Time-delayed Epidemic Model Research Articles

Modeling the transmission dynamics of a time-delayed epidemic model with saturated treatment rate

In this paper, an attempt has been made to explore a new delayed epidemiological model assuming that the disease is transmitted among the susceptible population and possessing nonlinear incidence function along with a saturated treatment rate. Due attention is paid to the positivity and boundedness of the solutions and the bifurcation of the dynamical system as well. Basic reproduction number is being calculated, and considering the latent period as a bifurcation parameter, it has been examined that a Hopf-bifurcation occurs near the endemic equilibrium point while the parameter attains critical values. We have also discussed the stability and direction of Hopf-bifurcation near the endemic equilibrium point, the global stability analysis and the optimal control theory. We conclude that the system reveals chaotic dynamics through a specific time-delay value. Numerical simulations are being performed in order to explain the accuracy and effectiveness of the acquired theoretical results.

A Markovian regime-switching stochastic hybrid time-delayed epidemic model with vaccination

This paper considers a stochastic hybrid time-delayed epidemic model with vaccination perturbed by both white noise and telegraph noise in the form of Markovian switching. Using Lyapunov method, we show existence and uniqueness of the global positive solution. Under some suitable conditions, we derive extinction with and without effect of delay. Moreover, the asymptotic bounds on solutions is also discussed. Finally, we carry out the numerical simulations, including two examples based on real life diseases to confirm our theoretical results.

Nonlinear dynamics of a time-delayed epidemic model with two explicit aware classes, saturated incidences, and treatment.

Whenever a disease emerges, awareness in susceptibles prompts them to take preventive measures, which influence individuals’ behaviors. Therefore, we present and analyze a time-delayed epidemic model in which class of susceptible individuals is divided into three subclasses: unaware susceptibles, fully aware susceptibles, and partially aware susceptibles to the disease, respectively, which emphasizes to consider three explicit incidences. The saturated type of incidence rates and treatment rate of infectives are deliberated herein. The mathematical analysis shows that the model has two equilibria: disease-free and endemic. We derive the basic reproduction number R_0 of the model and study the stability behavior of the model at both disease-free and endemic equilibria. Through analysis, it is demonstrated that the disease-free equilibrium is locally asymptotically stable when R_0<1, unstable when R_0>1, and linearly neutrally stable when R_0=1 for the time delay varrho >0. Further, an undelayed epidemic model is studied when R_0=1, which reveals that the model exhibits forward and backward bifurcations under specific conditions, which also has important implications in the study of disease transmission dynamics. Moreover, we investigate the stability behavior of the endemic equilibrium and show that Hopf bifurcation occurs near endemic equilibrium when we choose time delay as a bifurcation parameter. Lastly, numerical simulations are performed in support of our analytical results.

Threshold dynamics of a time-delayed epidemic model for continuous imperfect-vaccine with a generalized nonmonotone incidence rate.

In this paper, we study the dynamics of an infectious disease in the presence of a continuous-imperfect vaccine and latent period. We consider a general incidence rate function with a non-monotonicity property to interpret the psychological effect in the susceptible population. After we propose the model, we provide the well-posedness property and calculate the effective reproduction number {mathcal {R}}_E. Then, we obtain the threshold dynamics of the system with respect to {mathcal {R}}_E by studying the global stability of the disease-free equilibrium when {mathcal {R}}_E<1 and the system persistence when {mathcal {R}}_E>1. For the endemic equilibrium, we use the semi-discretization method to analyze its linear stability. Then, we discuss the critical vaccination coverage rate that is required to eliminate the disease. Numerical simulations are provided to implement a case study regarding data of influenza patients, study the local and global sensitivity of {mathcal {R}}_E<1, construct approximate stability charts for the endemic equilibrium over different parameter spaces and explore the sensitivity of the proposed model solutions.

Chimera states in multi-strain epidemic models with temporary immunity.

We investigate a time-delayed epidemic model for multi-strain diseases with temporary immunity. In the absence of cross-immunity between strains, dynamics of each individual strain exhibit emergence and annihilation of limit cycles due to a Hopf bifurcation of the endemic equilibrium, and a saddle-node bifurcation of limit cycles depending on the time delay associated with duration of temporary immunity. Effects of all-to-all and non-local coupling topologies are systematically investigated by means of numerical simulations, and they suggest that cross-immunity is able to induce a diverse range of complex dynamical behaviors and synchronization patterns, including discrete traveling waves, solitary states, and amplitude chimeras. Interestingly, chimera states are observed for narrower cross-immunity kernels, which can have profound implications for understanding the dynamics of multi-strain diseases.

Dynamics of vaccination in a time-delayed epidemic model with awareness

This paper investigates the effects of vaccination on the dynamics of infectious disease, which is spreading in a population concurrently with awareness. The model considers contributions to the overall awareness from a global information campaign, direct contacts between unaware and aware individuals, and reported cases of infection. It is assumed that there is some time delay between individuals becoming aware and modifying their behaviour. Vaccination is administered to newborns, as well as to aware individuals, and it is further assumed that vaccine-induced immunity may wane with time. Feasibility and stability of the disease-free and endemic equilibria are studied analytically, and conditions for the Hopf bifurcation of the endemic steady state are found in terms of system parameters and the time delay. Analytical results are supported by numerical continuation of the Hopf bifurcation and numerical simulations of the model to illustrate different types of dynamical behaviour.

A time-delayed epidemic model for Ebola disease transmission

In this paper, we propose a delayed mathematical model for the transmission of Ebola in humans. We consider the transmission of infection between the living humans and from infectious corpses to the living individuals in which the latent period of Ebola is incorporated. We identify the basic reproduction number R0 for the model, prove that the disease-free equilibrium is always globally asymptotically stable when R0 < 1, the disease is persistence and a unique endemic equilibrium exists when R0 > 1. We show that the endemic steady state is locally asymptotically stable under certain condition and globally asymptotically stable in a special case of the model. Numerical simulations are provided to demonstrate and complement the theoretical results.

A periodic age-structured epidemic model with a wide class of incidence rates

In this paper, a time-delayed epidemic model is formulated to describe the dynamics of seasonal diseases with age structure. By the method of the spectral radius of an integral operator, we define the basic reproduction number (R0) of the model. It is shown that the disease is uniformly persistent and there exists at least one positive periodic state when R0>1 while the disease will die out if R0<1. The presented case study not only confirms the theoretical results, but also demonstrates that the epidemic peak is very sensitive to the maturation period and the magnitude of seasonality, which is different from the dynamics of the model without considering age heterogeneities. These findings contribute to better understanding the epidemiological properties of the disease with age structure.

Threshold dynamics in a time-delayed epidemic model with dispersal

The global dynamics of a time-delayed model with population dispersal between two patches is investigated. For a general class of birth functions, persistence theory is applied to prove that a disease is persistent when the basic reproduction number is greater than one. It is also shown that the disease will die out if the basic reproduction number is less than one, provided that the initial size of the infected population is relatively small. Numerical simulations are presented using some typical birth functions from biological literature to illustrate the main ideas and the relevance of dispersal.

An Age-Structured Epidemic Model in a Patchy Environment

A time-delayed epidemic model is proposed to describe the dynamics of disease spread among patches. An age structure is incorporated in order to simulate the phenomenon that some diseases only occur in the adult population. Sufficient conditions are established for global extinction and uniform persistence of the disease. One example shows that the disease could undergo persistence-extinction-persistence switches as the migration rate of juveniles increases, although juveniles are immune and cannot transmit the disease. The second example indicates that this switching phenomenon also happens when juveniles migrate with susceptible adults.