Hopf bifurcation analysis of a delayed SEIR epidemic model with infectious force in latent and infected period

Bifurcation analysis in an epidemic model on adaptive networks

In this paper, we study a delayed adaptive network epidemic model in which the local spatial connections of susceptible and susceptible individuals have time-delay effects on the rate of demographic change of local spatial connections of susceptible and susceptible individuals. We prove that the Hopf bifurcation occurs at the critical value τ₀ with delay τ as the bifurcation parameter. Then, by using the normal form method and the central manifold theory, the criteria for the bifurcation direction and stability are derived. Finally, numerical simulations are presented to show the feasibility of our results.

Dynamical analysis of a stochastic delayed SIR epidemic model with vertical transmission and vaccination

In order to describe the dynamic process of epidemic transmission with vertical transmission and vaccination in more detail and to better track the factors that lead to the occurrence of epidemics, we construct a stochastic delayed model with a specific functional response to describe its epidemic dynamics. We first prove the existence and uniqueness of the positive solution of the model. Moreover, we analyze the sufficient conditions for the extinction and persistence of the model. Finally, numerical simulations are presented to illustrate our mathematical findings.

Incorporating global dynamics to improve the accuracy of disease models: Example of a COVID-19 SIR model

Mathematical models of infectious diseases exhibit robust dynamics, such as stable endemic, disease-free equilibriums or convergence of the solutions to periodic epidemic waves. The present work shows that the accuracy of such dynamics can be significantly improved by including global effects of host movements in disease models. To demonstrate improved accuracy, we extended a standard Susceptible-Infected-Recovered (SIR) model by incorporating the global dynamics of the COVID-19 pandemic. The extended SIR model assumes three possibilities for susceptible individuals traveling outside of their community:

• They can return to the community without any exposure to the infection.

• They can be exposed and develop symptoms after returning to the community.

• They can be tested positively during the trip and remain quarantined until fully recovered.

To examine the predictive accuracy of the extended SIR model, we studied the prevalence of the COVID-19 infection in six randomly selected cities and states in the United States: Kansas City, Saint Louis, San Francisco, Missouri, Illinois, and Arizona. The extended SIR model was parameterized using a two-step model-fitting algorithm. The extended SIR model significantly outperformed the standard SIR model and revealed oscillatory behaviors with an increasing trend of infected individuals. In conclusion, the analytics and predictive accuracy of disease models can be significantly improved by incorporating the global dynamics of the infection.

Mathematical analysis of a cancer model with time-delay in tumor-immune interaction and stimulation processes

In this study, we discuss a cancer model considering discrete time-delay in tumor-immune interaction and stimulation processes. This study aims to analyze and observe the dynamics of the model along with variation of vital parameters and the delay effect on anti-tumor immune responses. We obtain sufficient conditions for the existence of equilibrium points and their stability. Existence of Hopf bifurcation at co-axial equilibrium is investigated. The stability of bifurcating periodic solutions is discussed, and the time length for which the solutions preserve the stability is estimated. Furthermore, we have derived the conditions for the direction of bifurcating periodic solutions. Theoretically, it was observed that the system undergoes different states if we vary the system's parameters. Some numerical simulations are presented to verify the obtained mathematical results.

Mathematical modeling of intervention and low medical resource availability with delays: Applications to COVID-19 outbreaks in Spain and Italy

Infectious diseases have been one of the major causes of human mortality, and mathematical models have been playing significant roles in understanding the spread mechanism and controlling contagious diseases. In this paper, we propose a delayed SEIR epidemic model with intervention strategies and recovery under the low availability of resources. Non-delayed and delayed models both possess two equilibria: the disease-free equilibrium and the endemic equilibrium. When the basic reproduction number $ R_0 = 1 $, the non-delayed system undergoes a transcritical bifurcation. For the delayed system, we incorporate two important time delays: $ \tau_1 $ represents the latent period of the intervention strategies, and $ \tau_2 $ represents the period for curing the infected individuals. Time delays change the system dynamics via Hopf-bifurcation and oscillations. The direction and stability of delay induced Hopf-bifurcation are established using normal form theory and center manifold theorem. Furthermore, we rigorously prove that local Hopf bifurcation implies global Hopf bifurcation. Stability switching curves and crossing directions are analyzed on the two delay parameter plane, which allows both delays varying simultaneously. Numerical results demonstrate that by increasing the intervention strength, the infection level decays; by increasing the limitation of treatment, the infection level increases. Our quantitative observations can be useful for exploring the relative importance of intervention and medical resources. As a timing application, we parameterize the model for COVID-19 in Spain and Italy. With strict intervention policies, the infection numbers would have been greatly reduced in the early phase of COVID-19 in Spain and Italy. We also show that reducing the time delays in intervention and recovery would have decreased the total number of cases in the early phase of COVID-19 in Spain and Italy. Our work highlights the necessity to consider the time delays in intervention and recovery in an epidemic model.