Periodicity in an epidemic model with a generalized non-linear incidence

Relative prevalence-based dispersal in an epidemic patch model

In this paper, we propose a two-patch SIRS model with a nonlinear incidence rate: [Formula: see text] and nonconstant dispersal rates, where the dispersal rates of susceptible and recovered individuals depend on the relative disease prevalence in two patches. In an isolated environment, the model admits Bogdanov-Takens bifurcation of codimension 3 (cusp case) and Hopf bifurcation of codimension up to 2 as the parameters vary, and exhibits rich dynamics such as multiple coexistent steady states and periodic orbits, homoclinic orbits and multitype bistability. The long-term dynamics can be classified in terms of the infection rates [Formula: see text] (due to single contact) and [Formula: see text] (due to double exposures). In a connected environment, we establish a threshold [Formula: see text] between disease extinction and uniform persistence under certain conditions. We numerically explore the effect of population dispersal on disease spread when [Formula: see text] and patch 1 has a lower infection rate, our results indicate: (i) [Formula: see text] can be nonmonotonic in dispersal rates and [Formula: see text] ([Formula: see text] is the basic reproduction number of patch i) may fail; (ii) the constant dispersal of susceptible individuals (or infective individuals) between two patches (or from patch 2 to patch 1) will increase (or reduce) the overall disease prevalence; (iii) the relative prevalence-based dispersal may reduce the overall disease prevalence. When [Formula: see text] and the disease outbreaks periodically in each isolated patch, we find that: (a) small unidirectional and constant dispersal can lead to complex periodic patterns like relaxation oscillations or mixed-mode oscillations, whereas large ones can make the disease go extinct in one patch and persist in the form of a positive steady state or a periodic solution in the other patch; (b) relative prevalence-based and unidirectional dispersal can make periodic outbreak earlier.

Nonlinear dynamics of a SIRI model incorporating the impact of information and saturated treatment with optimal control

In this article, we propose and analyze an infectious disease model with reinfection and investigate disease dynamics by incorporating saturated treatment and information effect. In the model, we consider the case where an individual's immunity deteriorates and they become infected again after recovering. According to our findings, multiple steady states and backward bifurcation may occur as a result of treatment saturation. Further, if treatment is available for all, the disease will be eradicated providedR 0 < 1 ; however, because limited medical resources caused saturation in treatment, the disease may persist even ifR 0 < 1 . The global stability of the unique endemic steady state is established using a geometric approach. We also establish certain conditions on the transmission rate for the occurrence of periodic oscillations in the model system. Among nonlinear dynamics, we show supercritical Hopf bifurcation, bi-stability, backward Hopf bifurcation, and double Hopf bifurcation. To illustrate and validate our theoretical results, we present numerical examples. We found that when disease information coverage is high, infection cases fall considerably, and the disease persists when the reinfection rate is high. We then extend our model by incorporating two time-dependent controls, namely inhibitory interventions and treatment. Using Pontryagin's maximum principle, we prove the existence of optimal control paths and find the optimal pair of controls. According to our numerical simulations, the second control is less effective than the first. Furthermore, while implementing a single intervention at a time may be effective, combining both interventions is most effective in reducing disease burden and cost.

Modeling the effect of time delay in the increment of number of hospital beds to control an infectious disease

One of the key factors to control the spread of any infectious disease is the health care facilities, especially the number of hospital beds. To assess the impact of number of hospital beds and control of an emerged infectious disease, we have formulated a mathematical model by considering population (susceptible, infected, hospitalized) and newly created hospital beds as dynamic variables. In formulating the model, we have assumed that the number of hospital beds increases proportionally to the number of infected individuals. It is shown that on a slight change in parameter values, the model enters to different kinds of bifurcations, e.g., saddle-node, transcritical (backward and forward), and Hopf bifurcation. Also, the explicit conditions for these bifurcations are obtained. We have also shown the occurrence of Bogdanov-Takens (BT) bifurcation using the Normal form. To set up a new hospital bed takes time, and so we have also analyzed our proposed model by incorporating time delay in the increment of newly created hospital beds. It is observed that the incorporation of time delay destabilizes the system, and multiple stability switches arise through Hopf-bifurcation. To validate the results of the analytical analysis, we have carried out some numerical simulations.

