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Purpose: The susceptible-infectious-susceptible (SIS) infectious disease models without spatial heterogeneity have limited applications, and the numerical simulation without considering models' global existence and uniqueness of classical solutions might converge to an impractical solution. This paper aims to develop a robust and reliable numerical approach to the SIS epidemic model with spatial heterogeneity, which characterizes the horizontal and vertical transmission of the disease. Design/methodology/approach: This study used stability analysis methods from nonlinear dynamics to evaluate the stability of SIS epidemic models. Additionally, the authors applied numerical solution methods from diffusion equations and heat conduction equations in fluid mechanics to infectious disease transmission models with spatial heterogeneity, which can guarantee a robustly stable and highly reliable numerical process. The findings revealed that this interdisciplinary approach not only provides a more comprehensive understanding of the propagation patterns of infectious diseases across various spatial environments but also offers new application directions in the fields of fluid mechanics and heat flow. The results of this study are highly significant for developing effective control strategies against infectious diseases while offering new ideas and methods for related fields of research. Findings: Through theoretical analysis and numerical simulation, the distribution of infected persons in heterogeneous environments is closely related to the location parameters. The finding is suitable for clinical use. Originality/value: The theoretical analysis of the stability theorem and the threshold dynamics guarantee robust stability and fast convergence of the numerical solution. It opens up a new window for a robust and reliable numerical study. [ABSTRACT FROM AUTHOR]

To capture the underlying dynamics of the COVID-19 pandemic, we develop a stochastic SEIABR compartmental model, where the concentration of coronaviruses in the environment is considered. This paper is the first attempt to introduce the Black-Karasinski process as the random fluctuations in the modeling of epidemic transmission, and it is shown that Black-Karasinski process is a both mathematically and biologically reasonable assumption compared with existing stochastic modeling methods. We first obtain two critical values R 0 S and R 0 E related to the basic reproduction number R 0 of deterministic system. It is theoretically proved that (i) if R 0 S > 1 , the stochastic model has a stationary distribution ℓ (·) , which implies the long-term persistence of COVID-19; (ii) the disease will go extinct exponentially when R 0 E < 1 ; (iii) R 0 S = R 0 E = R 0 if there is no environmental noise in COVID-19 transmission. Then, we study the local stability of the endemic equilibrium P * of deterministic system under R 0 > 1. By developing an important lemma for solving the relevant Fokker-Planck equation, an approximate expression of probability density function of the distribution ℓ (·) around P * is further derived. Finally, several numerical examples are performed to substantiate our theoretical results. It should be mentioned that the techniques and methods of analysis in this paper can be applied to other complex high-dimensional stochastic epidemic systems. [ABSTRACT FROM AUTHOR]

This paper deals with the non-linear Susceptible–Infected–Hospitalized–Recovered model with Holling type II incidence rate, treatment with saturated type functional response for the prevention and control of disease with limited healthcare facilities. The well-posedness of the model is ensured with the help of the non-negativity and boundedness of the solution of the system. The feasibility of the model with DFE (Disease-free equilibrium) and EE (endemic equilibrium) is analysed. The local and global stability are discussed with the help of the computed basic reproduction number R 0. At R 0 = 1 , we use the Centre manifold theory to analyse the transcritical bifurcation exhibited by the system. It is found that the disease is not eradicated even if R 0 < 1 due to the occurrence of backward bifurcation. The occurrence condition of Hopf bifurcation is obtained. The optimal control theory is used to analyse the effects of the minimum possible medical facilities, hospital beds, and awareness creation on the population dynamics. The Hamiltonian function is constructed with the extended optimal control model and solved by Pontryagin's maximum principle to get the minimum possible expenditure. Different types of control strategies are shown by numerical simulation. The sensitivity analysis is discussed with the help of a crucial parameter that depends on the reproduction number. Further, the model is simulated numerically to support the theoretical studies. This paper emphasizes the significance of treatment intensity, the total number of hospital bed available and their occupancy rate as vital parameters for prevention of disease prevalence. • A nonlinear model is proposed to study the spread and control of infectious diseases with limited healthcare facilities. • Due to backward bifurcation, R 0 < 1 is insufficient to eliminate the disease, and the system exhibits periodic oscillation due to the emergence of Hopf bifurcation. • Using Pontryagin's maximum principle, effects of the limited medical facilities, hospital beds, and awareness-building initiatives are examined to determine the minimum possible expenses. • Sensitivity analysis is used to determine the appropriate control parameter and cost-effectiveness analysis is used for finding most optimal strategy. [ABSTRACT FROM AUTHOR]

In this paper, a class of virus-to-cell HIV model with intracellular delays, saturated incidence and Logistic growth is proposed to characterize the interaction between two types HIV strains, i.e., wild-type and drug-resistant strains. First, a series of threshold criteria on the locally and globally asymptotic stability of (infection-free, dominant, coexistence) equilibria are discussed based on the basic reproduction number R 0. Furthermore, a detailed Hopf bifurcation analysis is performed on the coexistence equilibrium using two delays as bifurcation parameters. We find that the Hopf bifurcations induced by double-strains are evidently different and more complicated than that of single strain, the former switches from stability (periodic branches) to un-stability (chaos) more frequently and earlier than the latter since double-strains would yield more pairs of imaginary roots in the characteristic equations. Meanwhile, the total viral load of double-strains would be higher than that of single-strain as well. The emergence of drug resistance imposes either negative or positive influences on the survival of wild-type strain, which would further facilitate the transmission of HIV. [ABSTRACT FROM AUTHOR]

The spreading of HIV or HTLV-I among the cells has received the great attention in recent modelling study to explore the virus infection dynamics. The co-infection of HIV and HTLV-I with the effect of Cytotoxic T-lymphocytes (CTLs) immune response is also important from epidemiological point of view. To identify the co-infection scenario of HIV and HTLV-I with the CTLs effect we proposed in this paper a six compartmental ODE-model with uninfected, HIV-infected, HTLV-I infected CD4+T cells and free HIV virus particles with HIV specific CTLs and HTLV-I specific CTLs. The rates of infection of the cases are considered here saturated type and proliferation rate of uninfected and HIV infected CD4+ T-cells are of logistic terms. To establish the well-posedness of the model we have shown that the solution of the proposed model is non-negative and bounded. We obtain the basic reproduction number which is the maximum of the HIV-related reproduction and the HTLV-I related reproduction number. Along with the disease free equilibrium point the system contains other seven endemic equilibrium points containing infection by single disease or both. Analytically, we establish the local and global stability conditions of the equilibrium points and also we establish that the system experiences transcritical bifurcation by the generation of only HIV or HTLV-I infected endemic equilibrium point. Using numerical simulations, we validate the theoretical results and found two infection paths, one initiating with HIV and other with HTLV-I, both cases ultimately become co-infected. Finally, using the optimal control analysis we found the optimal policy for treatment using AVR, RTI & PI for HIV or AZT for HTLV-I control and lastly concluded by some recommendations. • A co-infected HIV & HTLV-I infection model in vivo. • Proliferation of CD4+ T-cells are taken in logistic form. • Incidence rate is taken as saturated type. • Verified competitive exclusive principle. • Spreading path has been studied using Flow-diagram. • Three types of control strategies are investigated. [ABSTRACT FROM AUTHOR]

