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In this paper, we consider an epidemic model in well-mixed multiplex networks with distributed time delay. Specifically, the model consists of two layers of well-mixed networks in the physical and virtual worlds, respectively, where two diffusive processes interact and influence each other within the same individual. We assume that there is a distributed time delay for an individual to become infected, but no delay for an individual to transition from unawareness to awareness. Our main results are as follows: Let R 0 P and R 0 V represent the basic reproduction numbers in the physical and virtual worlds, respectively. First, we demonstrate that the disease will die out for any delay time, provided that R 0 P ≤ 1 or 1 < R 0 P ≤ R 0 V. The latter condition emphasizes the significance of effective information spreading in eradicating the disease. Secondly, in the case of R 0 P > max ⁡ { 1 , R 0 V } , we establish that the model exhibits an endemic and information saturated equilibrium, denoted as E 3. Additionally, we show that the model is uniformly persistent, indicating the sustained outbreak of the disease. • We explore an epidemic model influenced by human awareness within the framework of a multiplex network. • The latent period of the infectious disease is concerned. • The necessary and sufficient conditions for the uniform persistence of disease are derived. • Our results highlight the crucial role of information spreading in disease eradication. [ABSTRACT FROM AUTHOR]

We establish an SAIQR epidemic network model, in which asymptomatic infected people (A) are as contagious as infected people (I). The basic reproductive number R 0 is calculated, and the globally asymptotically stable of the disease-free equilibrium, the globally attractive and globally asymptotically stable of the endemic equilibrium are obtained. For the control of epidemic transmission, we take into account the non-instantaneous impulsive vaccination in the model, calculate the basic reproduction number R 0 ⁎ of the model, and demonstrate that the disease-free T -periodic solution is globally attractive and the model is permanent. Finally, we choose scale-free network to simulate numerically and validate the results of this paper. • An SAIQR epidemic network model is established, in which asymptomatic infected people are as contagious as infected people. • The globally asymptotically stable of the disease-free equilibrium and the endemic equilibrium are obtained. • Non-instantaneous impulsive vaccination is considered in the system, and the permanence of the system are investigated. [ABSTRACT FROM AUTHOR]

• The paper proposes the stochastic dynamics of SIQR model structure with Lévy jumps and three time delays. • The existence and uniqueness of the positive solution, the dynamic properties around the disease-free equilibrium and the endemic equilibrium are obtained by constructing some suitable Lyapunov functions. • The inuence of isolation and vaccination rate on the disease dynamics is also examined by sensitivity analysis. • The unknown parameters are estimated by the least-square method and the trend of COVID-19 epidemic was predicted. Isolation and vaccination are the two most effective measures in protecting the public from the spread of illness. The SIQR model with vaccination is widely used to investigate the dynamics of an infectious disease at population level having the compartments: susceptible, infectious, quarantined and recovered. The paper mainly aims to extend the deterministic model to a stochastic SQIR case with Lévy jumps and three-time delays, which is more suitable for modeling complex and instable environment. The existence and uniqueness of the global positive solution are obtained by using the Lyapunov method. The dynamic properties of stochastic solution are studied around the disease-free and endemic equilibria of the deterministic model. Our results reveal that stochastic perturbation affect the asymptotic properties of the model. Numerical simulation shows the effects of interested parameters of theoretical results, including quarantine, vaccination and jump parameters. Finally, we apply both the stochastic and deterministic models to analyze the outbreak of mutant COVID-19 epidemic in Gansu Province, China. [ABSTRACT FROM AUTHOR]

