizing effect of the introduction of a pure delay into the model becomes more significant as the distance in the variables space from the endemic steady state is increased. (f) For a fixed infectious period and very small subthreshold variation in the contact rate, two different types of solutions are found numerically: yearly small-amplitude oscillations about an endemic average and large-amplitude oscillations of a subharmonic period. The pattern seen depends on the initial conditions. For a sufficiently large initial deviation from the endemic level even very small seasonal variations lead to regular recurrent outbreaks of the disease. The effect of latent periods and of changing the form of the interaction are also considered.     

Basic Reproduction Number for HIV Model Incorporating Commercial Sex and Behavior Change

The basic reproduction number is obtained for an HIV epidemic model incorporating direct and indirect commercial sex as well as behavior change by the female commercial sex workers (CSWs) and their male customers in response to the proliferation of the disease in the community. A recent result by van den Driessche P., and Watmough J. (Math. Biosci. 180:29–48, 2002) is utilized to compute the threshold parameters for the local asymptotic stability of the Disease-Free Equilibrium (DFE), by considering the transfers in and out of the infective classes. Numerical examples are used to describe the uniqueness and global properties of the endemic equilibrium when DFE is unstable. Biological interpretation of the results obtained in this work is discussed, as are the implications of our results for the design of public health policies such as targeting strategy to target intervention and control measures toward specific high-risk population groups in order to reduce infections. We show that targeting any one sector of the commercial sex alone for prevention will be difficult to have a decided effect on eradicating the epidemic. However, if the aim of the targeted intervention policy is not eradication of the epidemic but decrease in HIV incidence of a particular high-risk group, then concentrated targeting strategy could be sufficient, if properly implemented. This work also demonstrates the usefulness of the theorem of van den Driessche and Watmough (Math. Biosci. 180:29–48, 2002) in obtaining threshold parameters for complicated infectious diseases models.     

Stage-structured transmission of phocine distemper virus in the Dutch 2002 outbreak

Heterogeneities in transmission among hosts can be very important in shaping infectious disease dynamics. In mammals with strong social organization, such heterogeneities are often structured by functional stage: juveniles, subadults and adults. We investigate the importance of such stage-related heterogeneities in shaping the 2002 phocine distemper virus (PDV) outbreak in the Dutch Wadden Sea, when more than 40 per cent of the harbour seals were killed. We do this by comparing the statistical fit of a hierarchy of models with varying transmission complexity: homogeneous versus heterogeneous mixing and density- versus frequency-dependent transmission. We use the stranding data as a proxy for incidence and use Poisson likelihoods to estimate the ‘who acquires infection from whom’ (WAIFW) matrix. Statistically, the model with strong heterogeneous mixing and density-dependent transmission was found to best describe the transmission dynamics. However, patterns of incidence support a model of frequency-dependent transmission among adults and juveniles. Based on the maximum-likelihood WAIFW matrix estimates, we use the next-generation formalism to calculate an R₀ between 2 and 2.5 for the Dutch 2002 PDV epidemic.     

Core recruitment effects in SIS models with constant total populations

We consider a set of SIS models for a heterosexually transmitted disease in which there is recruitment between core and non-core subpopulations as a function of prevalence of the disease. Behavior diverges from the traditional R0 threshold behavior and yields an extra pair of endemic equilibria in one case and a limit cycle in the other. Total at-risk population is constant.