In this paper, a stochastic HIV/AIDS epidemic model is presented to study the synthetic effect of ART (antiretroviral therapy) and PrEP (pre-exposure prophylaxis) treatments among MSM (men who have sex with men). Firstly, we give the global stability of disease-free equilibrium and the endemic equilibrium in terms of basic reproduction number R 0 for deterministic model. And then the existence of global positive solutions and the existence of unique ergodic stationary distribution under R 0 S > 1 for stochastic model are given. Further, the long-time stochastic dynamic of the model is investigated, including the criteria on the extinction and persistence in mean for the stochastic model. Finally, we give some numerical simulations to illustrate our theoretical results, and the sensitive analysis shows that ART (antiretroviral therapy) and PrEP (pre-exposure prophylaxis) treatments can effectively control the spread of AIDS among MSM population. [ABSTRACT FROM AUTHOR]

This paper aims to study an HIV model with age structure and latently in a spatially homogeneous environment. By applying the fixed point theorem, we obtain the existence of the global solution and the global attractor for the model. We also identify the explicit formula of the basic reproduction number by the mean of the Laplace transformation, and confirm that this number predicts whether the infection occurs or not. Through analyzing the root distribution of the characteristic equation, the local stability of the equilibrium is obtained. By appealing to the appropriate Lyapunov functionals, we further study the global stability of the equilibrium. Numerical simulations also validate our theoretical results. [ABSTRACT FROM AUTHOR]

Air pollution has a serious impact on the spread of respiratory infectious diseases. However, the concentration of air pollution is not the same in different places. In order to characterize the spatial heterogeneity and the diffusion of pollutants, this paper proposes a spatial heterogeneity model, where the transmission rate is closely related to air pollution. The dynamics are established in terms of the basic reproduction number R 0 : if R 0 ≤ 1 , the disease-free equilibrium is globally asymptotically stable; if R 0 > 1 , there exists at least one endemic equilibrium and the disease is uniformly persistent. Our findings suggest that spatial heterogeneity, increased pollutants emission, and limited movement of infected individuals will all contribute to the spread of the respiratory infectious diseases. Subsequently, the corresponding strategies are introduced into the model to control the prevalence of the diseases. Under the minimization of the objective function, the necessary condition for the optimal control is acquired with the help of Pontryagin's maximum principle. Numerical simulations are applied to validate the theoretical results of threshold dynamics and to demonstrate the influences of inflow and clearance rates of pollutants on threshold dynamics and optimal control. [ABSTRACT FROM AUTHOR]

Antiretroviral (ART) drugs have been used to treat HIV patients over last two decades. But, the daily use of these drugs in order to control HIV load in patients' body results different side effects and high cost over long period of time. Motivated by these, researchers have been trying to find other treatment strategy whose combination with antiretroviral treatments can control HIV without use of large quantities of drugs. Immunotherapy is one of such treatment which assist natural immune system of patients in order to get control over viral infection. Administration of immunotherapy in combination with antiretroviral drugs has been found safe and effective after different clinical trials on humans and other animals. In this paper, we propose a mathematical model to describe HIV dynamics under antiretroviral treatment and immunotherapy combination. We find that our model is biologically feasible and the findings of the clinical trials support the dynamics of our model. The stability analyses of HIV-free equilibrium point are performed and found the basic reproduction number as a threshold parameter. Also, we calculate a critical efficacy for overall antiretroviral treatment such that when the efficacy of overall treatment exceeds this value HIV will be completely removed. Also, we have found that critical drug efficacy for antiretroviral drugs decreases with increasing immunotherapy, which indicates combination of immunotherapy with antiretroviral treatments minimize the quantity of antiretroviral drugs for HIV eradication, hence minimize the side effects and cost of antiretroviral drugs. Numerical simulations are carried out to substantiate the analytical results and to illustrate different drug administration scenarios. Moreover, an optimal combination of drugs which can control HIV with lower costs and side effects, is obtained by solving an optimal control problem numerically. [ABSTRACT FROM AUTHOR]

In this paper, we present a delayed deterministic and stochastic S I R I C V models to investigate the effects of the white noise intensities and the waning immunity of vaccinated individuals in the evolution of the disease. For the deterministic S I R I C V model, the basic reproduction number R 0 and the equilibrium points are calculated. The local stability of equilibrium points is analyzed. Particularly, when R 0 < 1 the disease-free equilibrium is locally stable for any positive value of τ. Furthermore, when R 0 > 1 , the local stability and sufficient conditions to ensure the occurrence of Hopf bifurcation for the endemic equilibrium point are established by considering the time delay τ as a bifurcation parameter. For the stochastic S I R I C V model, the conditions of the extinction and persistence of the disease are given by using the stochastic basic reproduction numbers R 0 s and R 0 s ∗. Numerical simulations are presented to enhance our analytical results and contrast the deterministic and stochastic models. [ABSTRACT FROM AUTHOR]

In this paper, we discuss the numerical representation of the linearly implicit Euler method for an age-structured HIV infection model with a general transmission mechanism. We first define the basic reproduction number of the continuous model, and present the stability results of the equilibriums. For the numerical process, we establish the solvability of the system and the non-negativity and convergence of numerical solutions. In the analysis of the long-term dynamical behavior, this paper mainly focus on the existence of the infection equilibrium determined by the numerical reproduction number R 0 Δ t. To overcome the difficulty caused by the complexity of epidemic transmission mechanisms, the 1-order convergence analysis of numerical basic reproduction numbers R 0 Δ t is implemented by using the properties of the fundamental solution matrix. By a comparison principle, we show that the disease-free equilibrium is globally asymptotically stable if R 0 Δ t < 1. Moreover, for R 0 Δ t > 1 , a unique numerical endemic equilibrium exists, which converges to the exact one, is locally asymptotically stable. Hence, numerical processes visually represent the dynamic properties of nonlinear age-structured HIV models. Finally, some numerical experiments demonstrate the verification and the efficiency of our results. • The age-structured HIV model with general transmission is reviewed. • The exact basic reproduction number is recalled. • The linearly implicit Euler method is implemented to the model. • The theoretical and numerical threshold dynamics are investigated. • The convergence of the basic reproduction numbers is proved for general case. [ABSTRACT FROM AUTHOR]

In this paper, an age-structured infectious disease dynamical model that considers two diseases simultaneously but with limited medical resources is proposed and analyzed. The asymptotic smoothness and persistence of the solution semi-flow are investigated. Then conditions for the existence of a global attractor are derived, which means that disease persists when ℜ 0 > 1. By using a Lyapunov function, it is shown that the infection-free equilibrium is globally asymptotically stable if ℜ 0 < 1 and the infection equilibrium is globally asymptotically stable if ℜ 0 > 1. In the presence of limited medical resources, the results suggest that equitable distribution for the limited medical resources is significant when treating low-risk and high-risk diseases and that keeping a resource sharing coefficient at a moderate level helps to eliminate the disease. • We proposed an age-structured infectious disease model with limited medical resources. • some important parameters were included in ℜ 0. • ℜ 0 determines the global stability of the equilibria. • Discussions of the effects of the parameters and biological significance were provided. [ABSTRACT FROM AUTHOR]