A new perspective on infection forces with demonstration by a DDE infectious disease model

In this paper, we revisit the notion of infection force from a new angle which can offer a new perspective to motivate and justify some infection force functions. Our approach can not only explain many existing infection force functions in the literature, it can also motivate new forms of infection force functions, particularly infection forces depending on disease surveillance of the past. As a demonstration, we propose an SIRS model with delay. We comprehensively investigate the disease dynamics represented by this model, particularly focusing on the local bifurcation caused by the delay and another parameter that reflects the weight of the past epidemics in the infection force. We confirm Hopf bifurcations both theoretically and numerically. The results show that, depending on how recent the disease surveillance data are, their assigned weight may have a different impact on disease control measures.

Computer simulation of the dynamics of a spatial susceptible-infected-recovered epidemic model with time delays in transmission and treatment

BACKGROUND AND OBJECTIVE

In this work, we analyze the spatial-temporal dynamics of a susceptible-infected-recovered (SIR) epidemic model with time delays. To better describe the dynamical behavior of the model, we take into account the cumulative effects of diffusion in the population dynamics, and the time delays in both the Holling type II treatment and the disease transmission process, respectively.

METHODS

We perform linear stability analyses on the disease-free and endemic equilibria. We provide the expression of the basic reproduction number and set conditions on the backward bifurcation using Castillo's theorem. The values of the critical time transmission, the treatment delays and the relationship between them are established.

RESULTS

We show that the treatment rate decreases the basic reproduction number while the transmission rate significantly affects the bifurcation process in the system. The transmission and treatment time-delays are found to be inversely proportional to the susceptible and infected diffusion rates. The analytical results are numerically tested. The results show that the treatment rate significantly reduces the density of infected population and ensures the transition from the unstable to the stable domain. Moreover, the system is more sensible to the treatment in the stable domain.

CONCLUSIONS

The density of infected population increases with respect to the infected and susceptible diffusion rates. Both effects of treatment and transmission delays significantly affect the behavior of the system. The transmission time-delay at the critical point ensures the transition from the stable (low density) to the unstable (high density) domain.

Simplicial SIRS epidemic models with nonlinear incidence rates

Mathematical epidemiology that describes the complex dynamics on social networks has become increasingly popular. However, a few methods have tackled the problem of coupling network topology with complex incidence mechanisms. Here, we propose a simplicial susceptible-infected-recovered-susceptible (SIRS) model to investigate the epidemic spreading via combining the network higher-order structure with a nonlinear incidence rate. A network-based social system is reshaped to a simplicial complex, in which the spreading or infection occurs with nonlinear reinforcement characterized by the simplex dimensions. Compared with the previous simplicial susceptible-infected-susceptible (SIS) models, the proposed SIRS model can not only capture the discontinuous transition and the bistability of a complex system but also capture the periodic phenomenon of epidemic outbreaks. More significantly, the two thresholds associated with the bistable region and the critical value of the reinforcement factor are derived. We further analyze the stability of equilibrium points of the proposed model and obtain the condition of existence of the bistable states and limit cycles. This work expands the simplicial SIS models to SIRS models and sheds light on a novel perspective of combining the higher-order structure of complex systems with nonlinear incidence rates.

Melnikov analysis of chaos in a simple SIR model with periodically or stochastically modulated nonlinear incidence rate

In this paper, Melnikov analysis of chaos in a simple SIR model with periodically or stochastically modulated nonlinear incidence rate and the effect of periodic and bounded noise on the chaotic motion of SIR model possessing homoclinic orbits are detailed investigated. Based on homoclinic bifurcation, necessary conditions for possible chaotic motion as well as sufficient condition are derived by the random Melnikov theorem, and to establish the threshold of bounded noise amplitude for the onset of chaos.

