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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 cholera infection model with vaccination strategy is studied. • The basic reproduction number is obtained. • A threshold dynamics is established by using Lyapunov function technique. • Several reasonable optimal control strategies are suggested to the prevention and control of the cholera infection. In this paper, a cholera infection model with vaccination strategy is investigated. By analyzing corresponding characteristic equations, the local stability of each of feasible equilibria is established. By means of Lyapunov functions and LaSalle's invariance principle, it is proved that if the basic reproduction number is less than unity, the disease-free equilibrium is globally asymptotically stable. If the basic reproduction number is greater than unity, the endemic equilibrium is globally asymptotically stable. In addition, by using Pontryagin's maximum principle, several reasonable optimal control strategies are suggested to the prevention and control of the cholera infection. Numerical simulations are carried out to illustrate the theoretical results. [ABSTRACT FROM AUTHOR]

In this paper, we propose an age-since-infection virus model with Fickian diffusion and assume that all the parameters depend on the environment. Then we adopt the semigroup theory and the classical renewal process to compute the next generation operator R (x). The basic reproduction R 0 is the spectral radius of R (x). Furthermore, we clarify the relationship between R 0 and R (x) , and investigate asymptotical profile of the basic reproduction number R 0 associated with the diffusion rate d. In main part, we show that R 0 is a threshold value: the virus-free steady state E 0 is globally asymptotically stable if R 0 < 1 ; otherwise, the endemic steady state E * is globally attractive. Finally, we use backward Euler method to perform some numerical examples to illustrate the theoretical results. [ABSTRACT FROM AUTHOR]

• We study a Zika model with two transmission modes and comprehensive interventions. • We perform sensitivity test of the basic reproduction number on each control. • We formulate and solve the optimal control problem of the Zika model. • Using the cost effectiveness analysis, we obtain the most cost-effective strategy. Zika virus is a mosquito-borne virus and can spread to humans via bites by infected vectors or through sexual contact with infectious humans. Various protective and control measures such as the use of insecticide-treated nets, condoms, indoor residual spraying and medical treatment of infected humans may mitigate the infection. In this paper, we formulate and analyze a mathematical model that incorporates these comprehensive interventions to study the impact of mosquito-to-human and sexual transmission on the disease dynamics. We obtain the basic reproduction number, study when the disease dies out, and investigate the sensitivity of the basic reproduction number on each control. Numerical results suggest that vector transmission contributes more to the basic reproduction number than sexual transmission. We also formulate the optimal control problem and obtain the necessary conditions that minimize the population of infected individuals and the associated costs. Using the cost effectiveness analysis, we show that a combination of condom use and medical treatment of infected people provides the most cost-effective strategy to control the disease. [ABSTRACT FROM AUTHOR]

Abstract: This paper investigates an SIS model with seasonal changes in disease transmission. The model is formulated as a switched system and its long-term behavior is analyzed. A constant treatment strategy is then applied to a switched SIS model which takes media coverage into account. A pulse vaccination control strategy with vaccine failure is also considered. Throughout the paper, the stability of the disease-free solution is analyzed and threshold criteria involving the basic reproduction number are established which guarantee disease eradication. Sufficient conditions are also given for the endemic case. [Copyright &y& Elsevier]

Abstract: In this paper, we study the long-time behavior of a nonautonomous SEIRS epidemic model. We obtain new sufficient conditions for the permanence (uniform persistence) and extinction of infectious population of the model. By numerical examples we show that there are cases such that our results improve the previous results obtained in [T. Zhang, Z. Teng, On a nonautonomous SEIRS model in epidemiology, Bull. Math. Bio. 69 (2007) 2537–2559]. We discuss a relation between our results and open questions proposed in the paper. [Copyright &y& Elsevier]