In this paper, a novel COVID-19 reaction–diffusion model with virus mutation and media coverage is investigated. First, the solution's uniform boundedness for the system is established. Then, the basic reproduction numbers for ordinary and mutant viruses spread in heterogeneous environments are defined. Furthermore, the endemic equilibrium's asymptotic distribution for the system is explored. In addition, when one diffusion coefficient tends to zero and the other diffusion coefficients are greater than zero and fixed, the solution of the system will asymptotically approach endemic equilibrium. Next, a theoretical analysis of how high-frequency media coverage affects the development of the COVID-19 epidemic is conducted. Theoretical research shows that high-frequency media coverage will lead to the disappearance of the disease. Meantime, global sensitivity analysis on the basic reproduction numbers R 01 and R 02 are performed. Finally, theoretical simulations and instance predictions are carried out. Because of the complexity of the Shanghai epidemic and changes in management and control, the infection rates β 1 (t) , β 2 (t) are given in the form of a piecewise function with more practical significance, and they are used to predict the epidemic trend of COVID-19 in Shanghai. Through a series of numerical simulations and analysis, the key indicators of the Shanghai COVID-19 epidemic are as follows : (1) The basic reproduction numbers in the early, middle, and late stages of COVID-19 are R ¯ 0 (1 : 34) = 0. 9152 , R ¯ 0 (35 : 49) = 3. 1476 , and R ¯ 0 (50 : 140) = 0. 6547 , respectively; (2) This epidemic round in Shanghai will peak at 3,270 new daily confirmed cases on the 49th day (April 15); (3) The final size of the epidemic will reach 63,470 confirmed cases; (4) This round of COVID-19 epidemic in Shanghai, China, is expected to be fully cleared in late June to early July. The above conclusions are basically consistent with the facts. Of course, with the rise in temperature and strict control, the epidemic situation in Shanghai, China, is expected to be cleared earlier. Our results provide new ideas for preventing and controlling the COVID-19 epidemic. • A reaction–diffusion model with virus mutation and media coverage is proposed. • The asymptotic properties of endemic equilibrium with small diffusion are explored. • Global sensitivity analysis for R 01 and R 02 and are performed. • Applying piecewise functions β 1 (t) , β 2 (t) , to predict the Shanghai COVID-19 epidemic. • Key indicators of COVID-19, such as R 0 , peak, final scale, clear time are obtained. [ABSTRACT FROM AUTHOR]

In this paper, an epidemic model with homogeneous Dirichlet boundary condition is formulated to study the joint impact of spatial diffusion, infection age and treatment adherence on the HIV/AIDS transmission among humans. It is an interesting problem to understand the threshold dynamics of HIV/AIDS model with the above three factors. Since the complexity of the model and specificity of the boundary condition, there are two main mathematical challenges: i). the compactness of the solution map is not guaranteed; ii). the explicit expression of the basic reproduction cannot be given even if the parameters are spatially independent. We first discuss the well-posedness of the system, then we identify the basic reproduction number R 0 as the spectral radius of the next generation operator, followed by the global attractivity of disease-free steady state when R 0 < 1 , the uniform persistence of the disease and the existence of the endemic steady state when R 0 > 1. Numerical simulations are carried out to illustrate our theoretical results, which suggest that the diffusion of individuals has an opposite effect on the disease outbreaks, and improving the treatment compliance of HIV infected individuals can control HIV transmission among the population effectively. [ABSTRACT FROM AUTHOR]

In this paper, the dynamical behavior in a spatially heterogeneous reaction–diffusion SIS epidemic model with general nonlinear incidence and Dirichlet boundary condition is investigated. The well-posedness of solutions, including the global existence, nonnegativity, ultimate boundedness, as well as the existence of compact global attractor, are first established, then the basic reproduction number R 0 is calculated by defining the next generation operator. Secondly, the threshold dynamics of the model with respect to R 0 are studied. That is, when R 0 < 1 the disease-free steady state is globally asymptotically stable, and when R 0 > 1 the model is uniformly persistent and admits one positive steady state, and under some additional conditions the uniqueness of positive steady state is obtained. Furthermore, some interesting properties of R 0 are established, including the calculating formula of R 0 , the asymptotic profiles of R 0 with respect to diffusion rate d I , and the monotonicity of R 0 with diffusion rate d I and domain Ω. In addition, the bang–bang-type configuration optimization of R 0 also is obtained. This rare result in diffusive equation reveals that we can control disease diffusion at least at one peak. Finally, the numerical examples and simulations are carried out to illustrate the rationality of open problems proposed in this paper, and explore the influence of spatial heterogeneous environment on the disease spread and make a comparison on dynamics between Dirichlet boundary condition and Neumann boundary condition. [ABSTRACT FROM AUTHOR]

This paper introduces a deterministic fractional-order epidemic model (FOEM) for studying the transmission dynamics of the human immunodeficiency virus (HIV) and acquired immunodeficiency syndrome (AIDS). The model highlights the substantial role of unaware and undetected HIV-infected individuals in spreading the disease. Control strategies, such as wielding condoms, level of preventive measures to avoid infection, and self-strictness of susceptibles in sexual contact, have been incorporated into the study. The basic reproduction number ℛ 0 α has been derived, which suggests the conditions for ensuring the persistence and elimination of the disease. Further, to validate the model, actual HIV data taken from Mexico and India separately have been used. The disease dynamics and its control in both countries are analyzed broadly. The values of biological parameters are estimated at which numerical solutions better match the actual data of HIV patients in the case of fractional-order (FO) instead of integer-order (IO). Moreover, in the light of ℛ 0 α , our findings forecast that the disease will abide in the population in Mexico, and at the same time, it will die out from India after a long time. [ABSTRACT FROM AUTHOR]

In this paper, a three tier mathematical model describing the interactions between susceptible population, Covid-19 infected, diabetic population and Covid-19 infected, non diabetic population is proposed. Basic properties of such a dynamic model, namely, non negativity, boundedness of solutions, existence of disease-free and disease equilibria are studied and sufficient conditions are obtained. Basic reproduction number for the system is derived. Sufficient conditions on functionals and parameters of the system are obtained for the local as well as global stability of equilibria, thus, establishing the conditions for eventual prevalence of disease free or disease environment, as the case may be. The stability aspects are discussed in the context of basic reproduction number and vice versa. An important contribution of this article is that a novel technique is presented to estimate some key, influencing parameters of the system so that a pre-specified, assumed equilibrium state is approached eventually. This enables the society to prepare itself with the help of these key, influencing parameters so estimated. Several examples are provided to illustrate the results established and simulations are provided to visualize the examples. [ABSTRACT FROM AUTHOR]