Global dynamics of a fractional-order SIR epidemic model with memory

In this paper, an investigation and analysis of a nonlinear fractional-order SIR epidemic model with Crowley–Martin type functional response and Holling type-II treatment rate are established along the memory. The existence and stability of the equilibrium points are investigated. The sufficient conditions for the persistence of the disease are provided. First, a threshold value, [Formula: see text], is obtained which determines the stability of equilibria, then model equilibria are determined and their stability analysis is considered by using fractional Routh-Hurwitz stability criterion and fractional La-Salle invariant principle. The fractional derivative is taken in Caputo sense and the numerical solution of the model is obtained by L1 scheme which involves the memory trace that can capture and integrate all past activity. Meanwhile, by using Lyapunov functional approach, the global dynamics of the endemic equilibrium point is discussed. Further, some numerical simulations are performed to illustrate the effectiveness of the theoretical results obtained. The outcome of the study reveals that the applied L1 scheme is computationally very strong and effective to analyze fractional-order differential equations arising in disease dynamics. The results show that order of the fractional derivative has a significant effect on the dynamic process. Also, from the results, it is obvious that the memory effect is zero for [Formula: see text]. When the fractional-order [Formula: see text] is decreased from [Formula: see text] the memory trace nonlinearly increases from [Formula: see text], and its dynamics strongly depends on time. The memory effect points out the difference between the derivatives of the fractional-order and integer order.

Global Dynamics of a Susceptible-Infectious-Recovered Epidemic Model with a Generalized Nonmonotone Incidence Rate

A susceptible-infectious-recovered (SIRS) epidemic model with a generalized nonmonotone incidence rate kIS 1 + β I + α I 2 ( β > - 2 α such that 1 + β I + α I 2 > 0 for all I ≥ 0 ) is considered in this paper. It is shown that the basic reproduction number R 0 does not act as a threshold value for the disease spread anymore, and there exists a sub-threshold value R ∗ ( < 1 ) such that: (i) if R 0 < R ∗ , then the disease-free equilibrium is globally asymptotically stable; (ii) if R 0 = R ∗ , then there is a unique endemic equilibrium which is a nilpotent cusp of codimension at most three; (iii) if R ∗ < R 0 < 1 , then there are two endemic equilibria, one is a weak focus of multiplicity at least three, the other is a saddle; (iv) if R 0 ≥ 1 , then there is again a unique endemic equilibrium which is a weak focus of multiplicity at least three. As parameters vary, the model undergoes saddle-node bifurcation, backward bifurcation, Bogdanov-Takens bifurcation of codimension three, Hopf bifurcation, and degenerate Hopf bifurcation of codimension three. Moreover, it is shown that there exists a critical value α 0 for the psychological effect α , a critical value k 0 for the infection rate k, and two critical values β 0 , β 1 ( β 1 < β 0 ) for β that will determine whether the disease dies out or persists in the form of positive periodic coexistent oscillations or coexistent steady states under different initial populations. Numerical simulations are given to demonstrate the existence of one, two or three limit cycles.

Bifurcation analysis of an SIRS epidemic model with a generalized nonmonotone and saturated incidence rate

In this paper, we study a susceptible-infectious-recovered (SIRS) epidemic model with a generalized nonmonotone and saturated incidence rate k I 2 S 1 + β I + α I 2 , in which the infection function first increases to a maximum when a new infectious disease emerges, then decreases due to psychological effect, and eventually tends to a saturation level due to crowding effect. It is shown that there are a weak focus of multiplicity at most two and a cusp of codimension at most two for various parameter values, and the model undergoes saddle-node bifurcation, Bogdanov-Takens bifurcation of codimension two, Hopf bifurcation, and degenerate Hopf bifurcation of codimension two as the parameters vary. It is shown that there exists a critical value α = α 0 for the psychological effect, and two critical values k = k 0 , k 1 ( k 0 < k 1 ) for the infection rate such that: (i) when α > α 0 , or α ≤ α 0 and k ≤ k 0 , the disease will die out for all positive initial populations; (ii) when α = α 0 and k 0 < k ≤ k 1 , the disease will die out for almost all positive initial populations; (iii) when α = α 0 and k > k 1 , the disease will persist in the form of a positive coexistent steady state for some positive initial populations; and (iv) when α < α 0 and k > k 0 , the disease will persist in the form of multiple positive periodic coexistent oscillations and coexistent steady states for some positive initial populations. Numerical simulations, including the existence of one or two limit cycles and data-fitting of the influenza data in Mainland China, are presented to illustrate the theoretical results.

