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In this paper, we develop and analyze a mathematical model for spreading malaria, including treatment with transmission‐blocking drugs (TBDs). The paper's main aim is to demonstrate the impact the chosen model for demographic growth has on the disease's transmission and the effect of its treatment with TBDs. We calculate the model's control reproduction number and equilibria and perform a global stability analysis of the disease‐free equilibrium point. The mathematical analysis reveals that, depending on the model's demography, the model can exhibit forward, backward, and even some unconventional types of bifurcation, where disease elimination can occur for both small and large values of the reproduction number. We also conduct a numerical analysis to explore the short‐time behavior of the model. A key finding is that for one type of demographic growth, the population experienced a significantly higher disease burden than the others, and when exposed to high levels of treatment with TBDs, only this population succeeded in effectively eliminating the disease within a reasonable timeframe. [ABSTRACT FROM AUTHOR]

In this paper, we introduce a new mathematical epidemic model with the effect of vaccination. We formulate a Susceptible-High risk-Infective-Recovered-Vaccinated (SHIRV) model in which the susceptible individuals with a higher probability of being infected (H) are selected as a separate class. We study the dynamical behavior of this model and define the basic reproductive number, R0. It is proved that the disease-free equilibrium is asymptotically stable if R0 < 1, and it is unstable if R0 1. Also, we investigate the existence and stability of the endemic equilibrium point analytically. For the system of differential equations of the SHIRV model, we give an approximating solution by using the Legendre-Ritz-Galerkin method. Finally, we study the influence of vaccination on measles and smallpox, two cases of the epidemic, using the proposed method in this paper. Numerical results showed that choosing high-risk people for vaccination can prevent them from getting infected and reduce mortality in the community. [ABSTRACT FROM AUTHOR]

The protection of wireless sensor networks (WSN) against malware attacks is crucial. The paper discusses the issue of malware attacks in WSN, which are commonly used for monitoring and surveillance in various applications. Due to resource constraints, sensor nodes in WSN are vulnerable to malware attacks, which can spread rapidly and paralyze the network. The development of new technologies such as IoT, Industry 4.0 has increased the importance of WSN, and it has become essential to address the challenges posed by the resource‐constrained nature of sensor nodes and security concerns. In this paper, a model is considered with two exposed states to investigate the behaviour of malware spreading in WSN, and a SE1E2IR (Susceptible—Exposed State 1 ‐ Exposed State 2 ‐ Infectious—Recovered) model is proposed. The model is formulated as a system of differential equations, and its equilibrium and stability are examined. The basic reproduction number (R0) is also calculated as a key parameter that characterizes the spread of malware in the network. This parameter helps to identify the conditions under which the network will remain malware‐free or when it will experience an outbreak of malware. The proposed model provides a mechanism for the earlier detection of malware occurrences in WSN, and also discusses the effect of connectivity and coverages on the propagation of malware in the network. The paper also includes a comparative study of the proposed model with existing models; extensive theoretical study and computation analysis are performed to validate the proposed model. [ABSTRACT FROM AUTHOR]

This paper addresses an investigation into a leader‐following consensus problem in a novel multi‐agent‐based SIS (Susceptible‐Infected‐Susceptible) epidemic model. As infectious diseases are easily spread through travel, the model considers each agent to represent a city in an infected country, interacting with neighbouring cities towards achieving disease eradication through consensus. To make the model more realistic, the Effect Terms (ETs) arising from the effects of agent states on each other are also considered. The differential equations' positivity and existence of solutions are examined, and the proposed model's equilibrium points are obtained, with stability analysed using new local and global basic reproduction numbers defined in this study. Additionally, a leader‐following consensus control for the multi‐agent‐based system is proposed to drive each agent to follow a virtual leader agent. Finally, numerical simulations are included to validate the theoretical findings corresponding to the proposed scheme. [ABSTRACT FROM AUTHOR]

In this paper, the author proposes and studies a mathematical model for the spread of coronavirus 2019. The effect of preventive programs on the control of disease is also considered. The basic reproduction number R0$$ {R}_0 $$ was computed and the results showed that forward bifurcation occured. Local and global stability of the equilibrium points of the model was studied by geometric stability method and Lyapunov function. Finally, the particular data of China which was reported for Covid‐19 were used for numerical simulation. [ABSTRACT FROM AUTHOR]

This paper presents a spatiotemporal reaction–diffusion model for epidemics to predict how the infection spreads in a given space. The model is based on a system of partial differential equations with the Neumann boundary conditions. First, we study the existence and uniqueness of the solution of the model using the semigroup theory and demonstrate the boundedness of solutions. Further, the proposed model's basic reproduction number is calculated using the eigenvalue problem. Moreover, the dynamic behavior of the disease‐free steady states of the model for R0<1$$ {\mathcal{R}}_0<1 $$ is investigated. The uniform persistence of the model is also discussed. In addition, the global asymptotic stability of the endemic steady state is examined. Finally, the numerical simulations validate the theoretical results. [ABSTRACT FROM AUTHOR]

The diversity of transmission modes and the heterogeneity of populations in the transmission of infectious diseases are issues that have to be faced in the current disease protection. In this paper, an infectious disease model incorporating age structure and horizontal and environmental spread, along with asymptomatic infection, is proposed to describe diversification of disease transmission routes and population heterogeneity. The expression of the basic reproduction number R0$$ {\mathcal{R}}_0 $$ is derived by a linear approximation method, and it is concluded that if R0<1$$ {\mathcal{R}}_0<1 $$, the disease‐free steady state is globally asymptotically stable; if R0>1$$ {\mathcal{R}}_0>1 $$, the model has a unique endemic steady state which is locally asymptotically stable under certain conditions. Further, the uniform persistence of disease is proved using the persistence theory of the infinite‐dimensional dynamical system when R0>1$$ {\mathcal{R}}_0>1 $$. In additional, the issue of optimal control problem according to this model is investigated, and the representation of optimal control with respect to state variables and adjoint variables is obtained which also imply the existence and uniqueness of optimal control. Finally, numerical simulations are carried out to interpret the theoretical results and to discuss the impact of age and control measures in disease transmission. [ABSTRACT FROM AUTHOR]