This paper aims to explore stability and bifurcations with control analysis of a co-infected two serotypes dengue model in presence of three controls, namely protection control, treatment control and mosquito killing efforts. First we analyze the model incorporating with constant controls then find the control path considering variable control. We examine biological feasibility of the considered model along with existence and stability criteria of different equilibrium with respect to basic reproduction number. Stability of serotype-I, serotype-II free equilibrium and positive co-existence equilibrium are investigated. Center manifold theorem is used to prove the stability of the co-infected endemic equilibrium. The model experiences Transcritical bifurcation with respect to basic reproduction number. Sensitivity analysis has been employed to identify most influential model parameters to control the infection. The time dependent optimal control problem is solved analytically and numerically using Pontryagin's maximum principle. At last efficiency analysis has been carried out to find out more suitable control to combat the dengue disease. It is established from the analysis, protection control with treatment is more powerful than mosquito killing efforts by humans with treatment for controlling dengue. [ABSTRACT FROM AUTHOR]

Adaptive behavior on networks leads to bifurcation and other special phenomenon, which has been proved for differential systems. In this paper, one will discuss the stability, bifurcation and chaos of the discrete pair epidemic model on adaptive networks, which will bring some new challenges. The stability of the disease free equilibrium and endemic equilibrium with respect to basic reproduction number R 0 is studied. Under certain conditions, as the time step-size increases, flip bifurcation that occurs at the endemic equilibrium is discussed, the period-doubling bifurcation and the chaos phenomenon of the system are exhibited, some figures are provided for illustration. According to the results in this paper, the dynamical behaviors of the discrete model are multitudinous and quite different from the continuous model. [ABSTRACT FROM AUTHOR]

Rumors do the social's perception seal, leaving people untouched with the true thing. Since the harmfulness of rumors is well known, in order to cut off the net of rumors and make the society run in order, it is necessary to have a deeper understanding of the spread of rumors. In this paper, the non-smooth reaction–diffusion system with considering the encouraging effect of secondary propagation of Internet platform on rumor propagation is used to study the dynamic system of rumor propagation. Firstly, the existence of the non-negative solution is proved by using the theory of upper and lower solutions for mixed monotone system. Secondly, the value of the basic reproduction number is calculated and the existence of positive equilibrium are discussed. Thirdly, the backward bifurcation and the saddle-node bifurcation are investigated. Fourthly, the stability of rumor propagation equilibrium and Hopf bifurcation are analyzed theoretically. Finally, some numerical simulations which proves the validity of the above theory are provided. [ABSTRACT FROM AUTHOR]

In this paper, a mathematical model with a standard incidence rate is proposed to assess the role of media such as facebook, television, radio and tweeter in the mitigation of the outbreak of COVID-19. The basic reproduction number R 0 which is the threshold dynamics parameter between the disappearance and the persistence of the disease has been calculated. And, it is obvious to see that it varies directly to the number of hospitalized people, asymptomatic, symptomatic carriers and the impact of media coverage. The local and the global stabilities of the model have also been investigated by using the Routh–Hurwitz criterion and the Lyapunov's functional technique, respectively. Furthermore, we have performed a local sensitivity analysis to assess the impact of any variation in each one of the model parameter on the threshold R 0 and the course of the disease accordingly. We have also computed the approximative rate at which herd immunity will occur when any control measure is implemented. To finish, we have presented some numerical simulation results by using some available data from the literature to corroborate our theoretical findings. [ABSTRACT FROM AUTHOR]

In this paper, we proposed the stochastic perturbation to study the HIV viral dynamical model. The stochastic epidemic model of 2-LTR HIV infected patients is considered and we prove that the stochastic model admits a unique globally positive solution7 which is bounded and permanent. Then we analyze the necessary conditions for extinction and persistence of the disease by selecting the appropriate Lyapunov functions. The theoretical findings are confirmed by using suitable numerical experiments. [ABSTRACT FROM AUTHOR]

In this paper, the global dynamical behavior of non-linear infection-age SIR models is investigated visually by numerical processes. Firstly, for an age-independent SIR model with diseased mortality and isolation rates, it has been shown that the linearly implicit Euler method reproduces the local dynamical behavior. For the study of the numerical global stability, numerical removed individuals are introduced and then the global behavior is represented by numerical processes for any stepsize. Generally, the numerical global behavior of the linearly implicit Euler method applied to non-linear infection-age SIR model is investigated. Using an infinite-dimensional Leslie operator, the disease-free equilibrium point is globally asymptotically stable for numerical processes when the numerical basic reproduction number of R h < 1. The numerical endemic disease equilibriums are globally asymptotically stable when the numerical threshold R h > 1. It is much more important that the global behavior of the analytical solutions is visually displayed by the numerical processes for the sufficiently small stepsize h. Finally, numerical applications to some the influenza models including HIV models and SIR models with protections are shown to illustrate our analysis. [ABSTRACT FROM AUTHOR]

In this paper, we are concerned with the mathematical analysis of a host–pathogen model with diffusion, hyperinfectivity and nonlinear incidence. We define the basic reproduction number ℜ 0 by the spectral radius of the next generation operator, and study the relation between ℜ 0 and the principal eigenvalue of the problem linearized at the disease-free steady state (DFSS). Under some assumptions, we show the threshold property of ℜ 0 : if ℜ 0 < 1 , then the DFSS is globally asymptotically stable (GAS), whereas if ℜ 0 > 1 , then the system is uniformly persistent and a positive steady state (PSS) exists. Moreover, for the special case where all parameters are constants, we show that the PSS is GAS for ℜ 0 > 1. Numerical simulation suggests that the spatial heterogeneity could enhance the intensity of epidemic, whereas the diffusion effect could reduce it. [ABSTRACT FROM AUTHOR]

Understanding the role of age structure and the spatial heterogeneity on disease spreading and vanishing is a vital question in the transmission of diseases. In this paper we construct a reaction–diffusion vector-borne disease model on a bounded domain subject to the no-flux boundary condition, with two novel features: age-space structure and multiple transmission routes. The contribution of mathematical analysis in this paper is two-fold: (i) in a bounded domain, by using the Banach fixed point theory to solve the existence of the local solution, and then guaranteeing the existence of the global solution through the eventual boundedness, and (ii) in a homogeneous case, we establish the formula of the basic reproduction number, denoted by ℜ 0 , predicting whether the disease persist or not. In particular, by analyzing the characteristic equation and constructing new Lyapunov functionals, we achieve that the local and global stability of constant equilibria strictly depends on the sign of ℜ 0 − 1. Our threshold results were verified by numerical simulations in one and two-dimensional bounded domain. This paper also provides a framework of theoretical analysis in dealing with age-space structure disease model. • We propose a novel age-space structured vector-borne model. • We find the relationship between the next generator operator and the basic reproduction number. • We prove that the local and global stability of constant equilibria based on the basic reproduction number. [ABSTRACT FROM AUTHOR]