Highlights • A spreading disease model is set for a population in cellular automata. • Individuals' movements create groups where the disease is transmitted. • The average group size is a good indicator for disease spreading strength. • Based on evolutionary graph theory, the temperature of the lattice is proposed. • Temperature is also a good indicator of the disease spreading strength. Abstract Epidemiological modeling usually relies on contact between two individuals as the method for a contagious infection spread over the population. Although for some diseases the contact is indeed between only two individuals (HIV), many contagious diseases are favored to spread inside a group of individuals. For instance, consider flu: usually infected persons spread the disease on public transport, inside a room in their jobs or their homes. All these situations are related to groups of individuals that may get a disease due to the presence of an infected individual in the group. Here we use a population model structured by groups, where individuals move inside a neighborhood forming groups, where the disease may be transmitted. The population model is based on evolutionary graph theory and modeled by probabilistic cellular automata, and the disease is modeled by the classical SIR-model. A mean-field approach is presented in terms of ordinary differential equations, and population parameters are used to analyze the disease dynamic. The major results of this paper are: (1) the number of groups that an individual belongs and the lattice temperature can estimate the disease spread strength inside the population and; (2) when infected individuals do not move, this is not enough to control the disease spreading. [ABSTRACT FROM AUTHOR]

Abstract Transmission of Zika virus (ZIKV) is a serious problem in public health, which can make the babies suffer from microcephaly if their mothers are infected by ZIKV during pregnancy. In this paper, we develop a model of ZIKV transmission in Colombia on complex networks which considers both sexual transmission among humans and the transmission by an infective vector in the process of propagation. We estimate the basic reproduction number R 0 and prove that the disease-free equilibrium is globally asymptotically stable when R 0 < 1. In addition, we study the effects of sexual transmission and the transmission route by an infective vector on the process of propagation. Invasion regions of ZIKV were shown in two-parameters space. The obtained results may provide new insights for the control of ZIKV. [ABSTRACT FROM AUTHOR]

In this paper, we consider a transmission model of hepatitis B virus by taking into account media coverage. First, we formulate the model and find the basic reproduction number R 0 by using the next generation matrix method. We show that the disease free equilibrium is locally asymptotically stable for R 0 < 1 and unstable if R 0 > 1 . We also prove that the system is globally asymptotically stable for R 0 < 1 . In order to control the spread of this disease in community, we devise an optimal control problem by introducing three control functions, that is, the educational campaign, vaccination and the media coverage. To do this, we solve analytically the control problem with characterization of control variables using the Portraying’s Maximum Principle. Finally, we present some numerical illustrations. [ABSTRACT FROM AUTHOR]

In this paper, we investigate an SI epidemic model with feedback controls in a patchy environment where individuals in each patch can disperse among n ( n  ≥ 2) patches. We derive the basic reproduction number R 0 and prove that the disease-free equilibrium is globally asymptotically stable if R 0  ≤ 1. In the case of R 0  > 1, we derive sufficient conditions under which the endemic equilibrium is unique and globally asymptotically stable. Our proof of global stability utilizes the method of global Lyapunov functions and results from graph theory. Numerical simulations are carried out to support our theoretical results. [ABSTRACT FROM AUTHOR]

This paper is devoted to developing a nonlocal and time-delayed reaction–diffusion model which makes contributions to understanding the mechanism of block effect on HCV by high-mobility group box 1 (HMGB1). Then basic reproduction number R 0 is established. The existence of HMGB1 can make R 0 < 1 , that is, can induce the extinction of virus. Further, threshold dynamics are investigated in terms of R 0 in a bounded spatial domain. From the biological perspective, threshold-type result can predict whether the disease will die out or persist. To investigate the invasion speed of the infectious disease, the existence of traveling waves is investigated in an unbounded spatial domain. Our results imply that the nonlocal time delay cannot change threshold dynamics, while it can slow down the wave speed of virus. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the global dynamics of a general SIRS epidemic model with a ratio-dependent incidence rate and its corresponding stochastic differential equation version. For the deterministic model, we show that the basic reproduction number R 0 determines whether there is an endemic outbreak or not: if R 0 < 1 , the disease dies out; while if R 0 > 1 , the disease persists. For the stochastic model, we show that its related reproduction number R 0 S can determine whether there is a unique disease-free stationary distribution or a unique endemic stationary distribution. In addition, we provide analytic results regarding the stochastic boundedness and permanence/extinction. One of the most interesting findings is that random fluctuations introduced in our stochastic model can suppress disease outbreak, which can provide us some useful control strategies to regulate disease dynamics. [ABSTRACT FROM AUTHOR]