In this paper, we propose and analyze a reaction–diffusion vector–host disease model with advection effect in an one‐dimensional domain. We introduce the basic reproduction number (BRN) ℜ0$\Re _0$ and establish the threshold dynamics of the model in terms of ℜ0$\Re _0$. When there are no advection terms, we revisit the asymptotic behavior of ℜ0$\Re _0$ w.r.t. diffusion rate and the monotonicity of ℜ0$\Re _0$ under certain conditions. Furthermore, we obtain the asymptotic behavior of ℜ0$\Re _0$ under the influence of advection effects. Our results indicate that when the advection rate is large enough relative to the diffusion rate, ℜ0$\Re _0$ tends to be the value of local basic reproduction number (LBRN) at the downstream end, which enriches the asymptotic behavior results of the BRN in nonadvection heterogeneous environments. In addition, we explore the level set classification of ℜ0$\Re _0$, that is, there exists a unique critical surface indicating that the disease‐free equilibrium is globally asymptotically stable on one side of the surface, while it is unstable on the other side. Our results also reveal that the aggregation phenomenon will occur, namely, when the ratio of advection rate to diffusion rate is large enough, infected individuals will gather at the downstream end. [ABSTRACT FROM AUTHOR]

This paper concerns with the global threshold dynamics of a Tuberculosis (TB) model incorporating age‐space structure, treatment, and relapse. The original model is converted into a hybrid system comprising two Volterra integral equations and two partial differential equations by integrating along the characteristic line. The well‐posedness for the model is demonstrated by using the fixed point theory in conjunction with the induction method. In order to discuss whether a disease is persistent or extinct, we provide the explicit formulation of the basic reproduction number. By analyzing the distribution of the characteristic roots of the characteristic equations and constructing the proper Lyapunov functionals, the local and global stability for the steady states are addressed. Numerical simulations are conducted to confirm the conclusions of our analytical results and reveal that reduction of the TB transmission coefficient, reduction of infectiousness of treated individuals infected with TB, and increasing the treatment rate of infectious class are three feasible measures to control the transmission of TB. [ABSTRACT FROM AUTHOR]

In this paper, an susceptible–infected–susceptible (SIS) epidemic patch model with media delay is proposed at first. Then the basic reproduction number R0$\mathcal {R}_0$ is defined, and the threshold dynamics are studied. It is shown that the disease‐free equilibrium is globally asymptotically stable if R0<1$\mathcal {R}_0<1$ and the disease is uniformly persistent if R0>1$\mathcal {R}_0>1$. When the dispersal rates of susceptible and infected populations are identical and less than a critical value, it is proved that the limiting model has a unique positive equilibrium. Furthermore, the stability of the positive equilibrium and the existence of local and global Hopf bifurcations are obtained. Finally, some numerical simulations are performed. [ABSTRACT FROM AUTHOR]

Summary: This paper develops an individual‐based stochastic network SIR model for the empirical analysis of the Covid‐19 pandemic. It derives moment conditions for the number of infected and active cases for single as well as multigroup epidemic models. These moment conditions are used to investigate the identification and estimation of the transmission rates. The paper then proposes a method that jointly estimates the transmission rate and the magnitude of under‐reporting of infected cases. Empirical evidence on six European countries matches the simulated outcomes once the under‐reporting of infected cases is addressed. It is estimated that the number of actual cases could be between 4 to 10 times higher than the reported numbers in October 2020 and declined to 2 to 3 times in April 2021. The calibrated models are used in the counterfactual analyses of the impact of social distancing and vaccination on the epidemic evolution and the timing of early interventions in the United Kingdom and Germany. [ABSTRACT FROM AUTHOR]

The aim of this paper is to study the dynamics of a reaction‐diffusion SI (susceptible‐infectious) epidemic model with a nonlinear incidence rate describing the transmission of a communicable disease between individuals. We prove that the proposed model has two steady states under one condition. By analyzing the eigenvalues and using the linearization method and an appropriately constructed Lyapunov functional, we establish the local and global asymptotic stability of the non‐negative constant steady states subject to the basic reproduction number being greater than unity and of the disease‐free equilibrium subject to the basic reproduction number being smaller than or equal to unity in ODE case. By applying an appropriately constructed Lyapunov functional, we identify the condition of the global stability in the PDE case. Finally, we present some numerical examples illustrating and confirming the analytical results obtained throughout the paper. [ABSTRACT FROM AUTHOR]

Due to the spatial heterogeneity present in many aspects of disease transmission, the rate of transmission of infectious diseases, human birth/mortality rate, and the mobility ability of infected humans should be different in different geographical locations. On the other hand, infected humans may exhibit distinct differences in symptoms during the different stages of the disease transmission. This paper aims to study threshold dynamics of a reaction–diffusion host–pathogen model governed by two‐stage mechanism and no flux boundary condition. In the model, we use bilinear and saturated incidence mechanism to model the interactions between host populations and pathogens and assume that (i) the parameters are spatially dependent and (ii) infected individuals in the first stage can random walk, while in the second stage, the diffusion rates are neglected. By carrying out strict analysis, the paper establishes the threshold‐type results with the basic reproduction number. Specifically, in a homogeneous case that all parameters are constants, we establish the global attractivity of the endemic equilibrium. To investigate the effects of small or large diffusion rates of susceptible individuals on the disease transmission in a heterogeneous environment, we explore the asymptotic profile of positive steady state whenever it exists. Our numerical results validate the theoretical results and indicate the importance of the infected humans in the second stage and pathogens in the environment in disease transmission, which greatly contribute to disease spread in a bounded domain and should not be ignored. [ABSTRACT FROM AUTHOR]

