Global Hopf bifurcation of a cholera model with media coverage.

We propose a model for cholera under the impact of delayed mass media, including human-to-human and environment-to-human transmission routes. First, we establish the extinction and uniform persistence of the disease with respect to the basic reproduction number. Then, we conduct a local and global Hopf bifurcation analysis by treating the delay as a bifurcation parameter. Finally, we carry out numerical simulations to demonstrate theoretical results. The impact of the media with the time delay is found to not influence the threshold dynamics of the model, but is a factor that induces periodic oscillations of the disease.

Getting Jab or Regular Test: Observations from an Impulsive Epidemic COVID-19 Model.

Several safe and effective vaccines are available to prevent individuals from experiencing severe illness or death as a result of COVID-19. Widespread vaccination is widely regarded as a critical tool in the fight against the disease. However, some individuals may choose not to vaccinate due to vaccine hesitancy or other medical conditions. In some sectors, regular compulsory testing is required for such unvaccinated individuals. Interestingly, different sectors require testing at various frequencies, such as weekly or biweekly. As a result, it is essential to determine the optimal testing frequency and identify underlying factors. This study proposes a population-based model that can accommodate different personal decision choices, such as getting vaccinated or undergoing regular tests, as well as vaccine efficacies and uncertainties in epidemic transmission. The model, formulated as impulsive differential equations, uses time instants to represent the reporting date for the test result of an unvaccinated individual. By employing well-accepted indices to measure transmission risk, including the basic reproduction number, the peak time, the final size, and the number of severe infections, the study shows that an optimal testing frequency is highly sensitive to parameters involved in the transmission process, such as vaccine efficacy, disease transmission rate, test accuracy, and existing vaccination coverage. The testing frequency should be appropriately designed with the consideration of all these factors, as well as the control objectives measured by epidemiological quantities of great concern.

Basic reproduction ratios for periodic and time-delayed compartmental models with impulses.

Much work has focused on the basic reproduction ratio [Formula: see text] for a variety of compartmental population models, but the theory of [Formula: see text] remains unsolved for periodic and time-delayed impulsive models. In this paper, we develop the theory of [Formula: see text] for a class of such impulsive models. We first introduce [Formula: see text] and show that it is a threshold parameter for the stability of the zero solution of an associated linear system. Then we apply this theory to a time-delayed computer virus model with impulse treatment and obtain a threshold result on its global dynamics in terms of [Formula: see text]. Numerically, it is found that the basic reproduction ratio of the time-averaged delayed impulsive system may overestimate the spread risk of the virus.

A reaction-diffusion malaria model with seasonality and incubation period.

In this paper, we propose a time-periodic reaction-diffusion model which incorporates seasonality, spatial heterogeneity and the extrinsic incubation period (EIP) of the parasite. The basic reproduction number [Formula: see text] is derived, and it is shown that the disease-free periodic solution is globally attractive if [Formula: see text], while there is an endemic periodic solution and the disease is uniformly persistent if [Formula: see text]. Numerical simulations indicate that prolonging the EIP may be helpful in the disease control, while spatial heterogeneity of the disease transmission coefficient may increase the disease burden.

A periodic two-patch SIS model with time delay and transport-related infection.

In this paper, we propose a periodic SIS epidemic model with time delay and transport-related infection in a patchy environment. The basic reproduction number R0 is derived which determines the global dynamics of the model system: if R0 < 1, the disease-free periodic state is globally attractive while there exists at least one positive periodic state and the disease persists if R0 > 1. Numerical simulations are performed to confirm the analytical results and to explore the dependence of R0 on the transport-related infection parameters and the amplitude of fluctuations.

Threshold dynamics of a time-delayed SEIRS model with pulse vaccination.

In this paper, we consider a delayed SEIRS model with pulse vaccination and varying total population size. The basic reproduction number R0 is derived, and it is shown that the disease-free periodic solution is globally attractive if R0 < 1, while the disease is uniformly persistent when R0 > 1. Our results really improve the results by Gao et al. (2007) [8], where they left the open problem of finding a sharp threshold which determines the eradication and uniform persistence. Numerical simulations are conducted to illustrate the analytical results and explore the influences of pulse vaccination and time delay on the spread of the disease. To the best of our knowledge, it is the first work to have the sharp threshold dynamics for impulsive epidemic models with the delay in the infected compartments.

Threshold dynamics of a periodic SIR model with delay in an infected compartment.

Threshold dynamics of epidemic models in periodic environments attract more attention. But there are few papers which are concerned with the case where the infected compartments satisfy a delay differential equation. For this reason, we investigate the dynamical behavior of a periodic SIR model with delay in an infected compartment. We first introduce the basic reproduction number R0 for the model, and then show that it can act as a threshold parameter that determines the uniform persistence or extinction of the disease. Numerical simulations are performed to confirm the analytical results and illustrate the dependence of R0 on the seasonality and the latent period.

Positive periodic solutions of an epidemic model with seasonality.

An SEI autonomous model with logistic growth rate and its corresponding nonautonomous model are investigated. For the autonomous case, we give the attractive regions of equilibria and perform some numerical simulations. Basic demographic reproduction number R d is obtained. Moreover, only the basic reproduction number R 0 cannot ensure the existence of the positive equilibrium, which needs additional condition R d > R 1. For the nonautonomous case, by introducing the basic reproduction number defined by the spectral radius, we study the uniform persistence and extinction of the disease. The results show that for the periodic system the basic reproduction number is more accurate than the average reproduction number.