Mobile devices are considerably pervasive in society, but also increase their vulnerability to worm attacks from mobile networks. In this paper, we propose a new Vulnerable-Exposed-Infectious-Quarantined-Secured worm propagation model with saturated incidence and strategies of both vaccination and quarantine. We obtain that the basic reproduction number R 0 is a sharp threshold parameter such that the worm-free equilibrium is asymptotically stable for R 0 ≤ 1, implying that the worm dies out eventually and its attack remains under control; the worm-existence equilibrium is asymptotically stable when R 0 > 1, namely, the worm is always persistent and spreading within a population. This paper provides some novel insights to cyber security by that (a) the stability of worm-free equilibrium establishes the control strategies to reduce the intensity of worm attacks, and the optimal control strategy is proposed by using Pontryagins Minimum Principle; (b) the stability of worm-existence equilibrium predicts the tendency of worm propagation in a long run and assesses the level of the worm popularity by the final scale of infected devices. Numerical simulations are implemented to illustrate the feasibility of the theoretical results and the effectiveness of the control strategies. [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]

Currently, more than one million children suffer from hand, foot and mouth disease (HFMD) every year in China. And there are no effective vaccines or antiviral drugs specifically targeted HFMD. In this paper, a two stage-structured model is constructed to fit the HFMD data from 2009 to 2014 in China and obtain its optimal parameter values by the Chi-square test of statistical inference. Using the parameter values obtained can precisely and reliably forecast the clinical cases. The basic reproduction numbers for each year are more than unit, which suggests that HFMD will persist in China under the current conditions. Sensitivity analysis of the basic reproduction number is conducted to evaluate the effectiveness of HFMD control measures. Using Pontryagins maximum principle, we derive the necessary conditions for optimal control of HFMD. Further, the impact of combinations of the strategies on HFMD transmission is investigated. Finally, we find out the most cost-effective strategy though carrying out cost-effectiveness analysis. [ABSTRACT FROM AUTHOR]

In this paper, an SEIR epidemiological model with saturation incidence and a time delay describing the latent period of the disease is investigated, where it is assumed that the susceptible population is subject to logistic growth in the absence of the disease. By analyzing the corresponding characteristic equations, the local stability of a disease-free equilibrium and an endemic equilibrium is discussed. The existence of Hopf bifurcations at the endemic equilibrium is established. By means of Lyapunov functionals and LaSalle’s invariance principle, it is proved that if the basic reproduction number is less than unity, the disease-free equilibrium is globally asymptotically stable and the disease dies out; if the basic reproduction number is greater than unity, sufficient conditions are obtained for the global stability of the endemic equilibrium. Numerical simulations are carried out to illustrate some theoretical results. [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]

In this paper, a model for the use of heroin with treat-age is formulated based on the principles of mathematical epidemiology. The model accounts for relapse rate that depends on how long the host has been in treatment for heroin addiction. An explicit formula for the reproductive number of the heroin spread is obtained. By using the method of Lyapunov functional, we established the dynamical properties of the heroin epidemic model, and the results show that the global dynamics of the model is completely determined by the basic reproduction number. It is shown that the drug-free equilibrium is locally and globally asymptotically stable if the basic reproduction number is less than one. In addition, the heroin spread system is uniform persistence and the unique drug spread equilibrium is locally and globally asymptotically stable if the basic reproduction number is greater than one. [ABSTRACT FROM AUTHOR]

In this paper, to study the spread of vector-borne diseases in human population, we build two coupled models for human population and vector population respectively on bipartite networks. By taking approximate expression for the density of infective vectors, we reduce the coupled models to a delayed SIS model describing the spread of diseases in human population. For the delayed dynamic model, we analyze its dynamic behavior. The basic reproduction number R 0 is given. And based on the Lyapunov–LaSalle invariance principle, we prove the global asymptotic stability of the disease-free equilibrium and the endemic equilibrium. Finally we carry out simulations to verify the conclusions and reveal the effect of the topology structure of networks and the time delay on the transmission process. Our results show that the basic reproduction number depends on the topology structure of bipartite networks and the time delay. It is also pointed out that the time delay can reduce the basic reproduction number. Furthermore, when the disease will disappear, the delay speeds up the disappearing process; when disease will become endemic, the delay slows the disease spreading down and reduces the density of infective humans. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the disease dynamics of a reaction–diffusion Feline Panleukopenia virus model incorporating direct and indirect transmissions as well as spatially environmental heterogeneity. We derive a basic reproduction number and show that the disease-free equilibrium will be locally stable if the basic reproduction number is below one, which means that the disease will be extinct. Furthermore, we perform some numerical simulations to explore the impact of the heterogeneous environment on the disease dynamics. [ABSTRACT FROM AUTHOR]