In this paper, we investigate an SIRI epidemic model with nonlinear incidence rate and high‐order stochastic perturbation. First, we obtain a stochastic threshold R0P related to the basic reproduction number R0. A key contribution of our paper is to derive the existence and uniqueness of an ergodic stationary distribution of the stochastic model if R0P>1. Next, by solving the corresponding Fokker‐Planck equation, the exact expression of probability density function of the stochastic model is obtained. Moreover, we establish the sufficient condition R0Q<1 for disease extinction in a long term. Finally, several empirical examples and numerical simulations are provided to verify the above theoretical results. [ABSTRACT FROM AUTHOR]

Human immunodeficiency virus (HIV) is a serious disease that threatens and affects capital stock, population composition, and economic growth. This research paper aims to study the mathematical modeling and disease dynamics of HIV/acquired immunodeficiency syndrome (AIDS) with memory effect. We propose two fractional models in the Caputo sense for HIV/AIDS with and without migration. First, we prove the existence and positivity of both models and calculate the basic reproduction number R0$$ {\mathcal{R}}_0 $$ using the next generation method. Then, we study the local and global stability of the obtained equilibria. In addition, numerical simulations are provided for different values of the fractional order ρ$$ \rho $$ using the Adams–Bashforth–Moulton fractional scheme, to verify the theoretical results. Moreover, a sensitivity analysis of the parameters for the model with migration is carried out. [ABSTRACT FROM AUTHOR]

This paper examines an epidemic patch model with mass-action transmission mechanism and asymmetric connectivity matrix. Results on the global dynamics of solutions and the spatial structures of endemic equilibrium (EE) solutions are obtained. In particular, we show that when the basic reproduction number R0 is less than one and the dispersal rate of the susceptible population ds is large, the population would eventually stabilize at the disease-free equilibrium. However, the disease may persist if ds is small, even if R0 < 1. In such a scenario, explicit conditions on the model parameters that lead to the existence of multiple EE are identified. These results provide new insights into the dynamics of infectious diseases in multipatch environments. Moreover, results in Li and Peng (Stud Appl Math. 2023;150(3):650-704), which is for the same model but with symmetric connectivitymatrix, are generalized and improved. [ABSTRACT FROM AUTHOR]

Age heterogeneity in influenza transmission leads to significant differences in infection risk among different age groups. In this paper, an influenza model with vaccination and discrete age structure is constructed. First, we obtain the basic reproduction number R0$$ {R}_0 $$ and prove that the disease‐free equilibrium E0$$ {E}_0 $$ is globally stable at R0<1$$ {R}_0<1 $$. When R0>1$$ {R}_0>1 $$, the disease will persist uniformly, and there is a unique positive equilibrium, which is globally asymptotically stable under certain conditions. The model is used to analyze influenza data from 2016 to 2017 in China. By comparing various prevention and control measures, it can be found that by wearing masks to reduce contact between different age groups, while increasing vaccine coverage and reducing vaccine failure rates, the prevention and control goals can be achieved in the shortest time. However, if only the vaccine coverage rate is increased and the vaccine protection rate is strengthened, the time to reach the control target is 8 months later than the above case, and the cost is the lowest. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the qualitative behavior of a class of fractional SEIR epidemic models with a more general incidence rate function and time delay to incorporate latent infected individuals. We first prove positivity and boundedness of solutions of the system. The basic reproduction number R0$$ {\mathcal{R}}_0 $$ of the model is computed using the method of next generation matrix, and we prove that if R0<1$$ {\mathcal{R}}_0<1 $$, the healthy equilibrium is locally asymptotically stable, and when R0>1$$ {\mathcal{R}}_0>1 $$, the system admits a unique endemic equilibrium which is locally asymptotically stable. Moreover, using a suitable Lyapunov function and some results about the theory of stability of differential equations of delayed fractional‐order type, we give a complete study of global stability for both healthy and endemic steady states. The model is used to describe the COVID‐19 outbreak in Algeria at its beginning in February 2020. A numerical scheme, based on Adams–Bashforth–Moulton method, is used to run the numerical simulations and shows that the number of new infected individuals will peak around late July 2020. Further, numerical simulations show that around 90% of the population in Algeria will be infected. Compared with the WHO data, our results are much more close to real data. Our model with fractional derivative and delay can then better fit the data of Algeria at the beginning of infection and before the lock and isolation measures. The model we propose is a generalization of several SEIR other models with fractional derivative and delay in literature. [ABSTRACT FROM AUTHOR]

In this paper, an SIR model with a nonlinear incidence rate is studied. A disease‐free equilibrium E0$$ {E}_0 $$, an endemic equilibrium E1$$ {E}_1 $$, and the basic reproduction number of the model R0$$ {R}_0 $$ are obtained. If R0<1,E0$$ {R}_0&lt;1,{E}_0 $$ is locally asymptotically stable and if R0>1,E1$$ {R}_0&gt;1,{E}_1 $$ is locally asymptotically stable. By Barbalat's lemma, we study the global stability of the model. Transcritical bifurcation analysis is investigated by using the Sotomayor theorem. As the infection rate increases, the asymptotic behavior of the system near E0$$ {E}_0 $$ approaches E1$$ {E}_1 $$ and the system has a transcritical bifurcation. Also, we check the existence of Hopf bifurcation for the given system. In addition, a sensitivity analysis is provided for the basic reproduction number. Our results are supported with numerical simulations. [ABSTRACT FROM AUTHOR]