In this paper, we examine a stochastic vegetation-water model, where the Black–Karasinski process is introduced to characterize the random fluctuations in vegetation evolution. It turns out that Black–Karasinski process is a both mathematically and biologically reasonable assumption by comparison with existing stochastic modeling approaches. First, it is theoretically proved that the solution of the stochastic model is unique and global. Then two critical values ℛ 0 E and ℛ 0 S are obtained to classify the dynamical behavior of vegetation. It is shown that: (i) If ℛ 0 S > 1 , the stochastic model has a stationary distribution ℓ (⋅) , which reflects the long-term coexitence of vegetation and the water environment. (ii) The vegetation will go extinct exponentially if ℛ 0 E < 1. (iii) ℛ 0 E = ℛ 0 S = ℛ 0 if there are no random noises in vegetation dynamics, where ℛ 0 is the basic reproduction number of deterministic model. Furthermore, by solving the associated Fokker–Planck equation, the approximate expression for probability density function of the distribution ℓ (⋅) around a quasi-positive equilibrium is studied. Finally, several numerical examples are provided to support our theoretical findings. • We develop a stochastic vegetation-water model, where the surface water and the soil water are considered. • This paper is the first mathematical attempt to introduce the Black–Karasinski process to characterize the random fluctuations in vegetation dynamics. • We prove that the stochastic system has a stationary distribution if R0S > 1. • We obtain that the vegetation will go extinct exponentially if R0E < 1. • An approximate expression of a local density function of the stationary distribution is derived. • Our techniques and theoretical methods used can be successfully applied to many complex ecological models perturbed by Black–Karasinski process. [ABSTRACT FROM AUTHOR]

• A novel reaction-diffusion COVID-19 model with direct and aerosol transmission is proposed. • The basic reproduction number R 0 of spatial diffusion system is obtain and confirm that R 0 is the threshold for predicting whether COVID-19 is prevalent. • Lyapunov methods and some novel differential inequality techniques for PDE. • Develop a new numerical simulation method to accurately fit the development of COVID-19 in Putian and Yangzhou, China, respectively. The COVID-19 epidemic has infected millions of people and cast a shadow over the global economic recovery. To explore the epidemic's transmission law and provide theoretical guidance for epidemic prevention and control. In this paper, we investigate a novel SEIR-A reaction-diffusion COVID-19 system with direct and aerosol transmission. First, the solution's positivity and boundedness for the system are discussed. Then, the system's the basic reproduction number is defined. Further, the uniform persistence of disease when R 0 > 1 is explored. In addition, the system equilibrium's global stability based on R 0 is demonstrated. Next, the system's NSFD scheme is investigated and the discrete system's positivity, boundedness, and global properties are studied. Meantime, global sensitivity analysis on threshold R 0 is investigated. Interestingly, the effects of three strategies, including vaccination, receiving treatment, and wearing a mask, are evaluated numerically. The results suggest that the above three strategies can effectively control the peak and final scale of infection and shorten the duration of the epidemic. Finally, theoretical simulations and instance predictions are used to give several key indicators of the epidemic, including threshold R 0 , peak, time to peak, time to clear cases, and final size. The instance prediction results are as follows: (1) The basic reproduction numbers of Yangzhou and Putian in China are R 0 = 2.5107 and R 0 = 1.8846 , respectively. (2) This epidemic round in Yangzhou will peak at 56 new daily confirmed cases on the 9th day (August 5), and Putian will peat at 37 new daily confirmed cases on the 6th day (September 15). (3) The final scale of infections in Yangzhou and Putian reached 570 and 205 cases, respectively. (4) The Yangzhou epidemic is expected to be completely cleared on the 25th day (August 21). In addition, the Putian epidemic will continue for 15 days and be cleared on September 24. The analysis results mean that we should improve our immunity by actively vaccinating, reducing the possibility of aerosol transmission by wearing masks. In particular, people should maintain proper social distance, and the government should strengthen medical investment and COVID-19 project research. [ABSTRACT FROM AUTHOR]

In this paper, a susceptible–exposed–infectious–recovered (SEIR) epidemic patch model with nonlinear incidence rate, vaccination and quarantine strategies is presented. According to the properties of nonnegative matrices, the range of basic reproduction number is obtained. Moreover, the stability of disease-free equilibrium is studied. Some numerical simulations are given to validate the theoretical results. The effects of vaccination, quarantine strategies and population migration on disease transmission are also investigated by numerical simulations. [ABSTRACT FROM AUTHOR]

Based on the distinctive spatial diffusion characteristics observed in syphilis transmission patterns, this paper introduces a novel reaction–diffusion model for syphilis disease dynamics, incorporating general incidence functions within a heterogeneous environment. We derive the basic reproduction number essential for threshold dynamics and investigate the uniform persistence of the model. We validate the model and estimate its parameters by employing the multi-objective Markov Chain Monte Carlo (MCMC) method, using real syphilis data from the years 2004 to 2018 in China. Furthermore, we explore the impact of spatial heterogeneity and intervention measures on syphilis transmission. Our findings reveal several key insights: (1) In addition to the original high-incidence areas of syphilis, Xinjiang, Guizhou, Hunan and Northeast China have also emerged as high-incidence regions for syphilis in China. (2) The latent syphilis cases represent the highest proportion of newly reported cases, highlighting the critical importance of considering their role in transmission dynamics to avoid underestimation of syphilis outbreaks. (3) Neglecting spatial heterogeneity results in an underestimation of disease prevalence and the number of syphilis-infected individuals, undermining effective disease prevention and control strategies. (4) The initial conditions have minimal impact on the long-term spatial distribution of syphilis-infected individuals in scenarios of varying diffusion rates. This study underscores the significance of spatial dynamics and intervention measures in assessing and managing syphilis transmission, which offers insights for public health policymakers. • Formulating a reaction-diffusion model to understand spatiotemporal syphilis dynamics • Validating the model using real syphilis data • Uncovering new high-incidence regions beyond the original hotspots • Emphasizing the significance of latent stage in syphilis transmission • Testing intervention measures for effective syphilis containment strategies [ABSTRACT FROM AUTHOR]

In this paper, a stochastic HBV infection model with virus-to-cell infection, cell-to-cell transmission and CTL immune response is proposed. The model has a general form of infection rate, in which the contact rate is governed by the log-normal Ornstein–Uhlenbeck process. First, it is proved that the stochastic model has a unique positive global solution. The dynamic properties of the solutions around the equilibrium points are also analysed. It further follows that the disease-free equilibrium is globally asymptotically stable if R 0 < 1 , while the endemic equilibrium is globally asymptotically stable if R 0 > 1. Then, we establish sufficient conditions for the stationary distribution and extinction of the model by constructing suitable Lyapunov functions, respectively. After that, we calculate the exact analytical expression for the probability density function of stationary distribution near the quasi-endemic equilibrium. Finally, the effect of the Ornstein–Uhlenbeck process on the dynamical behaviour of the model is verified by numerical simulations. One of the most interesting findings is that larger regression speeds and smaller volatility intensities can significantly reduce major outbreaks of HBV infection within the host, which may have important implications for future HBV therapeutic regimens. • A HBV infection model disturbed by Ornstein–Uhlenbeck process is presented. • Dynamic properties of the solutions around the equilibrium points are analysed. • Stationary distribution and extinction are studied. • The density function around the quasi-endemic equilibrium is discussed. • Comprehensive numerical simulations are represented. [ABSTRACT FROM AUTHOR]