In this paper, we intend to understand the influences of the spatial heterogeneity, crowding effect and non-local infection caused by the movements of the latent mosquitoes on the dynamics of dengue transmission. For this purpose, we modify the homogeneous system provided in Esteva and Vargas (1998) to obtain a nonlocal and time-delayed reaction–diffusion system with the Neumann condition on the boundary. Then the basic reproduction number R 0 is defined for the model system, and it can be obtained explicitly when all model parameters are constants. Finally, we show that the global threshold dynamics of the model system can be determined by R 0 . [ABSTRACT FROM AUTHOR]

The aim of this paper is to study the effects of awareness and mosquito control on dengue epidemic. A non-linear host–vector model has been developed to study the dynamics of primary and secondary infection in the presence of two serotypes of dengue virus. The existence and local stability of equilibrium points have been discussed. The global stability of disease-free equilibrium point has been established. The thresholds for both the serotypes have been computed. The mosquito control and awareness parameters decrease the threshold and thereby enhancing the stability of disease-free state. [ABSTRACT FROM AUTHOR]

In this paper seasonal epidemiological processes are considered and a strategy of periodic vaccination is proposed. The invariant formulations associated with an N-periodic system and the reproduction numbers associated with them are considered. A collection of measures to study the stability of the system is introduced. Moreover, the collection of s-basic reproduction number at time j help us to establish conditions on the periodic vaccination rates in the vaccination program. Finally, an SIR model is showed and a comparison between the results obtained using constant or periodic vaccination program is analyzed. [ABSTRACT FROM AUTHOR]

Abstract: Brucellosis is one of the major public health problems in China, since there is a very large number of cattle and sheep. While the mixed feeding between cattle and sheep in public farm is one of the most typical characteristics in brucellosis outbreak regions, which has a large influence on the spreading of brucellosis. In this paper, we propose a multi-group SEIRV dynamical model with bidirectional mixed cross infection between cattle and sheep, and aim to investigate the influence of cross infection of mixed feeding on the brucellosis transmission. Firstly, we obtain the basic reproduction number , confirm the uniform existence and unique positive equilibrium of dynamic model when and prove the global stability of endemic equilibrium when . In addition, sensitivity analysis of the infectious herds with different system in term of some parameter values are given. It is found that if there exists the bidirectional of mixed cross infection between cattle and sheep, the disease cannot disappear even though and . These findings have important implication for brucellosis control: if we want to eliminate the brucellosis, firstly we should take measures to remove mixed cross infection, i.e., to prohibit mixed feeding. [Copyright &y& Elsevier]

Abstract: As drug treatment allows more and more people with HIV/AIDS to live longer, the trade-off between benefits to drug treatment and potential threat infectivity individuals needs to be carefully evaluated. In this paper, we shall extend and investigate an HIV/AIDS treatment model (Cai et al., 2009) [14]. The infection force of the extended model is assumed to be of density dependent form. The resulting incidence term contains, the bilinear and the standard incidence. Mathematical analysis establishes that the global dynamics of the HIV infectious disease is completely determined by the basic reproduction number . If , the disease always dies out and the disease-free equilibrium is globally stable. If , the disease persists and the unique endemic equilibrium is globally asymptotically stable in the interior of the feasible region. [Copyright &y& Elsevier]

Abstract: In this paper, a multi-group SEI model with the bilinear incidence and variable susceptibility and infectivity is analyzed. The susceptible individuals are divided into n groups according to their susceptibility, and infected individuals are divided into m groups according to infectivities. Each group of infected individuals has an exposed period. The basic reproduction number is obtained, and globally asymptotic stability of disease-free equilibrium is proved if . If , the disease-free equilibrium is unstable and the endemic equilibrium exists and is unique, which is globally asymptotically stable. Finally, some numerical examples are given to illustrate the above analysis. [Copyright &y& Elsevier]

• 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]

Abstract: In this paper, an SEIS epidemic model with a saturation incidence rate and a time delay describing the latent period of the disease is investigated. By analyzing the corresponding characteristic equations, the local stability of a disease-free equilibrium and an endemic equilibrium is discussed. It is shown that if the basic reproduction number is greater than unity, the disease is permanent. By comparison arguments, it is proved that if the basic reproduction number is less than unity, the disease-free equilibrium is globally asymptotically stable. Sufficient conditions are derived for the global asymptotic stability of the endemic equilibrium by means of an iteration scheme. [Copyright &y& Elsevier]