The deadly coronavirus disease 2019 (COVID‐19) has recently affected each corner of the world. Many governments of different countries have imposed strict measures in order to reduce the severity of the infection. In this present paper, we will study a mathematical model describing COVID‐19 dynamics taking into account the government action and the individuals reaction. To this end, we will suggest a system of seven fractional deferential equations (FDEs) that describe the interaction between the classical susceptible, exposed, infectious, and removed (SEIR) individuals along with the government action and individual reaction involvement. Both human‐to‐human and zoonotic transmissions are considered in the model. The well‐posedness of the FDEs model is established in terms of existence, positivity, and boundedness. The basic reproduction number (BRN) is found via the new generation matrix method. Different numerical simulations were carried out by taking into account real reported data from Wuhan, China. It was shown that the governmental action and the individuals' risk awareness reduce effectively the infection spread. Moreover, it was established that with the fractional derivative, the infection converges more quickly to its steady state. [ABSTRACT FROM AUTHOR]

Background: Streptococcus equi spp. equi (S. equi), the cause of strangles in horses, is considered a highly contagious pathogen affecting equines and the equine industry worldwide. Fundamental epidemiological characteristics of outbreaks, such as the basic reproduction number (R0), are not well described. Objectives: Estimate R0 for S. equi in equine populations from outbreak data. Study design: Systematic review and meta‐analysis of published and unpublished data. Methods: A literature search for outbreak reports was carried out. Depending on data available in the reports, the early epidemic growth rate or final attack rate (AR) approach was used to estimate the basic reproduction number for that outbreak. Other recorded outbreak characteristics were the type of housing (group vs. individual). An overall estimate for R0 was computed by meta‐analysis. Results: Data from eight outbreaks were extracted from peer‐reviewed publications. Data from two additional, non‐published outbreaks was also included in the meta‐analysis. A conservative estimate for R0 was 2.2 (95% confidence interval [CI] 1.9–2.5). A less conservative estimate, including outbreaks with a 100% AR for which a lower limit R0 was estimated, was 2.7 (95% CI 2.1–3.3). Main limitations: Few papers describing longitudinal incidence data were found so most estimates were based on the outbreaks' final size. Several outbreaks had a 100% attack rate and could therefore only be included as a lower limit estimate in the meta‐analysis. The reported result therefore may be an underestimation. Conclusions: This estimate for R0 for S. equi informs parameters for future mathematical modelling, quantifies desired preventive vaccine coverage and helps evaluate the effect of prevention strategies through future modelling studies. [ABSTRACT FROM AUTHOR]

Due to the diversity and inaccuracy in media communications, the effect of awareness programs includes positive and negative emotions which are aroused by the prevalence of media coverage on infectious diseases. In this paper, an epidemic model with positive and negative information is proposed on heterogeneous network. Firstly, the basic reproduction number of infectious disease transmission is carried on, and the global stability of disease‐free equilibrium and endemic equilibrium is analyzed. Secondly, three time‐varying control schemes are offered to contain the contact infection rate and enhance the implementation rate of positive information for improving people's awareness and reducing the number of infected and the cost of control. In addition, under the principle of the Pontryagin minimization, the combination strategy is analyzed to form a solution for optimal control. Finally, the numerical simulations are performed to verify the analytical findings, that is, the optimal control effectively reduces the final infected scale and control cost. [ABSTRACT FROM AUTHOR]

In most HIV models, the emergence of backward bifurcation means that the control for basic reproduction number less than one is no longer effective for HIV treatment. In this paper, we study an HIV model with CTL response and cell‐to‐cell transmission by using the dynamical approach. The local and global stability of equilibria is investigated, the relations of subcritical Hopf bifurcation and supercritical bifurcation points are revealed, especially, the so‐called new type bifurcation is also found with two Hopf bifurcation curves meeting at the same Bogdanov–Takens bifurcation point. Forward and backward bifurcation, Hopf bifurcation, saddle‐node bifurcation, Bogdanov–Takens bifurcation are investigated analytically and numerically. Two limit cycles are also found numerically, which indicates that the complex behavior of HIV dynamics. Interestingly, the role of cell‐to‐cell interaction is fully uncovered, it may cause the oscillations to disappear and keep the so‐called new type bifurcation persist. Finally, some conclusions and discussions are also given. [ABSTRACT FROM AUTHOR]

In this paper, we study an SIS (susceptible–infected–susceptible) epidemic model with mass action infection mechanism in a patchy environment. We first analyze the long‐time dynamics of the model in terms of the basic reproduction number R0$\mathcal {R}_0$, and prove under certain conditions that the disease‐free equilibrium (DFE) is globally attractive if R0≤1$\mathcal {R}_0\le 1$ whereas the endemic equilibrium (EE) is globally attractive if R0>1$\mathcal {R}_0>1$. Then we establish the existence and uniqueness of EE provided R0>1$\mathcal {R}_0>1$, which is achieved, among other ingredients, by reducing the equilibrium problem into the one involving only the component of infected population. We further investigate the asymptotic profile of EE as the motility rate of the susceptible and/or infected population becomes small. We also perform the theoretical analysis on the corresponding model with linear birth/death effect. By comparing the findings of the models considered here with those of other closely related ones, our results suggest that the variation of total population, individual motility, infection mechanism, and spatial heterogeneity all can play significant roles in the dynamics of disease transmission, which may shed light on the practical implications for predicting the patterns of disease outbreak and designing optimal control strategies. [ABSTRACT FROM AUTHOR]

In this paper, the ABC fractional derivative is used to provide a mathematical model for the dynamic systems of substance addiction. The basic reproduction number is investigated, as well as the equilibrium points' stability. Using fixed point theory and nonlinear analytic techniques, we verify the theoretical results of solution existence and uniqueness for the proposed model. A numerical technique for getting the approximate solution of the suggested model is established by using the Adams type predictor‐corrector rule for the ABC‐fractional integral operator. There are several numerical graphs that correspond to different fractional orders. Furthermore, we present a numerical simulation for the transmission of substance addiction in two scenarios with fundamental reproduction numbers greater than and fewer than one. [ABSTRACT FROM AUTHOR]