• A model for HTLV-I infection that includes uninfected & infected CD4+T cells and HTLV-I-specific CD8+T cells. • To better understand the dynamics of HTLV-I infection, we incorporate a CTL immune response delay. • Applying Lyapunov functional procedures to determine global stability by the reproduction numbers R0 and R1. • The model with delay experiences a destabilization of the infected steady state leading to Hopf bifurcation and periodic solutions. • We estimate the length of delay to preserve the stability of bifurcating limit cycle using Nyquist criteria. This paper deals with a four dimensional mathematical model for Human T-cell leukemia virus type-I (HTLV-I) infection that includes delayed CD8+ cytotoxic T-cells (CTLs) immune response. The proposed system has three biologically feasible steady states, namely disease-free steady state, CTL-inactive steady state and an interior steady state. Our theoretical analysis demonstrates that local and global stability analysis are established by the two critical parameters R 0 and R 1 , basic reproduction numbers due to viral infection and due to CTLs immune response, respectively. The disease-free steady state E 0 is globally stable if R 0 ≤ 1 , and the HTLV-I infections are eliminated. The asymptotic-carrier steady state E 1 is globally stable if R 1 ≤ 1 < R 0 , which indicates HTLV-I infection is chronic without persistence of CTLs immune response. The interior steady state E 2 is globally asymptotic stable if R 1 > 1 , which implies that the HTLV-I infection is choric in persistence of CTLs immune response. Due to immune response delay, our proposed model undergoes a destabilization of the interior steady state leading to Hopf bifurcation and periodic oscillations. We estimate the length of time delay that preserve the stability of period-1 limit cycle. We also derived the direction and stability of Hopf bifurcation around the interior steady state by center manifold theory and normal form method. To determine the robustness of the model, we performed normalized forward sensitivity analysis with reference to R 0 and R 1. Our proposed model undergoes Hopf bifurcation with respect to the production rate of uninfected CD4+T cells h , removal rate of virus-specific CTLs d 4 , spontaneous infected CD4+T cell activation d 2 and transmissibility coefficient β. Implications of our numerical illustrations to the pathogenesis of HTLV-I infection and the development of HTLV-I related HAM/TSP are explored. [ABSTRACT FROM AUTHOR]

• An SVEITR model with vaccination, fast and slow progression, incomplete treatment and relapse for TB is proposed. • Based on the data from 1988 to 2019, the basic reproduction number (R 0)of TB in the United States is 1.3763. • If R 0 > 1, the unique endemic equilibrium is globally asymptotically stable under certain conditions. • Optimal control strategies are prevention and control education, timely treatment and enhanced efficacy. • The most cost-effective strategies are timely treatment and enhanced efficacy. Tuberculosis is a chronic infectious disease with a high death rate, which has attracted worldwide attention. In this paper, we focus on the annual data of tuberculosis in the United States from 1988 to 2019. Firstly, we propose an S V E I T R dynamical model with vaccination, fast and slow progression, incomplete treatment, and relapse. Mathematical analyses show that the disease-free equilibrium is globally asymptotically stable (unstable) if R 0 ≤ 1 (if R 0 > 1). There exists a unique endemic equilibrium that is globally asymptotically stable when R 0 > 1. Next, we use the Markov-chain Monte-Carlo (MCMC) method to acquire the model's optimal parameters. From R 0 = 1.3763 , tuberculosis will not be eliminated without control measures taken in the United States. Through partial rank correlation analysis, the most sensitive parameters to the basic reproduction number are found. Based on this, we take the actual epidemic situation of tuberculosis in the United States into considerations to get the required conditions for optimal control using Pontryagin's maximum principle and cost-effectiveness analysis. Implementation of three public health measures: tuberculosis prevention and control education, timely treatment, and enhanced efficacy can reduce the prevalence of tuberculosis in the United States. But at present, it is challenging for us to achieve the goal of eliminating tuberculosis by 2030 as called for by WHO. [ABSTRACT FROM AUTHOR]

• A hybrid stochastic SEQIHR model is studied for disease dynamics with media effect. • We obtain the sufficient conditions for disease extinction of stochastic model. • We obtain the criteria for the existence of an ergodic stationary distribution. • The developed θ -stochastic method can be greatly applied to other hybrid models. Combining a deterministic SEQIHRS model proposed by Sahu et al. (2015) and the present mathematical modelling of media coverage effect (2020), a hybrid stochastic SEQIHR model, perturbed by both nonlinear white noise and colored noise, is formulated and studied for the transmission dynamics of an infectious disease with media coverage, quarantine strategies and pre-existing immunity in a community. First, we prove that the stochastic model possesses a unique global positive solution. Second, by means of the basic reproduction number R 0 of the corresponding deterministic model, two relevant critical values which include R ¯ 0 and R 0 C are derived. Next, we obtain the disease extinction under R 0 < 1 and R ¯ 0 < 1. Moreover, we further establish the sufficient condition R 0 C > 1 for the existence and uniqueness of an ergodic stationary distribution of the stochastic model, which means the infectious disease will be prevailing and persistent in a community. Finally, several numerical simulations are performed to validate the above theoretical results. Besides, the impact of media coverage and nonlinear hybrid noises on the dynamical behavior of the stochastic model are studied at the end of this paper. [ABSTRACT FROM AUTHOR]