Asymptotic periodicity in a diffusive West Nile virus model in a heterogeneous environment

This paper is concerned with a diffusive West Nile virus model (WNv) in a heterogeneous environment. The basic reproduction number [Formula: see text] for spatially homogeneous model is first introduced. We then define a threshold parameter [Formula: see text] for the corresponding diffusive WNv model in a heterogeneous environment. It is shown that if [Formula: see text], the model admits at least one nontrivial T-periodic solution, whereas if [Formula: see text], the model has no nontrivial T-periodic solution. By means of monotone iterative schemes, the true solution can be obtained and the asymptotic behavior of periodic solutions is presented. The paper is closed with some numerical simulations to illustrate our theoretical results.

Global stability of SEIVR epidemic model with generalized incidence and preventive vaccination

In this paper, an SEIVR epidemic model with generalized incidence and preventive vaccination is considered. First, we formulate the model and obtain its basic properties. Then, we find the equilibrium points of the model, the disease-free and the endemic equilibrium. The stability of disease-free and endemic equilibrium is associated with the basic reproduction number [Formula: see text]. If the basic reproduction number [Formula: see text], the disease-free equilibrium is locally as well as globally asymptotically stable. Moreover, if the basic reproduction number [Formula: see text], the disease is uniformly persistent and the unique endemic equilibrium of the system is locally as well as globally asymptotically stable under certain conditions. Finally, the numerical results justify the analytical results.

The effect of incidence function in backward bifurcation for malaria model with temporary immunity

This paper addresses the effect of the choice of the incidence function for the occurrence of backward bifurcation in two malaria models, namely, malaria model with standard incidence rate and malaria model with nonlinear incidence rate. Rigorous qualitative analyzes of the models show that the malaria model with standard incidence rate exhibits the phenomenon of backward bifurcation whenever a certain epidemiological threshold, known as the basic reproduction number, is less than unity. The epidemiological consequence of this phenomenon is that the classical epidemiological requirement of making the reproductive number less than unity is no longer sufficient, although necessary, for effectively controlling the spread of malaria in a community. For the malaria model with nonlinear incidence rate, it is shown that this phenomenon does not occur and the disease-free equilibrium of the model is globally-asymptotically stable whenever the reproduction number is less than unity. When the associated basic reproduction number is greater than unity, the models have a unique endemic equilibrium which is globally asymptotically stable under certain conditions. The sensitivity analysis based on the mathematical technique has been performed to determine the importance of the epidemic model parameters in making strategies for controlling malaria.

Disease control of delay SEIR model with nonlinear incidence rate and vertical transmission

The aim of this paper is to develop two delayed SEIR epidemic models with nonlinear incidence rate, continuous treatment, and impulsive vaccination for a class of epidemic with latent period and vertical transition. For continuous treatment, we obtain a basic reproductive number ℜ0 and prove the global stability by using the Lyapunov functional method. We obtain two thresholds ℜ* and ℜ∗ for impulsive vaccination and prove that if ℜ* < 1, then the disease-free periodic solution is globally attractive and if ℜ∗ > 1, then the disease is permanent by using the comparison theorem of impulsive differential equation. Numerical simulations indicate that pulse vaccination strategy or a longer latent period will make the population size infected by a disease decrease.

CHAOS ANALYSIS AND CONTROL FOR A CLASS OF SIR EPIDEMIC MODEL WITH SEASONAL FLUCTUATION

In this paper, the problems of chaos and chaos control for a class of susceptible-infected-removed (SIR) epidemic model with seasonal fluctuation are investigated. The seasonality in outbreak is natural among infectious diseases, as the common influenza, A type H1N1 influenza and so on. It is shown that there exist chaotic phenomena in the epidemic model. Furthermore, the tracking control method is used to control chaotic motions in the epidemic model. A feedback controller is designed to achieve tracking of an ideal output. Thus, the density of infected individuals can converge to zero, in other words, the disease can be disappeared. Finally, numerical simulations illustrate that the controller is effective.