In this paper, we formulate and investigate a stochastic one‐prey and two‐predator model with Holling II functional response and disease in the prey, in which the predators only feed on infected prey. The existence and uniqueness of global positive solution is proved by using conventional methods. The corresponding deterministic model has a disease‐free equilibrium point if the basic reproduction number R0<1$$ {R}_0<1 $$, and it has three boundary equilibrium points and one positive equilibrium point if R0>1$$ {R}_0>1 $$. For the stochastic model, we investigate the asymptotic behavior around all of the five equilibrium points and prove that there is a unique ergodic stationary distribution under certain conditions. Moreover, we obtain the condition on which the population of the infected prey and the two predators will die out in the time mean sense. Finally, numerical simulations are conducted to illustrate our analysis results. [ABSTRACT FROM AUTHOR]

During the recent pandemic caused by COVID‐19, media awareness played a crucial role in educating people about social distancing, wearing masks, quarantine, vaccination, and medication. Media awareness brought individual behavioral changes among the people, which in turn helped reduce the infection rate. Motivated by this, we have formulated a mathematical model introducing a media compartment to mitigate measles disease transmission. In this paper, the SEIR model is used to study measles disease in three cases: one with a delay in vaccination, the second with regular vaccination, and the third with the impact of media awareness on the spreading of measles disease. Further, the dynamical behavior of the models is studied in terms of positivity, boundedness, equilibrium, and basic reproduction number (BRN). The sensitivity analysis of the models is conducted, which verifies the importance of the BRN (R0$$ {R}_0 $$) to be less than one for disease eradication. The numerical study confirms the impact of media awareness on exposed and infected populations. [ABSTRACT FROM AUTHOR]

This paper is concerned with a degenerate and time delayed virus infection model with spatial heterogeneity and general incidence. The well‐posedness of the system, including global existence, uniqueness, and ultimately boundedness of the solutions, as well as the existence of a global attractor, is discussed. The basic reproduction number R0$\mathcal {R}_0$ is defined and a characterization of R0$\mathcal {R}_0$ is presented. Without the compactness of the solution semiflow, we establish the global dynamics of the system based on R0$\mathcal {R}_0$. In addition, when the system is spatially homogeneous, the unique infection steady state is globally asymptotically stable. Simulations are presented to illustrate our theoretical results. [ABSTRACT FROM AUTHOR]

In this paper, to investigate the synthetic effect of PrEP (pre‐exposure prophylaxis) and ART (antiretrovial therapy) on HIV transmission among MSM (men who have sex with men) in heterogenous environment, a realistic HIV epidemic model with spatial diffusion is established. Here, HIV infectious people are divided into three immunity based compartments, that is, CD4+ T cell count less than 350, between 350 and 500, and more than 500, respectively. The basic reproduction number R0 is established and proved as a threshold parameter: The global asymptotic stability of the disease‐free steady state holds for R0<1, and the disease will be present if R0>1. Considering the substantial advantages of PrEP and ART in controlling HIV transmissions among MSM, the optimal control problem is presented for the case of positive constant diffusion coefficients, which minimize the total population of susceptible individual and HIV infected individual, the cost of PrEP and ART therapy. As an illustration of our theoretical results, we conduct numerical simulations. We also conduct an optimal control case study where model parameters are estimated from the demographic and epidemiological data from China. This work suggests the following: (1) Spatial factors cannot be ignored during the HIV intervention; (2)taking the PrEP intervention measure for HIV transmissions among MSM as early as possible will help to improve the control efficiency and reduces its cost; (3) reducing the PrEP drug costs will promote the efficiency of PrEP treatment in preventing the spread of HIV among MSM. [ABSTRACT FROM AUTHOR]

As the first‐wave COVID‐19 has passed in 2020, people's awareness of self‐protection began to decline gradually. How to prevent and control the second‐wave COVID‐19 has become an important issue in many countries and regions. By analyzing the transmission of the second‐wave COVID‐19 caused by an imported case in Tonghua City, Jilin Province, China, in January 2021, we establish a new mathematical COVID‐19 model to simulate the transmission characteristics of the second‐wave COVID‐19. First, we analyze the basic properties of the model, prove the existence of the equilibrium point, and obtain the expression of the basic reproduction number with important biological significance. Secondly, we use the weighted nonlinear least square estimation method to fit the cases in Tonghua City of Jilin Province in January 2021, and get the estimated value of the parameters. The basic reproduction number of the second‐wave COVID‐19 in Tonghua City is R0=1.0695, which is much smaller than that of the first‐wave COVID‐19 in Wuhan in 2020. Finally, in the optimal control part, we consider two control methods (keeping social distance and nucleic acid detection of all people in the city) to simulate the control of the disease. The results show that the control intensity of the two control methods needs to be dynamically changed and adjusted, so that the cost can be minimized with the least infection. The results of this paper can not only provide suggestions for health management departments, but also provide a reference for the analysis of the second‐wave COVID‐19 in other countries or regions. [ABSTRACT FROM AUTHOR]

In this paper, a reaction–diffusion SIRE epidemic model in contaminated environments is proposed, in which the effect of protection for susceptible individuals is included by the nonlinear incidence functions b(S)E$b(S)E$ and g(S)I$g(S)I$. When the space is heterogeneous, the basic reproduction number R0$\mathcal {R}_{0}$ is derived, by which we find that if R0≤1$\mathcal {R}_{0}\le 1$, the disease‐free steady state is globally asymptotically stable, while R0>1$\mathcal {R}_{0}>1$, the disease is uniform persistent. Furthermore, when R0>1$\mathcal {R}_{0}>1$ and additional conditions hold, the global asymptotic stability of special endemic steady state is obtained in homogeneous space. Finally, the theoretical results are validated by numerical simulations, some open questions are illustrated. [ABSTRACT FROM AUTHOR]

