In this work, we propose a stochastic Human Papillomavirus (HPV) epidemic model with two kinds of delays and media influences. These two time delays are the delay time caused by media receiving the disease information and the delay time of public feedback after the media coverage. In addition, media coverage not only has a negative impact on the infection rate, but it also has a positive impact on the vaccination rate of disease. We discuss the existence and uniqueness of the positive solution for the HPV epidemic model, and then put forward a positively invariant set. The sufficient conditions of the extinction and persistence for the HPV epidemic are given. For the optimal control problem of the HPV epidemic, we obtain an optimal strategy. Our numerical simulations validate the theoretical results of this paper, showing that appropriate media coverage can help control the development of the disease.

Invasive species can inflict significant harm on biodiversity and disrupt ecosystem stability. Introducing pathogens to combat invasive species is recognized as an effective method for preserving biodiversity. Against this backdrop, we examine a predator–prey model involving diseased predators, taking into account the impact of environmental noise, and then propose a stochastic predator–prey model where predators carry diseases. In this model, we assume that the intrinsic growth rate of the prey and the disease transmission rate of the predator follow an Ornstein–Uhlenbeck process. Our contributions in this study can be summarized as follows: First, we establish the existence and uniqueness of the global solution for the stochastic predator–prey model, along with an analysis of moment estimation for the global solution. Second, we provide sufficient conditions for the existence of stationary distribution and population extinction within the stochastic predator–prey model. Furthermore, we derive the precise expression of the probability density function near the quasi-positive equilibrium [Formula: see text] of the stochastic predator–prey model. Lastly, we validate our analytical findings through numerical simulations.

This study investigates the dynamics of an epidemic employing an (Susceptible-Infected-Hospitalized-Recovered-Susceptible (SIHRS) epidemiological model highlighting the crucial importance of hospital bed availability and a non-monotone incidence function, which incorporates the influences of stringent governmental measures, social behavior dynamics and public responses in both autonomous and non-autonomous scenarios. The analysis investigates the conditions for existence of infection-free and endemic steady states based on the basic reproduction number as it surpasses unity. Sensitivity analysis has been conducted to evaluate the impact of different system parameters on disease transmission. This work also investigates alterations in stability of the system caused by transcritical, Hopf and saddle-node bifurcations. Additionally, two-parameter bifurcation identifies the regions where the stability of both the equilibrium points has been examined. Numerical simulations are shown to illustrate all the theoretically obtained results. Also, the model examines the dynamical behavior of the epidemic when the quantity of available hospital beds varies periodically. This aspect of the study highlights the significant impact of hospital bed availability. Such factors are crucial in preventing disease spread during an epidemic. The results provide valuable insights into how dynamic patterns of disease transmission are influenced by healthcare infrastructure and public health interventions. This comprehensive exploration underscores the importance of integrated approaches combining medical resources and societal measures in managing and mitigating the effects of epidemics.

The impact of multiple time delays and the variation of populations between patches make population dynamics research more realistic and challenging. The main goal of this work is to study the effect of delay on the asymptotic behavior of tick population dynamics model with patch structure and multiple time-varying delays. We make use of the fluctuation lemma and inequality techniques to tackle the positivity, persistence and the global attractivity of the considered high-dimensional system. Finally, we give an example with three numerical simulations to verify that our theoretical results are completely new and complement some earlier existing ones. Our work is conducive to further understanding the more complex laws of tick population dynamics.

In this study, we establish a sophisticated compartmental Caputo fractional-order Susceptible-Vaccinated-Exposed-Infected-Hospitalized-Recovered Pathogens mathematical model for analyzing the dynamical behavior of COVID-19 transmission in the perspective of COVID-19 pandemic situation of Bangladesh considering the data up to 2nd dose vaccination. The frame of Caputo fractional-order derivative has been used to fractionalize the model’s equations. A total of 10 distinct compartments have been used to represent the dynamics of human and novel coronavirus. The Next-generation matrix approach has been used for finding the basic reproduction number of considered model. For establishing the existence and uniqueness of the model’s solutions, here we apply Banach fixed point theorem. Routh–Hurwitz, Ulam–Hyers and generalized Ulam–Hyers stability criteria have been applied for stability analysis. We execute a sensitivity analysis to check the comparative prominence of the model’s parameters. For the numerical simulation, data have been accumulated from the COVID-19 dashboard of Directorate general of health services of Bangladesh. The Fractional Adams–Bashforth–Moulton method along with MATLAB code fde12.m has also been applied to perform the numerical simulation. Furthermore, here we use some Maple codes for finding the model’s equilibrium points, basic reproduction number and sensitivity indices. Finally, we include a discussion on the influence of new rate of infections through environment and interaction with infected persons as well as the effect of 1st and 2nd dose vaccinations.