Towards uncertainty quantification and inference in the stochastic SIR epidemic model

In this paper we address the problem of estimating the parameters of Markov jump processes modeling epidemics and introduce a novel method to conduct inference when data consists on partial observations in one of the state variables. We take the classical stochastic SIR model as a case study. Using the inverse-size expansion of van Kampen we obtain approximations for the first and second moments of the state variables. These approximate moments are in turn matched to the moments of an inputed Generic Discrete distribution aimed at generating an approximate likelihood that is valid both for low count or high count data. We conduct a full Bayesian inference using informative priors. Estimations and predictions are obtained both in a synthetic data scenario and in two Dengue fever case studies.

DYNAMICS OF SIS EPIDEMIC MODEL WITH THE STANDARD INCIDENCE RATE AND SATURATED TREATMENT FUNCTION

An SIS epidemic model with the standard incidence rate and saturated treatment function is proposed. The dynamics of the system are discussed, and the effect of the capacity for treatment and the recruitment of the population are also studied. We find that the effect of the maximum recovery per unit of time and the recruitment rate of the population over some level are two factors which lead to the backward bifurcation, and in some cases, the model may undergo the saddle-node bifurcation or Bogdanov–Takens bifurcation. It is shown that the disease-free equilibrium is globally asymptotically stable under some conditions. Numerical simulations are consistent with our obtained results in theorems, which show that improving the efficiency and capacity of treatment is important for control of disease.

Analysis of SIR epidemic models with nonlinear incidence rate and treatment

This paper deals with the nonlinear dynamics of a susceptible-infectious-recovered (SIR) epidemic model with nonlinear incidence rate, vertical transmission, vaccination for the newborns of susceptible and recovered individuals, and the capacity of treatment. It is assumed that the treatment rate is proportional to the number of infectives when it is below the capacity and constant when the number of infectives reaches the capacity. Under some conditions, it is shown that there exists a backward bifurcation from an endemic equilibrium, which implies that the disease-free equilibrium coexists with an endemic equilibrium. In such a case, reducing the basic reproduction number less than unity is not enough to control and eradicate the disease, extra measures are needed to ensure that the solutions approach the disease-free equilibrium. When the basic reproduction number is greater than unity, the model can have multiple endemic equilibria due to the effect of treatment, vaccination and other parameters. The existence and stability of the endemic equilibria of the model are analyzed and sufficient conditions on the existence and stability of a limit cycle are obtained. Numerical simulations are presented to illustrate the analytical results.

EXOGENOUS REINFECTION AND RESURGENCE OF TUBERCULOSIS: A THEORETICAL FRAMEWORK

The importance of exogenous reinfection versus endogenous reactivation for the resurgence of tuberculosis (TB) has been a matter of ongoing debate. Previous mathematical models of TB give conflicting results on the possibility of multiple stable equilibria in the presence of reinfection, and hence the failure to control the disease even when the basic reproductive number is less than unity. The present study reconsiders the effect of exogenous reinfection, by extending previous studies to incorporate a generalized rate of reinfection as a function of the number of actively infected individuals. A mathematical model is developed to include all possible routes to the development of active TB (progressive primary infection, endogenous reactivation, and exogenous reinfection). The model is qualitatively analyzed to show the existence of multiple equilibria under realistic assumptions and plausible range of parameter values. Two examples, of unbounded and saturated incidence rates of reinfection, are given, and simulation results using estimated parameter values are presented. The results reflect exogenous reinfection as a major cause of TB emergence, especially in high prevalence areas, with important public health implications for controlling its spread.

ANALYSIS ON AN EPIDEMIC MODEL WITH A RATIO-DEPENDENT NONLINEAR INCIDENCE RATE

In this paper, we study the global dynamics of an epidemic model with nonlinear incidence rate of saturated mass action which depends on the ratio of the number of infectives to that of the susceptibles. The model has set up a challenging issue regarding its dynamics at the R-axis since it is not well defined on it. By carrying out a global qualitative analysis of the model and studying the stabilities of the disease-free equilibrium and the endemic equilibrium, it is shown that either the number of infective individuals tends to zero as time evolves or the disease persists. Computer simulations are presented to illustrate the results.