Recent experimental evidence suggests that spatial heterogeneity plays an important role in within‐host infections caused by different viruses including hepatitis B virus (HBV), hepatitis C virus (HCV), and human immunodeficiency virus (HIV). To examine the spatial effects of viral infections, in this paper we study the asymptotic spreading in a within‐host viral infection model, which describes the spatial expansion speeds of viruses and infected cells within an infected host. We first establish the boundedness of solutions to the Cauchy problem via local Lp$L^p$‐estimates and dual arguments. Then the spreading speed is estimated when the basic reproduction number of the corresponding kinetic system is larger than one. More precisely, the upper bounds of the spreading speed are given by constructing suitable upper solutions while the lower bounds of the spreading speed are obtained by introducing an auxiliary equation with nonlocal delay. When the basic reproduction number of the corresponding kinetic system is less than or equal to one, the virus dies out uniformly. Finally, we present some numerical simulations to illustrate our theoretical findings and discuss the biological relevance of these results. [ABSTRACT FROM AUTHOR]

This paper is concerned with the spreading or vanishing of an epidemic disease which is characterized by a nonlocal diffusion SIR model with nonlocal incidence rate and double free boundaries. We prove that the disease will vanish if the basic reproduction number R0<1$$ {R}_0<1 $$, or the initial area h0$$ {h}_0 $$, the initial datum S0$$ {S}_0 $$, and the expanding ability μ$$ \mu $$ are sufficiently small when R0>1$$ {R}_0>1 $$. Otherwise, the disease will spread to the whole area if R0>1$$ {R}_0>1 $$, when h0$$ {h}_0 $$ is suitably large or h0$$ {h}_0 $$ is small but μ$$ \mu $$ is large enough. Moreover, we also show the long‐time asymptotic limit of the solution when vanishing happens. [ABSTRACT FROM AUTHOR]

This paper investigates the asymptotic analysis of spatially heterogeneous viral transmission, incorporating cell‐to‐cell transmission, virus nonlocal dispersal, and intracellular delay. Due to the noncompactness of the semiflow, we used the Kuratowski measure of noncompactness to demonstrate the existence of a global compact attractor. This noncompactness issue generates difficulties in calculating the basic reproduction number R0$$ {R}_0 $$, which is the principal eigenvalue of the next‐generation operator. The threshold role of this number is determined, where we derived two different cases: (i) the global stability of the virus‐free steady state, which is globally stable for R0<1$$ {R}_0<1 $$ by the Lyapunov direct method, and (ii) the global stability of the virus steady state for R0>1$$ {R}_0>1 $$. Indeed, the second case is demonstrated through several steps that include uniform persistence, the existence of a virus steady state, and the global stability of the virus steady state. The results are supported by different graphical representations with proper biological justifications. [ABSTRACT FROM AUTHOR]

In this paper, a stochastic susceptible‐infective‐recovered‐susceptible (SIRS) model with density‐dependent demographics is proposed to study the dynamics of transmission of infectious diseases under stochastic environmental fluctuations. We demonstrate that the position of the basic reproduction number R0s$R_0^s$ with respect to 1 is the threshold between extinction and persistence of the disease under mild extra conditions. That is, under mild extra conditions, when R0s<1$R_0^s<1$, the disease is eradicated with probability 1; when R0s>1$R_0^s>1$, the disease is persistent almost surely and the Markov process has a unique stationary distribution and is polynomial ergodic. As an application, we use the 2017 influenza A data from Western Asia to estimate the parameter values of the model and based on that investigate the effect of random noises on the dynamics of the model. Our study reveals that the basic reproduction number R0s$R_0^s$ is negatively correlated with the noise intensity for the infected but positively correlated with that for the susceptible population, which are different from the findings obtained in the existing literature. [ABSTRACT FROM AUTHOR]

Many countries worldwide have been affected by the outbreak of the novel coronavirus (COVID‐19) that was first reported in China. To understand and forecast the transmission dynamics of this disease, fractional‐order derivative‐based modeling can be beneficial. We propose in this paper a fractional‐order mathematical model to examine the COVID‐19 disease outbreak. This model outlines the multiple mechanisms of transmission within the dynamics of infection. The basic reproduction number and the equilibrium points are calculated from the model to assess the transmissibility of the COVID‐19. Sensitivity analysis is discussed to explain the significance of the epidemic parameters. The existence and uniqueness of the solution to the proposed model have been proven using the fixed‐point theorem and by helping the Arzela–Ascoli theorem. Using the predictor–corrector algorithm, we approximated the solution of the proposed model. The results obtained are represented by using figures that illustrate the behavior of the predicted model classes. Finally, the study of the stability of the numerical method is carried out using some results and primary lemmas. [ABSTRACT FROM AUTHOR]

The present paper studies a fractional‐order SEIR epidemic model for the transmission dynamics of infectious diseases such as HIV and HBV that spreads in the host population. The total host population is considered bounded, and Holling type‐II saturation incidence rate is involved as the infection term. Using the proposed SEIR epidemic model, the threshold quantity, namely, basic reproduction number R0, is obtained that determines the status of the disease, whether it dies out or persists in the whole population. The model's analysis shows that two equilibria exist, namely, disease‐free equilibrium (DFE) and endemic equilibrium (EE). The global stability of the equilibria is determined using a Lyapunov functional approach. The disease status can be verified based on obtained threshold quantity R0. If R0<1, then DFE is globally stable, leading to eradicating the population's disease. If R0>1, a unique EE exists, and that is globally stable under certain conditions in the feasible region. The Caputo type fractional derivative is taken as the fractional operator. The bifurcation and sensitivity analyses are also performed for the proposed model that determines the relative importance of the parameters in disease transmission. The numerical solution of the model is obtained by the generalized Adams–Bashforth–Moulton method. Finally, numerical simulations are performed to illustrate and verify the analytical results. From the obtained results, it is concluded that the order of the fractional derivative plays a significant role in the dynamic process. Also, from the sensitivity analysis results, it is seen that the parameter δ (infection rate) is positively correlated to R0, while the parameter μ (inhibition rate) is negatively correlated to R0, which means these parameters are more sensitive than others in disease dynamics. [ABSTRACT FROM AUTHOR]