Abstract: In this paper, an epidemic model with nonlinear incidences is proposed to describe the dynamics of diseases spread by vectors (mosquitoes), such as malaria, yellow fever, dengue and so on. The constant human recruitment rate and exponential natural death, as well as vector population with asymptotically constant population, are incorporated into the model. The stability of the system is analyzed for the disease-free and endemic equilibria. The stability of the system can be controlled by the threshold number . It is shown that if is less than one, the disease free equilibrium is globally asymptotically stable and in such a case the endemic equilibrium does not exist; if is greater than one, then the disease persists and the unique endemic equilibrium is globally asymptotically stable. Our results imply that the threshold condition of the system provides important guidelines for accessing control of the vector diseases, and the spread of vector epidemic in an efficient way can be prevented. The contribution of the nonlinear saturating incidence to the basic reproduction number and the level of the endemic equilibrium are also analyzed, respectively. [Copyright &y& Elsevier]

Abstract: In this paper we develop a mathematical model to study the dynamics of visceral leishmaniasis in the Sudan. To develop this model we consider the dynamics of the disease between three different populations, human, reservoir and vector populations. The model is analyzed at equilibrium and the stability of the equilibria is analyzed. The basic reproduction number is derived, and the threshold conditions for disease elimination established. Results show that the disease can be eliminated under certain conditions. Simulations of the model show that human treatment helps in disease control, and its synergy with vector control will more likely result in the elimination of the disease. [Copyright &y& Elsevier]

Abstract: In an epidemiological model, time spent in one compartment is often modeled by a delay in the model. In general the presence of delay in differential equations can change the stability of an equilibrium to instability and causes the appearance of oscillatory solutions. In this paper we consider a SIS epidemiological model with demographic effects: birth, mortality and mortality caused by infection. The delay is the period of infection. We define the concept of oscillation in the sense that solutions of the model studied fluctuate around a steady state. Our goal is to show that in this model, there are oscillating solutions for certain parameters values. We determine a large set of initial data for which solutions of this model are slowly oscillating. [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]

Abstract: Human T-cell lymphotropic virus I (HTLV-I) infection is linked to the development of adult T-cell leukemia/lymphoma (ATL), among many illness. The healthy CD4+ T cells infect HTLV-I through cell-to-cell contact with infected T-cells. The infected T cells can remain latent and harbor virus for several years before virus production occurs. Actively infected T cells can infect other T cells and can convert to ATL cells, whose growth is assumed to follow a classical logistic growth function. In this paper, we consider the classical mathematical model with saturation response of the infection rate. By stability analysis we obtained the condition for the infected T cells die out and the condition for HTLV-I infection becomes chronic. At the same time, we also obtained the condition for a unique endemic equilibrium is globally stable in the interior of the feasible region. [Copyright &y& Elsevier]

This paper considers an SI1I2R epidemic model that incorporates two classes of infectious individuals with differential infectivity, and the incidence rate is nonlinear. The basic reproduction number R0 is found. If R0⩽1, the disease-free equilibrium is globally asymptotically stable and the disease always dies out eventually. If R0>1, a unique endemic equilibrium is locally asymptotically stable for general assumption. For a special case the global stability of the endemic equilibrium is proved. [Copyright &y& Elsevier]

• 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]

• A fully discretization scheme by the implicit Euler method for stochastic population models. • The preservation of the total population with a "suitable" numerical boundary condition. • The proposed numerical basic reproduction number R h through embedded infinite stochastic Leslie operators. • The preservation and detection of the analytic stability through numerical solutions for small stepsize. In this paper, a fully discretization scheme based on the implicit Euler method (IM) is considered for stochastic age-structured population models. The preservation of the total population with a suitable numerical boundary condition according to the biological meanings are shown. An explicit formula of the numerical basic reproductive number R h is proposed by the technique that the numerical process is embedded into an l 1 (R) -valued integrable stochastic process with infinite stochastic Leslie operators. Furthermore, the convergence and connection between R h and the stability of numerical solution is analyzed. The preservation and detection of the analytic stability through the numerical solutions are discussed for small stepsize. Finally, some numerical experiments including an infection-age model for modified SARS epidemic illustrate the verification and efficiency of our analysis. [ABSTRACT FROM AUTHOR]