In this paper, mathematical modeling of the dynamics of Human T-cell lymphotropic virus type I (HTLV-1) infection and the development of adult T-cell leukemia (ATL) cells is investigated by a time scale approach. The proposed models, constructed by nonlinear systems of first-order difference equations and h -difference equations, characterize the relationship among uninfected, latently infected, actively infected CD4 + cells, and ATL cells, where the growth of leukemia cells is described by discrete logistic curves. The stability results are established based on basic reproduction number ℛ 0. When ℛ 0 < 1 , infected T-cells always die out and there exist two disease-free equilibria depending on the proliferation rate and the death rate of leukemia cells. When ℛ 0 > 1 , HTLV-1 infection becomes chronic and spreads, and there exists a unique endemic equilibrium point. The stability results of disease-free and endemic equilibrium points are obtained when ℛ 0 < 1 and ℛ 0 > 1 , respectively. Furthermore, the sensitivity analysis discovers the key parameters of the models related to ℛ 0. Estimated parameters are applied based on the experimental observation. The numerical analysis also shows the equilibrium level of ATL cell proliferation is higher when the HTLV-I infection of T-cells is chronic than when it is acute. Moreover, our mathematical modeling by a time scale approach yields a new parameter to an HTLV-1 infection model which determines data frequency. • ATL development associated with HTLV-1 infection. • Mathematical modeling by a time scale approach. • The number of leukemia cells described by implicit discrete logistic curve. • Determining data frequency by h-difference equations. [ABSTRACT FROM AUTHOR]

In addition to the traditional transmission route via the biting-and-defecating process, non-human host predation of triatomines is recognized as another significant avenue for Chagas disease transmission. In this paper, we develop an eco-epidemiological model to investigate the impact of predation on the disease's spread. Two critical thresholds, R v p (the basic reproduction number of triatomines) and R 0 p (the basic reproduction number of the Chagas parasite), are derived to delineate the model's dynamics. Through the construction of appropriate Lyapunov functions and the application of the Bendixson–Dulac theorem, the global asymptotic stabilities of the equilibria are fully established. The vector-free equilibrium E 0 is globally stable when R v p < 1. E 1 , the disease-free equilibrium, is globally stable when R v p > 1 and R 0 p < 1 , while the endemic equilibrium E ∗ is globally stable when both R v p > 1 and R 0 p > 1. Numerical simulations highlight that the degree of host predation on triatomines, influenced by non-human hosts activities, can variably increase or decrease the Chagas disease transmission risk. Specifically, low or high levels of host predation can reduce R 0 p to below unity, while intermediate levels may increase the infected host populations, albeit with a reduction in R 0 p. These findings highlight the role played by non-human hosts and offer crucial insights for the prevention and control of Chagas disease. • A novel model integrating systemic and non-human predation transmission is developed. • Two thresholds of R v p and R 0 p are derived to delineate the model dynamics. • The global stabilities of the equilibria for the high-dimensional model are shown. • The work offers the crucial insight for the prevention and control of Chagas disease. [ABSTRACT FROM AUTHOR]

Researchers at the Harbin Institute of Technology in China have conducted a study on the numerical analysis of an age-structured HIV infection model with a general transmission mechanism. The study focuses on the numerical representation of the linearly implicit Euler method and the stability of equilibriums. The researchers also analyze the long-term dynamical behavior of the model and demonstrate the verification and efficiency of their results through numerical experiments. This research provides insights into the dynamic properties of nonlinear age-structured HIV models. [Extracted from the article]

The dynamics of COVID-19 pandemic varies across countries and it is important for​ researchers to study different kind of phenomena observed at different stages of the waves during the epidemic period. Our interest in this paper is not to model what happened during the endemic state but during the epidemic state. We proposed a continuous formulation of a unique maximum reproduction number estimate with an assumption that the epidemic curve is in form of the Gaussian curve and then compare the model with the discrete form and the observed basic reproduction number during the contagiousness period considered. Furthermore, we estimated the transmission rate from identification of the first inflection point of a wave of the curve of daily new infectious cases using the Bernoulli S–I (Susceptible–Infected) equation. We applied this new method to the real data from Cameroon COVID-19 outbreak both at national and regional levels. High correlation was observed between the socio-economic parameters and epidemiology parameters at regional level in Cameroon. Also, the method was applied to the second wave COVID-19 outbreak for the world data which is a period the phenomena we are considering were observed. Lastly, it was observed that the models presented results correspond with the epidemic dynamics in Cameroon and World data. We recommend that it is important to study what happened during the growth inflection point as some countries data did not climax. [ABSTRACT FROM AUTHOR]

In this paper, we perform a complete analysis on a degenerated reaction–diffusion cholera model with stabilizing total humans and non-mobility of cholera bacteria in a spatially heterogeneous bounded domain. The existence of a global attractor is established through introducing the Kuratowski measure of non-compactness. The basic reproduction number and its equivalent characterizations have been used to analyze the threshold-type results, where the persistence and extinction of cholera can also be characterized by the dispersal rate of infected humans. In the homogeneous case, the global attractivity of the unique positive equilibrium is achieved by the Lyapunov function. Moreover, we compare the basic reproduction numbers for the models without and with considering the mobility of cholera bacteria. Our results suggest that: the basic reproduction numbers attain different values as the dispersal rates of infected humans and cholera approach to infinity, while they attain the same value as the dispersal rates of infected humans and cholera approach to zero. Numerical simulations support our analytical results and discuss the impact of the dispersal rate of infected humans on the transmission of cholera. [ABSTRACT FROM AUTHOR]

• Under the multi-lingual environment, a new SIR rumor spreading model, which divides the population with different languages into distinct groups, is first proposed in this paper. Meanwhile, the dynamical behaviors are systematically studied based on the Lyapunov function, graph-theoretic approach and uniform persistence theory. • To further illustrate the influence of model parameters on the rumor spreading, we evaluate the sensitivity index of the basic reproduction number and analyze how crucial each parameter is to the rumor transmission and prevalence. • Since the rumor assumed in this paper can cross-transmit between different groups, the results obtained here are more general and practical compared with the existing papers [11,12,14,16]. The dynamical behaviors of the multi-lingual SIR rumor spreading model are investigated in this paper. In view of the actual background, we consider the rumor spreading under the multi-lingual environment, and further establish a new model with cross-transmitted mechanism. Employing the theory of infectious diseases, both the basic reproduction number and the stability of the rumor-free equilibrium are systematically discussed. In addition, the global asymptotic stability of endemic equilibrium of the proposed model is analyzed via applying Lyapunov functions and graph theories. In addition, we perform sensitivity analysis on threshold conditions to determine the relative importance of model parameters to rumor transmission. Finally, the effectiveness of theoretical results is illustrated through the numerical examples. [ABSTRACT FROM AUTHOR]