This paper investigates an infection age‐space structured susceptible‐vaccinate‐infectious‐recovered (SVIR) epidemic model. The model is described by reaction‐diffusion models involving the spatial dispersal of human populations in the same domain Ω∈ℝn. We first give the well‐posedness of the model by converting the model to a reaction‐diffusion model and two Volterra integral equations and obtain two constant equilibria. Our result suggests that the basic reproduction number determines the dichotomy of disease persistence and extinction, which is achieved by studying the local stability, disease persistence and global attractivity of equilibria. [ABSTRACT FROM AUTHOR]

Considering the effect of stochasticity including white noise and colored noise, this paper aims to study a hybrid stochastic cholera epidemic model with waning vaccine‐induced immunity and nonlinear telegraph perturbations. First, we derive a critical value ℛ0C related to the basic reproduction number ℛ0 of the deterministic model. The key aim of this paper is to generalize the θ‐stochastic criterion method proposed by the recent work (Han et al. in Chaos Solit Fract 140:110238, 2020) to eliminate nonlinear telegraph perturbations. Next, via constructing several θ‐stochastic Lyapunov functions and using the generalized method, we further prove that the stochastic model have a unique ergodic stationary distribution under ℛ0C>1. Results show that the prevention and control of cholera epidemic depend on low transmission rate and small telegraph perturbations. Finally, the corresponding numerical simulations are performed to illustrate our analytical results and a practical application on the Somalia cholera outbreak is shown at the end of this paper. [ABSTRACT FROM AUTHOR]

This paper presents an epidemic model with varying population, incorporating a new vaccination strategy and time delay. It investigates the impact of vaccination with respect to vaccine efficacy and the time required to see the effects, followed by determining how to control the spread of the disease according to the basic reproduction ratio of the disease. Some numerical simulations are provided to illustrate the theoretical results. [ABSTRACT FROM AUTHOR]

This paper is concerned with the existence and nonexistence of traveling waves of a virus infection model with humoral immunity, cell‐to‐cell transmission, and general nonlinear incidence. Our results show that the existence of traveling wave solutions is determined not only by the basic reproduction number R0$$ {\mathcal{R}}_0 $$ of virus infection and the antibody response reproduction number R1$$ {\mathcal{R}}_1 $$ but also by the critical wave speed c∗$$ {c}^{\ast } $$. More precisely, we obtain the existence of traveling wave solution connecting infection‐free equilibrium and antibody‐free infection equilibrium for R0>1,R1<1$$ {\mathcal{R}}_0>1,{\mathcal{R}}_1<1 $$, and c>c∗$$ c>{c}^{\ast } $$ and connecting infection‐free equilibrium and antibody‐present infection equilibrium for R0>1,R1>1$$ {\mathcal{R}}_0>1,{\mathcal{R}}_1>1 $$, and c>c∗$$ c>{c}^{\ast } $$. The existence of traveling wave solution connecting infection‐free equilibrium and antibody‐present infection equilibrium is discussed. Some numerical simulations are carried out to illustrate our analytical results. [ABSTRACT FROM AUTHOR]

We introduce, in this paper, a delayed hybrid epidemic model describing the evolution of heroin addiction in a given population. The main objective of our mathematical analysis will be to provide the global behavior of the solutions. To the best of our knowledge, there are no results about the global dynamics of such a model containing both age structure and distributed time delay. We will prove that our model has threshold dynamics in terms of the basic reproduction number R0$$ {R}_0 $$. It will be established that for R0≤1$$ {R}_0\le 1 $$, the addiction‐free equilibrium is globally stable and for R0>1$$ {R}_0&gt;1 $$, the endemic equilibrium is globally stable. The theoretical results are checked numerically. Especially, we focus on the influence of the addiction rate on the population dynamic. [ABSTRACT FROM AUTHOR]

In this paper, we develop a diffusive and delayed hepatitis B virus (HBV) infection model, which incorporates both virus‐to‐cell and cell‐to‐cell transmissions along with cytotoxic T lymphocyte (CTL) immune response. Two threshold parameters ℜ0$$ {\mathfrak{\Re}}_0 $$ (basic reproduction number) and ℜ1$$ {\mathfrak{\Re}}_1 $$ (immune response reproduction number) are defined, which determine the dynamic behavior of the model. Employing the approach of Lyapunov functionals, we show that the infection‐free steady state is globally asymptotically stable when ℜ0≤1$$ {\mathfrak{\Re}}_0\le 1 $$; the immune‐free infected steady state is globally asymptotically stable when ℜ1≤1<ℜ0$$ {\mathfrak{\Re}}_1\le 1<{\mathfrak{\Re}}_0 $$; and the immune‐present steady state is globally asymptotically stable when ℜ0>1,ℜ1>1$$ {\mathfrak{\Re}}_0>1,{\mathfrak{\Re}}_1>1 $$, and φ(I)=I$$ \varphi (I)=I $$. We perform the sensitivity and uncertainly analysis on ℜ0$$ {\mathfrak{\Re}}_0 $$ with respect to parameters and find that the most important parameter that drives the HBV infection is the mortality of HBV DNA capsids. Moreover, the theoretical results are supported with numerical simulations. [ABSTRACT FROM AUTHOR]

The basic reproduction number (R0) often cannot be explicitly computed when dealing with continuously age‐structured epidemic models. In this paper, we numerically compute R0 of a PDE model for a human immunodeficiency virus (HIV) infection, defined as the spectral radius of a next‐generation operator. Since R0 cannot be analytically obtained, on the one hand, we discretize the linearized PDE model into a system of linear ODEs (Euler method), and on the other hand, we discretize the eigenvalue problem (pseudo‐spectral method). In both cases, we approximate R0 by the largest eigenvalue R0, n of a next‐generation matrix, and we show the convergence of R0, n to R0 as the discretization index n increases to infinity. Finally, we present several tests to check/compare the accuracy of both numerical methods. [ABSTRACT FROM AUTHOR]