The epidemiological dynamics of infectious trachoma may facilitate elimination

INTRODUCTION

Trachoma programs use mass distributions of oral azithromycin to treat the ocular strains of Chlamydia trachomatis that cause the disease. There is debate whether infection can be eradicated or only controlled. Mass antibiotic administrations clearly reduce the prevalence of chlamydia in endemic communities. However, perfect coverage is unattainable, and the World Health Organization's goal is to control infection to a level where resulting blindness is not a public health concern. Here, we use mathematical models to assess whether more ambitious goals such as local elimination or even global eradication are possible.

METHODS

We fit a class of non-linear, stochastic, susceptible-infectious-susceptible (SIS) models which allow positive or negative feedback, to data from a recent community-randomized trial in Ethiopia, and make predictions using model averaging.

RESULTS

The models predict that reintroduced infection may not repopulate the community, or may do so sufficiently slowly that surveillance might be effective. The preferred model exhibits positive feedback, allowing a form of stochastic hysteresis in which infection returns slowly after mass treatment, if it returns at all. Results for regions of different endemicity suggest that elimination may be more feasible than earlier models had predicted.

DISCUSSION

If trachoma can be eradicated with repeated mass antibiotic distributions, it would encourage similar strategies against other bacterial diseases whose only host is humans and for which effective vaccines are not available.

First principles modeling of nonlinear incidence rates in seasonal epidemics

In this paper we used a general stochastic processes framework to derive from first principles the incidence rate function that characterizes epidemic models. We investigate a particular case, the Liu-Hethcote-van den Driessche's (LHD) incidence rate function, which results from modeling the number of successful transmission encounters as a pure birth process. This derivation also takes into account heterogeneity in the population with regard to the per individual transmission probability. We adjusted a deterministic SIRS model with both the classical and the LHD incidence rate functions to time series of the number of children infected with syncytial respiratory virus in Banjul, Gambia and Turku, Finland. We also adjusted a deterministic SEIR model with both incidence rate functions to the famous measles data sets from the UK cities of London and Birmingham. Two lines of evidence supported our conclusion that the model with the LHD incidence rate may very well be a better description of the seasonal epidemic processes studied here. First, our model was repeatedly selected as best according to two different information criteria and two different likelihood formulations. The second line of evidence is qualitative in nature: contrary to what the SIRS model with classical incidence rate predicts, the solution of the deterministic SIRS model with LHD incidence rate will reach either the disease free equilibrium or the endemic equilibrium depending on the initial conditions. These findings along with computer intensive simulations of the models' Poincaré map with environmental stochasticity contributed to attain a clear separation of the roles of the environmental forcing and the mechanics of the disease transmission in shaping seasonal epidemics dynamics.

Parameter estimation of some epidemic models. The case of recurrent epidemics caused by respiratory syncytial virus

The research presented in this paper addresses the problem of fitting a mathematical model to epidemic data. We propose an implementation of the Landweber iteration to solve locally the arising parameter estimation problem. The epidemic models considered consist of suitable systems of ordinary differential equations. The results presented suggest that the inverse problem approach is a reliable method to solve the fitting problem. The predictive capabilities of this approach are demonstrated by comparing simulations based on estimation of parameters against real data sets for the case of recurrent epidemics caused by the respiratory syncytial virus in children.

The basic reproduction number in epidemic models with periodic demographics

Patterns of contact in social behaviour and seasonality due to environmental influences often affect the spread and persistence of diseases. Models of epidemics with seasonality and patterns in the contact rate include time-periodic coefficients, making the systems nonautonomous. No general method exists for calculating the basic reproduction number, the threshold for disease extinction, in nonautonomous epidemic models. However, for some epidemic models with periodic coefficients and constant population size, the time-averaged basic reproduction number has been shown to be a threshold for disease extinction. We extend these results by showing that the time-averaged basic reproduction number is a threshold for disease extinction when the population demographics are periodic. The results are shown to hold in epidemic models with periodic demographics that include temporary immunity, isolation, and multiple strains.

The Allee effect and infectious diseases: extinction, multistability, and the (dis-)appearance of oscillations

Infectious diseases that affect their host on a long timescale can regulate the host population dynamics. Here we show that a strong Allee effect can lead to complex dynamics in simple epidemic models. Generally, the Allee effect renders a population bistable, but we also identify conditions for tri- or monostability. Moreover, the disease can destabilize endemic equilibria and induce sustained oscillations. These disappear again for high transmissibilities, with eventually vanishing host population. Disease-induced extinction is thus possible for density-dependent transmission and without any alternative reservoirs. The overall complexity suggests that the system is very sensitive to perturbations and control methods, even in parameter regions with a basic reproductive ratio far beyond R(0) = 1. This may have profound implications for biological conservation as well as pest management. We identify important threshold quantities and attribute the dynamical behavior to the joint interplay of a strong Allee effect and infection.