Infectious disease transmission, mainly affected by media coverage and stochastic perturbations, has imposed great social financial burden on the community in the past few decades and even threatened public health. However, there are few studies devoted to the theoretical dynamics of epidemic models with media coverage and biologically reasonable stochastic effect yet. In this sense, this paper mainly formulates and studies a stochastic epidemic model with media coverage and two mean-reverting Ornstein–Uhlenbeck processes. We first illustrate the biological implication and mathematically reasonable assumption of introducing the Ornstein–Uhlenbeck process as stochastic effect. It is theoretically proved that the solution to the stochastic model is unique and global, as well as the existence of an ergodic stationary distribution. After that, by solving the corresponding Fokker–Planck equation and using our developed algebraic equation theory, it is derived that the above global solution around the endemic equilibrium follows a unique probability density function. For completeness, the sufficient criteria for extinction exponentially of the model are established. Finally, several numerical simulations are provided to verify our theoretical results. Besides, the impact of stochastic noises and media coverage on epidemic transmission is studied by comparison with the previous results of a deterministic model. [ABSTRACT FROM AUTHOR]

It is well known that the mathematical biology and dynamical systems give very important information for the study and research of viral infection models such as HIV, HBV, HCV, Ebola and Influenza. This paper deals with the global dynamics of generalized virus model with logistic growth rate for target cells, general incidence rate and cellular immunity. The results will be obtained by using Lyapunov's second method and LaSalle's invariance principle. We prove the global stability of the rest points of the system by the value of basic reproduction number (R 0) and the immune response reproduction number (R CTL). We have found that if R 0 < 1 , then the infection-free equilibrium is globally asymptotically stable. For R 0 > 1 and R CTL < 1 , under certain conditions on incidence rate function, immune-free equilibrium is globally asymptotically stable. Finally, we prove that if R 0 > 1 and R CTL > 1 , then under certain conditions on incidence rate function the endemic equilibrium is globally asymptotically stable. Since the logistic growth rate for target cells and general incidence rate have been included in this manuscript, our obtained results are the generalization of those in the previous literatures. Moreover, the results have been obtained with weaker assumptions in comparison with the previous ones. Numerical simulations are presented to support and illustrate our analytical results. [ABSTRACT FROM AUTHOR]

Travel between patches and behavior changes of susceptible individuals are two crucial factors affecting the disease dynamics for any emerging infectious disease. Therefore studies of their impacts on disease dynamics are of great importance, especially to public health policy-makers. In this paper, we propose an SIR model that incorporates these two factors into consideration. For our two-patch SIR model with infection force mediated by (past) disease surveillance, we identify the basic reproduction number and show how it is related to travel modified basic reproduction numbers. We also carry out stability and Hopf bifurcation analysis and obtain conditions under which the disease persists in one patch or both patches. Numerical simulations are carried out to illustrate how travel between two patches and disease surveillance mediated infection force affect the disease dynamics and thus provide us some helpful insights on proper control of emerging infectious diseases. • Build a two-patch SIR model with infection force mediated by (past) disease surveillance. • Identify the overall basic reproduction number and related it with several travel modified basic reproduction numbers (See Table 1). • Find that the travel of susceptible individuals alone can cause an endemic even if the disease dies out in each isolated patch. • Study the effects of the past disease surveillance on the spread of the disease. [ABSTRACT FROM AUTHOR]

The outbreak of crown-of-thorns starfish (CoTS) is one of the most critical biological disturbances in coral reef systems and contributes greatly to the degradation of coral. Although many studies have focused on coral-CoTS interactions, there is no effective method for suppressing CoTS outbreaks in the long term, in part because of incomplete knowledge of their life cycle. In this paper, a stochastic system is formulated to characterize the interactions among coral, immature CoTS, and mature CoTS, where environmental noise is introduced into the growth of coral. The existence and uniqueness of a global positive solution with any given initial value are investigated, sufficient conditions are established for the extinction of CoTS, and the existence and uniqueness of a stable stationary distribution are explored by applying the Markov semigroup theory. In particular, by solving the Fokker–Planck equation, the approximate expression of the probability density function of the distribution around its quasi positive equilibrium is expounded under certain parametric restrictions. By employing Milstein's higher-order method, numerical simulations are presented to validate the theoretical results. This study reveals that coral are more vulnerable and are more likely to go extinct in the presence of strong noise, while weak noise is more conducive to the existence of a stable stationary distribution, which implies the long-term persistence of coral and CoTS. • A new coral-CoTS interaction model with environmental noise for coral and with stage-structure for CoTS is proposed and is well explored. • The basic reproduction number is formulated to establish threshold criteria for the extinction of CoTS and the existence of a stable stationary distribution. • The approximate expression of probability density function is explicitly derived for the stationary distribution around its quasi-positive equilibrium. • The stage structure of CoTS has a significant impact on the capability of CoTS to survive and to become persistent. • The ability for coral to adapt to the external environmental fluctuations is limited and the strong environmental noise is devastating to coral reef system. [ABSTRACT FROM AUTHOR]

• An SIVS epidemic model with two degree-dependent vaccination rates on scale-free networks is developed. • Spatial immunity and time-varying optimal vaccination schemes are performed. • Effects of primary and booster vaccinations on spread of infectious diseases are addressed. • Effects of different weights on optimal controls are discussed. Understanding effectiveness of control policies can be helpful to prevent and control diseases. In this paper, we evaluate effects of control strategies containing two vaccinations, which are given to the newly born and the susceptible individuals. An SIVS epidemic model with two degree-dependent vaccination rates on scale-free networks is developed. We argue the effects of the vaccinations on the basic reproduction number R 0. And spatial immunity and time-varying optimal vaccination schemes are performed. Particularly, we discuss effects of different weights in the objective function on disease control. We find that, two vaccinations can give rise to decreasing of R 0 ; especially for R 0 > 1 , the effect of the booster vaccination is larger than that of the primary. Thus, strengthening the booster vaccination, or enlarging costs for the infected individuals and the booster vaccination, or reducing costs for the primary vaccination can be beneficial to stop and control diseases more effectively. These results can provide a new insight into controlling infectious disease on networks. [ABSTRACT FROM AUTHOR]

Malaria, a parasite based infectious disease spread by anopheles mosquitos, is widespread, affecting people of all ages. Malaria blood-borne pathogens cause approximately 110 million clinical cases of malaria and between one and two million deaths associated with Plasmodium falciparum each year worldwide, including Bangladesh. In this paper, we develop a human–mosquito transmission dynamics malaria model and analyse of the system properties and solutions. Both analytical and numerical results suggest that if the basic reproduction number R 0 < 1 , the disease-free equilibrium is asymptotically stable, meaning malaria naturally dies out. Further, if R 0 > 1 , the malaria disease persists in the population. We also provide the model calibration to estimate parameters with year-wise malaria incidence data from 2001 to 2014 in Bangladesh. Sensitivity analysis also performs to identify the most critical parameters through the partial rank correlation coefficient method. We found that the contact rate of both humans and mosquitoes had the most extensive influence on malaria prevalence. Finally, the impacts of progression rate, disease-related death rate, recovery rate and the rate of losing immunity are examined through numerical simulations and graphical analysis. [ABSTRACT FROM AUTHOR]