• A drug-sensitive and drug-resistant mixed strains of HIV infection model is proposed. • The local stability of equilibria for the model are obtained. • The global stability of equilibria for the model are stated and proved. • The uniform persistence of HIV infection model is established. In this paper, a drug-sensitive and drug-resistant mixed strains of HIV infection model with saturated incidence and distributed infection delays is proposed. The drug-sensitive strain could mutate and become drug-resistant strain during the process of reverse transcription (i.e., SR conversion in short). The nonnegativity and boundedness of solutions are discussed, by introducing two kinds of principle reproduction numbers, i.e., R s (the reproduction number induced by drug-sensitive strain of HIV virus) and R r (the reproduction number induced by drug-resistant strain of HIV virus), the existence of equilibria is also acquired. Using the linearization method and constructing the suitable Lyapunov functionals, the criteria are established on the local and global stability of equilibria (including infection-free, drug-sensitive strain, drug-resistant strain and endemic) based on the composite effects of R s and R r. We find that there exist competitive exclusion and coexistence phenomenon between drug-sensitive and drug-resistant strains without SR conversion, while uniform persistence between the two kinds of strains with SR conversion. The theoretical results are illustrated by numerical simulations. [ABSTRACT FROM AUTHOR]

• A stochastic SEI epidemic model with general distributed delay is established. • The existence of a unique global positive solution for the stochastic model are proved. • We prove that there is a stationary solution to the stochastic system. • Two probability density functions with respect to the stationary solution are obtained. • Some sufficient conditions are established for the disease extinction of stochastic system. A stochastic SEI (Susceptible-Exposed-Infected) epidemic model with general distributed delay is studied in this paper. First, we prove the existence and uniqueness of a global positive solution to the stochastic system. By means of the Lyapunov method, we verify the existence of a stationary distribution of the positive solution under a stochastic criterion R 0 p > 1 , which is known as stationary solution. Moreover, if R 0 p > 1 , two exact probability density functions around the quasi-stable equilibrium are obtained by solving the corresponding Fokker-Planck equation. Notably, both the explicit expression of density function and the existence of stationary distribution suggest the disease persistence in biological sense. For completeness, some sufficient conditions for disease extinction are established. At last, several numerical simulations are provided to verify our analytical results and reveal the impact of stochastic perturbations on disease transmission. [ABSTRACT FROM AUTHOR]

• The vaccination strategies analysis from mandatory to voluntary vaccination. • Establishes a SEIRV model considering birth rate and vaccination rate. • The evolutionary analysis of vaccination on heterogeneous network. • A simulation experiment is conducted using the SARS-Cov-2 pandemic data. Nowadays, vaccination is the most effective way to control the epidemic spreading. In this paper, an epidemic SEIRV (susceptible-exposed-infected-removed -vaccinated) model and an evolutionary game model are established to analyze the difference between mandatory vaccination method and voluntary vaccination method on heterogeneous networks. Firstly, we divide the population into four categories, including susceptible individuals, exposed individuals, infected individuals and removed individuals. Based on the mean field approximation theory, differential equations are developed to characterize the changes of the proportions of the four groups over time under mandatory vaccination. Then through the analysis of the differential equations, the disease-free equilibrium point (DFE) and the endemic disease equilibrium point (EDE) are obtained. Also, the basic reproduction number is obtained by the next-generation matrix method and the stability analysis of the equilibrium points is performed. Next, by considering factors such as vaccination cost, treatment cost and government subsidy rate, differential equations are established to represent the change of vaccination rate over time. By analyzing the final vaccination coverage rate, we can get the minimum vaccination cost to make infectious disease disappear. Finally, the Monte Carlo method is used for numerical simulation to verify the results obtained from the theoretical analysis. Using the SARS-Cov-2 pandemic data from Wuhan, China, the experimental results show that when the effectiveness rate of vaccination is 0.75, the vaccination cost is not higher than 0.886 so that the vaccination strategy can be spread among the population. If mandatory vaccination is adopted, the minimum vaccination rate is 0.146. [ABSTRACT FROM AUTHOR]

In this paper, we propose a generalized SEI epidemic model with a general nonlinear incidence rate and death rate functions. Furthermore, we consider that those who spread the disease are the infected and the exposed. Applying the direct Lyapunov method, we prove that the endemic equilibrium is globally asymptotically stable when the basic reproduction number R 0 is greater than unity and the disease free equilibrium is globally asymptotically stable when R 0 is lower than unity. We conclude that in order to obtain global stability and uniqueness of the equilibrium points, monoticity is necessary but not differentiability in the functions present in the model. [ABSTRACT FROM AUTHOR]