• An SIS epidemiological model is considered with disease stage progression and varying infectivity. • Curiously, the stability of the endemic equilibrium depends on how many stages are included in the model. • A classical result on the stability of SIRS models is generalized. • It is pointed out that different discretizations of the infectious period may lead to different predictions. In this paper we consider multistage SIS models of infectious diseases, where infected individuals are passing through infectious stages I 1 , I 2 , ⋯ I n and then return to the susceptible compartment. First we calculate the basic reproduction number R 0 , and prove that the disease dies out for R 0 ≤ 1 , while a unique endemic equilibrium exists for R 0 > 1. Our main result is that the stability of the endemic equilibrium depends on the number of stages: the endemic equilibrium is always stable when n ≤ 3, while for any n > 3 it can be either stable or unstable, depending on the particular choice of the parameters. We generalize previous stability results for SIRS models as well and point out a mistake in the literature for multistage SEIRS models. Our results have important implications on the discretization of infectious periods with varying infectivity. [ABSTRACT FROM AUTHOR]

Recurrent malaria constitutes one of the greatest setbacks to the realization of malaria disease elimination from the population. In this paper, a deterministic model governed by a system of nonlinear differential equations is developed to assess the effects of recurrent malaria – reinfection and relapse on the transmission dynamics of the disease. The model is distributed into autonomous and non-autonomous systems. Analysis of the autonomous model shows that reinfection, which is the recurrence of malaria symptoms due to new parasites infection, has the potential to trigger the existence of two endemic equilibrium points when the basic reproduction number is below unity. Consequently without reinfection, global asymptotic dynamics of the autonomous model is established in the presence of relapse with the aid of carefully constructed Lyapunov functions for both the disease-free and endemic equilibria. The non-autonomous model with time-dependent control strategies is analyzed using Pontryagin's Maximum Principle to find the optimal solutions to the malaria control problem. Cost-effectiveness analysis is conducted to buttress the results of the optimal control problem by using the average cost-effectiveness ratio (ACER) and incremental cost-effectiveness ratio (ICER) methods. Finally, numerical simulations are demonstrated to enhance the theoretical results. [ABSTRACT FROM AUTHOR]

The emergence of rumors is subjective and aggressive which will bring disturbance and misleading to people's lives. This paper is dedicated to comprehensively investigating the influence of forced silence function, time delay and network topology on rumor propagation in online social networks. Considering the network supervision such as forced silence and network topology, a novel delayed SIR(susceptible-propagating-recovery) epidemic-like rumor propagation model in both homogeneous and heterogeneous networks is proposed. By using the mean-field theory, we analyze the propagation dynamics of the model in detail. Firstly, we demonstrate the boundedness of the solutions, then we obtain the basic reproduction number and the existence of the rumor-prevailing equilibrium points. Secondly, the forward and backward bifurcation as well as the local and global stabilities of the equilibrium points in both homogeneous and heterogeneous networks are theoretically analyzed and revealed. Finally, the possible impact is analyzed through numerical simulations, and the factors affecting rumor propagation are theoretically analyzed, which prove the validity of the theoretical analysis. [ABSTRACT FROM AUTHOR]

Dengue is one of the most widely spread mosquito-borne viral diseases which is transmitted by mosquitoes of Aedes species. It is endemic in all states and union territories (UTs) of India. Dengue virus is one of the major cause of illness and death in the tropical and subtropical regions. There is no vaccine available which can prevent infection with dengue virus. The best way to reduce the transmission of this disease is to avoid mosquito bites. Treatment of dengue infected individuals is an important factor that contributes significantly in the reduction of transmission of this disease. But sometimes in the developing countries it is not always possible to give treatment to each infected individuals. That is why we include a treatment function in our model. In the present paper, a mathematical model for dengue virus transmission with treatment by considering logistic growth of mosquito is formulated and analyzed. It is assumed that the treatment rate is proportional to the number of infectives below the capacity and is constant when the number of infectives is greater than the capacity. We find that the system exhibits backward bifurcation if the capacity is small. When the basic reproduction number (R 0) is greater than one, the endemic equilibrium point exists and is locally asymptotically stable under some restriction on parameters. We estimate the parameter corresponding to transmission of dengue using real data from different states of India by the least square method. We also perform sensitivity analysis to identify the key parameters that influence the basic reproduction number and hence regulate the transmission dynamics of dengue. [ABSTRACT FROM AUTHOR]