Information-related changes in contact patterns may trigger oscillations in the endemic prevalence of infectious diseases

It is well known that behavioral changes in contact patterns may significantly affect the spread of an epidemic outbreak. Here we focus on simple endemic models for recurrent epidemics, by modelling the social contact rate as a function of the available information on the present and the past disease prevalence. We show that social behavior change alone may trigger sustained oscillations. This indicates that human behavior might be a critical explaining factor of oscillations in time-series of endemic diseases. Finally, we briefly show how the inclusion of seasonal variations in contacts may imply chaos.

Saturation recovery leads to multiple endemic equilibria and backward bifurcation

The number of patients need to be treated may exceed the carry capacity of local hospitals during the spreading of a severe infectious disease. We propose an epidemic model with saturation recovery from infective individuals to understand the effect of limited resources for treatment of infectives on the emergency disease control. It is shown that saturation recovery from infective individuals leads to vital dynamics, such as bistability and periodicity, when the basic reproduction number R(0) is less than unity. An interesting dynamical behavior of the model is a backward bifurcation which raises many new challenges to effective infection control.

Bifurcations of an epidemic model with non-monotonic incidence rate of saturated mass action

In this paper, we study the bifurcations of an epidemic model with non-monotonic incidence rate of saturated mass action, which describes the psychological effects of the community on certain serious diseases when the number of infective is getting larger. By carrying out the bifurcation analysis of the model, we show that there exist some values of the model parameters such that numerous kinds of bifurcation occur for the model, such as Hopf bifurcation, Bogdanov-Takens bifurcation.

Global analysis of an epidemic model with nonmonotone incidence rate

In this paper we study an epidemic model with nonmonotonic incidence rate, which describes the psychological effect of certain serious diseases on the community when the number of infectives is getting larger. By carrying out a global analysis of the model and studying the stability of the disease-free equilibrium and the endemic equilibrium, we show that either the number of infective individuals tends to zero as time evolves or the disease persists.

Neimark–Sacker bifurcations in a non-standard numerical scheme for a class of positivity-preserving ODEs

We discuss the nature of Neimark–Sacker bifurcations occurring in a non-standard numerical scheme, for a class of positivity-preserving systems of ordinary differential equations (ODEs) which undergoes a corresponding Hopf bifurcation. Extending previous work (Alexander & Moghadas 2005 a Electron. J. Diff. Eqn. Conf . 12 , 9–19), it is shown that the type of Neimark–Sacker bifurcation (supercritical or subcritical) may be affected by the integration time-step . The general form of the scheme in the vicinity of a fixed point is given, from which the expression for the first Lyapunov coefficient for two-dimensional systems, valid for arbitrary time-step, is explicitly derived. The analysis shows that this coefficient undergoes an shift with respect to the corresponding coefficient of the original ODE. This could lead to a type of bifurcation which differs from the corresponding Hopf bifurcation in the ODE, due to changes in the sign of the first Lyapunov coefficient as varies. This is especially problematic in the vicinity of certain types of degenerate Hopf bifurcation, at which this coefficient may vanish. We also present a general method to eliminate the possible shift in the bifurcation parameter of the scheme; however, the first Lyapunov coefficient may still be subjected to an shift, leading to a possibly erroneous type of bifurcation. Examples are given to illustrate the theoretical results of the paper with applications to mathematical biology.

Dynamical behaviour of epidemiological models with sub-optimal immunity and nonlinear incidence

In this paper we analyze the dynamics of two families of epidemiological models which correspond to transitions from the SIR (susceptible-infectious-resistant) to the SIS (susceptible-infectious-susceptible) frameworks. In these models we assume that the force of infection is a nonlinear function of density of infectious individuals, I. Conditions for the existence of backwards bifurcations, oscillations and Bogdanov-Takens points are given.