An SVEIR epidemic model with continuous age‐dependent vaccination, latency, infection, and disease relapse is proposed and analyzed in this paper. The dynamical behaviors including the derivation of basic reproduction number R0, the existence and the stability of steady states, and the uniform persistence of the model are investigated. The results indicate that if R0≤1, the disease‐free steady state is globally asymptotically stable, and the disease dies out, whereas if R0>1, the disease is uniformly persistent, and the endemic steady state is also globally asymptotically stable, and the disease remains at the endemic level. The research shows the global dynamics of the model are sharply determined by its basic reproduction number R0. Finally, numerical examples support our main theoretical results. [ABSTRACT FROM AUTHOR]

We consider the transmission of visceral leishmaniasis in three populations: humans, dogs and sandflies. To investigate the effects of spatial heterogeneity and temporal periodicity on disease transmission, a nonlocal periodic reaction‐diffusion equation model is developed. We introduce the basic reproduction number R0$$ {\mathcal{R}}_0 $$ and establish the global dynamics. Numerical simulations show that (i) R0$$ {\mathcal{R}}_0 $$ is overestimated in the absence of spatial heterogeneity; (ii) R0$$ {\mathcal{R}}_0 $$ is underestimated without considering the effect of temporal seasonality; and (iii) shortening periodic delays of visceral leishmaniasis reduces the risk of disease transmission. [ABSTRACT FROM AUTHOR]

The emergence of COVID‐19 pandemic has been a major social as well as economic challenges around the globe. Infections from the infected surfaces have also been identified as drivers of COVID‐19 transmission, but most of the epidemic models do not include the effect of environmental contamination to account for the indirect transmission of the disease. The present study is devoted to the investigation of the effect of environmental contamination on the spread of the coronavirus pandemic by means of a mathematical model. We also consider the impact of vaccination coverage as an effective control measure against COVID‐19. The proposed model is analyzed to discuss the feasibility as well as stability of the disease‐free and endemic equilibria; an epidemic threshold in the form of basic reproduction number is obtained. Further, we incorporate the effect of seasonal periodic changes by letting the rate of direct transmission of disease as time dependent, and find sufficient conditions for the global attractivity of the positive periodic solution. We employ sophisticated techniques of sensitivity analysis to identify model parameters which significantly alter the epidemic threshold and the disease prevalence. We find that by enhancing the vaccination of the susceptible population and hospitalization of the symptomatic/asymptomatic individuals, the basic reproduction number can be lowered to a value less than unity. The findings show that the prevalence of disease can be potentially suppressed by increasing the vaccination of susceptible population, hospitalization of infected people and depletion of environmental contamination. Moreover, we observe that seasonal pattern in the disease transmission causes persistence of the pandemic in the population for a longer period. [ABSTRACT FROM AUTHOR]

We study the global dynamics of a susceptible‐vaccinated‐infected‐recovered model that incorporates nonlocal diffusion. By identifying the basic reproduction number ℛ0$$ {\mathrm{\mathcal{R}}}_0 $$ of the model, we obtain the following threshold‐type results: (i) If ℛ0<1$$ {\mathrm{\mathcal{R}}}_0<1 $$, then the epidemic becomes extinct in the sense that the infection‐free equilibrium is globally attractive; (ii) if ℛ0>1$$ {\mathrm{\mathcal{R}}}_0>1 $$ and the diffusion coefficients are the same for all classes, then the epidemic persists in the sense that the system is uniformly persistent; and (iii) if ℛ0>1$$ {\mathrm{\mathcal{R}}}_0>1 $$, the diffusion coefficients for susceptible, and the vaccinated classes are zero, then the system admits a unique endemic equilibrium, and the omega‐limit set is included in the singleton of the endemic equilibrium. Our results show that ℛ0$$ {\mathrm{\mathcal{R}}}_0 $$ is an essential value for determining global epidemic dynamics in our model. [ABSTRACT FROM AUTHOR]

In this manuscript, we have studied the dynamical behavior of a deadly COVID‐19 pandemic which has caused frustration in the human community. For this study, a new deterministic SEIHR fractional model is developed for the first time. The purpose is to perform a complete mathematical analysis and the design of an optimal control strategy for the proposed Caputo–Fabrizio fractional model. We have proved the existence and uniqueness of solutions by employing principle of mathematical induction. The positivity and the boundedness of solutions is proved using comprehensive mathematical techniques. Two main equilibrium points of the pandemic model are stated. The basic reproduction number for the model is computed using next generation technique to handle the future dynamics of the pandemic. We develop an optimal control problem to find the best controls for the quarantine and hospitalization strategies employed on exposed and infected humans, respectively. For numerical solution of the fractional model, we implemented the Adams–Bashforth method to prove the importance of order. A general fractional‐order optimal control problem and associated optimality conditions of Pontryagin type are discussed, with the goal to minimize the number of exposed and infected humans. The extremals are obtained numerically. [ABSTRACT FROM AUTHOR]

The article deals with the analysis of the fractional COVID‐19 epidemic model (FCEM) with a convex incidence rate. Keeping in view the fading memory and crossover behavior found in many biological phenomena, we study the coronavirus disease by using the noninteger Caputo derivative (CD). Under the Caputo operator (CO), existence and uniqueness for the solutions of the FCEM have been analyzed using fixed point theorems. We study all the basic properties and results including local and global stability. We show the global stability of disease‐free equilibrium using the method of Castillo‐Chavez, while for disease endemic, we use the method of geometrical approach. Sensitivity analysis is carried out to highlight the most sensitive parameters corresponding to basic reproduction number. Simulations are performed via first‐order convergent numerical technique to determine how changes in parameters affect the dynamical behavior of the system. [ABSTRACT FROM AUTHOR]