In this paper, we present a stochastic hepatitis B virus (HBV) infection model and the dynamic behaviors of the model are investigated. When the fraction of vertical transmission μ ω ν C is not considered to be new infections, the existence and ergodicity of the stationary distribution of the model are obtained by constructing a suitable Lyapunov function, which determines a critical value ρ 0 s corresponding to the basic reproduction number of ODE system. This implies the persistence of the diseases when ρ 0 s > 1. Meanwhile, the sufficient conditions for the extinction of the diseases are derived when ρ 0 T < 0. What is more, we give the specific expression of the probability density function of the stochastic model around the unique endemic quasi-equilibrium by solving the Fokker–Planck equation. Finally, the numerical simulations are illustrated to verify the theoretical results and match the HBV epidemic data in China. • A stochastic HBV infection model is proposed and studied. • Existence and ergodicity of stationary distribution are proved. • Critical value ρ 0 s corresponding to basic reproduction number ρ 0 is obtained. • Density function of stochastic model near quasi-equilibrium is given. • Simulations match the HBV epidemic data in China. [ABSTRACT FROM AUTHOR]

The spread of online games has become more and more serious in recent years, which not only endangers the healthy growth of young people, but also has a certain potential harm to society. Therefore, how to effectively control the spread of games has become an urgent problem to be solved. To solve this problem, a new fractional model of online game addiction is developed in this paper. In the aspect of qualitative analysis, the non-negativity and boundedness of the model solutions are demonstrated. All equilibria and basic reproduction number are solved. The global asymptotic stability of all equilibria are proved by constructing ingenious Lyapunov functions. In terms of quantitative analysis, different from previous fractional order literature, we use a new method to fit the real data of game users in China from 2010 to 2021, and obtain the optimal fractional order α = 0. 98942 and model parameters. The estimated results show that the value of the basic reproduction number during this period is R 0 = 7. 1045. By setting different values of the parameters, the effects of all parameters on the model are shown. With the Latin hypercube sampling (LHS) and partial rank correlation coefficients (PRCC) techniques, the influence of all parameters in the basic reproduction number R 0 are quantified and the means to control the spread of the game were generalized: isolation and treatment. Finally, the Fractional Euler Method (FEM) and the fractional Forward Backward Sweep Method (FBSM) are used together to solve the fractional optimal control problem and the optimal control strategy is determined. The results of this paper provide a new research idea for parameter estimation of fractional equation, and a more reliable result for controlling the spread of online games. • In Caputo sense, a new fractional model of online game addiction is developed. • The nonlinear dynamics of the new model are investigated. • Through a new parameter estimation method, the estimates are much closer to the real data are obtained. • An efficient algorithm combining Fractional Euler Method (FEM) and Forward Backward Sweep Method (FBSM) is shown. [ABSTRACT FROM AUTHOR]

Online social networks (OSNs), as multilingual communication platforms, provide great convenience for rumor spreading. However, a great deal of negative impacts on individuals, society and economy will be caused during the spread of rumor. In order to explore the rules of rumor propagation on online social networks and formulate the corresponding control strategies, in this paper, we propose new 2I2SR rumor propagation models with and without time-delay based on multilingual environment. Firstly, we calculate the basic reproduction number which completely determines the local and global stability of the rumor free equilibrium. Secondly, our theoretical analysis reveals the existence and stability of the rumor equilibria, and some critical conditions are obtained for the existence of Hopf bifurcation by selecting critical parameters and delay as bifurcation parameters. Furthermore, by using Pontryagin's Maximum principle, a real-time optimization method that minimizes the cost of restraining rumor is proposed to eliminate the rumor within an expected time period. Finally, some numerical simulations and a practical application are given to prove the correctness of the theoretical results. [ABSTRACT FROM AUTHOR]

In this paper, we study a Susceptible–Vaccinated–Infected–Recovered (SVIR) epidemic model in a spatially heterogeneous environment under the Dirichlet boundary condition. We define the basic reproduction number ℜ 0 by the spectral radius of the next generation operator, and show that it is a threshold parameter. The disease extinction and persistence in the case of a bounded domain are considered. More precisely, we show that the disease-free equilibrium is globally asymptotically stable if ℜ 0 < 1 ; the system is uniformly persistent and an endemic equilibrium exists if ℜ 0 > 1. To verify our theoretical results, we perform some numerical simulations, using the Fredholm discretization method to identify ℜ 0. [ABSTRACT FROM AUTHOR]

In this paper, we study a wide spectrum of dynamical features of a prey–predator model with infection in prey population only. In the model formulation, we consider nonlinear incidence for infection transmission and Holling type II as response function of predator. We analyze the model for local stability and Hopf bifurcation, and the theoretical results are validated and extrapolated by numerical simulations. Also, we perform sensitivity analysis to explore the significance of the crucial parameters influencing the densities of prey and predator populations. Our numerical results show that disease persists in the system even if the basic reproduction number is below unity. The rates of predation on susceptible and infected prey populations are shown to dramatically change the behavior of eco-epidemiological system. We extend our model by considering the fact that the newly infected prey becomes productively infectious after the effective contact between susceptible and infectious preys, and some time lag is involved in this biological process; also the conversion process conceals some time. We analyze our eco-epidemiological model with two time delays (incubation delay and predator's gestation delay) for stability and bifurcation. From numerical simulations, we observe that both the delay parameters substantially affect the stable configuration of the system. [ABSTRACT FROM AUTHOR]

This paper aims to explore the behavior of malware propagation on mobile wireless sensor networks (MWSNs). A new malware propagation model with nonlinear incidence rate and sigmoid type removal rate is established, and its global stability, spatiotemporal stability, and optimal control are analyzed. Specifically, the theoretical analysis shows that (i) a forward transcritical bifurcation occurs when the basic reproduction number R 0 > 1 ; (ii) time and space affect the spreading behavior of malware due to spatial distribution; (iii) the optimization technology can effectively control the malware spreading on MWSNs. Finally, some numerical simulations are performed to verify the obtained theoretical results, and the experimental results confirm that the generated patterns are consistent with the field observations of actual MWSNs. Our study helps in controlling the propagation of malware and applicable to design and prediction of the security and robustness of a sensor network. • A new malware propagation model with nonlinear incidence rate and sigmoid type removal rate is studied. • Global stability, spatiotemporal stability, and optimal control are analyzed. • Optimal control technique is employed to reduce the malware propagation. • Stability switching phenomena are obtained via Hopf bifurcation. • Spatio-temporal dynamics depends on transmission rate, time movement of nodes and its initial deployment. [ABSTRACT FROM AUTHOR]

Rumors affect a variety of human affairs, which may cause a serious social disorder. In order to accurately simulate the propagation dynamics of rumors in real social networks and effectively ameliorate the harm of rumors to society, a new rumor propagation model is proposed in this paper. The model considers the influence of discussants, and divides the total population into four groups: ignorances, discussants, spreaders, and removers. The equilibria and basic reproduction number of the model are calculated, and the local and global asymptotic stability and the transcritical bifurcation of the equilibria are analyzed and proved. The theoretical analysis reveals the dynamic behavior and mechanism of rumor propagation. The parameters of the model are estimated by least-squares fitting, and the rumor spreading process is predicted according to the fitted parameters. The model is verified by using a rumor actual dataset. Simulation results show that the R-squared is 0.9544, which means that the proposed model can accurately simulate the rumor propagation in real social networks. By comparing the proposed model and the existing ones, the outcome indicates that discussants have an important impact on the rumor propagation process. [ABSTRACT FROM AUTHOR]