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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]

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]

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]

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]

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, 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]

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]

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]

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]

Considering the media coverage and age‐dependent education, a deterministic and a stochastic class‐age‐structured rumor propagation models are studied, respectively. First, the deterministic rumor propagation model is characterized by a coupled system of ordinary and partial differential equations. The positivity and boundness of solutions are proved, and the basic reproduction number is derived. Second, the stochastic rumor propagation model is formulated by stochastic differential equations. The existence of global positive solutions in model is discussed with the Itô's formula and stochastic Lyapunov function. Additionally, by utilizing comparison principle of stochastic differential equations and the strong law of large numbers, several sufficient conditions for extinction and persistence of the rumor are derived. Finally, numerical simulations are carried out for illustrating the results. [ABSTRACT FROM AUTHOR]

We formulate susceptible–infectious–susceptible (SIS) mathematical models to assess health education roles in enhancing human papillomavirus (HPV) vaccines uptake in both childhood and adult ages. The model is considered in logistically growing populations. We derive the basic reproduction number for the model and use it to describe the local‐asymptotic stability of the disease‐free equilibrium (DFE). The contribution of educated and uneducated in basic reproduction number is estimated too. We estimate the time series solution of the model for uneducated, low educated and high educated scenarios. The higher the education level the more the vaccine uptake, the less the risky behavior, and the shorter the time needed to control HPV. Late education for adults is more effective than early education for children as late education is applied with vaccine and that could increase vaccine uptake for parents and their kids. Moreover, we introduce an optimal problem to search for the best introduction of early and late educations that maximize the vaccines uptake and control HPV with the least possible cost. In the time period 20 years, the time‐dependent optimal educations are as follows: Full coverage of early education for the full period regardless of vaccine efficacy used. This needs to be supported with maximum education rate (one unit) for the first 13 years for vaccine with 90% efficacy, which needs to be extended for three more years when vaccine efficacy is reduced to 65%. The optimal educations have almost the same cost and the same effect in reducing HPV cases regardless of vaccine efficacy. This means that increasing health education is a viable low‐cost strategy to support vaccines that have low efficacy. [ABSTRACT FROM AUTHOR]

A malaria transmission disease model with host selectivity and insecticide‐treated bed nets (ITNs), as an intervention for controlling the disease, is formulated. Since the vector is an insect, the vector time scale is faster than the host time scale. This leads to a singularly perturbed model with two distinctive intrinsic time scales, two‐slow for the host and one‐fast for the vector. The basic reproduction number ℜo$$ {\mathfrak{\Re}}_{\mathfrak{o}} $$ is calculated, and the local stability analysis is performed at equilibria of the model when the perturbation parameter ϵ>0$$ \epsilon >0 $$. The model is analyzed when ϵ→0$$ \epsilon \to 0 $$ using asymptotic expansions technique. The results show that if over 30% of humans use ITNs, then ℜo$$ {\mathfrak{\Re}}_{\mathfrak{o}} $$ can be reduced below 1, and hence, malaria disease can be eliminated. In addition, the dynamics on the slow surface indicate that the infected vectors decay fast when ϵ=0.001$$ \epsilon =0.001 $$ according to the numerical simulations. [ABSTRACT FROM AUTHOR]

Tuberculosis was ranked No. 3 among 29 types of infectious diseases in January 2022 of Jiangsu Province, in the southeast of China, and it has strong seasonality whose incidence rate has been high in springs and low in winters. This comprehensive study extends the spreading dynamics to the control strategies of TB in Jiangsu Province with time‐varying transmission rate, which is the first one in this type. The periodic transmission rate is incorporated to the classic SEIR model with susceptible, latent, infectious, and recovered compartments. The analysis shows that when the basic reproduction number R0$$ {R}_0 $$ is less than one, the epidemic will eventually disappear but, if R0$$ {R}_0 $$ is larger than one, the disease may persist with a periodic solution. The R0$$ {R}_0 $$ value of Jiangsu TB is calculated as 1.297. In addition, the model predicts that the epidemic will not disappear over time and there will be a major outbreak after certain years. To eradicate TB, this work introduces three control strategies on susceptible, latent, and infectious populations. It turns out that the optimal control solutions can reduce the incidence rate in 2035 by 90% from 2015, the goal of World Health Organization. Simultaneously, we find that the optimal controls have the same seasonality as the transmission rate. This indicates that Jiangsu Province needs to invest more control efforts in springs than in winters. In addition, the cost of control strategies is analyzed in terms of the unit cost, total cost, and incremental cost‐effectiveness. Summarizing all results, we recommend targeting the latent patients, or both latent and infectious individuals, with the transmission rate‐synchronized control strength as the most effective control strategy, to mitigate or even eradicate TB disease in Jiangsu Province. [ABSTRACT FROM AUTHOR]

Vector–host diseases outbreak is a major public health concern, and it has greatly affected human health and economy in various regions around the globe. Different approaches have been adopted to investigate the dynamical behavior and possible control of these diseases. In this study, we present a compartmental transmission model in order to explore the dynamics of vector–host infectious diseases. The saturated incidence rate instead of bilinear (or standard) and saturated treatment function is considered in model formulation which enhance the biological suitability of the proposed model. We first formulate the model based on nonlinear classical integer‐order differential equations. Then, the proposed integer‐order model is reformulated using the fractional‐order operator in Caputo–Fabrizio sense with nonsingular kernel. We investigate the model equilibria and evaluate the expression for the most important threshold parameter known as the basic reproduction number. Furthermore, the existence and uniqueness are presented via the fixed point approach. Additionally, using an efficient numerical scheme, the iterative solution of the model is obtained. Finally, we present the model simulations to illustrate the impact of arbitrary fractional order and some of other important parameters involved in the model on the disease dynamics and minimization. [ABSTRACT FROM AUTHOR]

Health education directly affects the different behaviors of individuals who visit and do not visit after illness by reporting the number of patients and their changing rate. As a result, visiting and non‐visiting patients have different infection rates, cure rates, and disease‐related mortality. In response to this impact, this article establishes a model to further analyze the impact of health education on the spread of diseases. We divide patients into visiting and non‐visiting patients and introduce a weighting function that depends on the number of visiting patients and their changing rate. We constructs an implicit differential equations and use the properties of Lambert W function and its inverse function to transform it into piecewise smooth systems with explicit definitions. According to the analysis method of switched system, the two subsystems are analyzed qualitatively. The results show that health education cannot affect the basic reproduction number of disease transmission, but it causes the system to produce many types of equilibria. In addition, the difference in the healing rate and self‐healing rate between visiting and non‐visiting patients will affect the number of visiting and non‐visiting patients and the stability of the equilibria. Moreover, the system will generate fold branch. In addition, the patient's healing/self‐healing rate and the implementation of health education will affect the appearance of virtual or false equilibrium. Numerical simulation and sensitivity analysis further confirm that the total number of visiting and non‐visiting patients will decrease with the increase of health education. Finally, our research results show that health education can reduce the infection rate only if there are enough susceptible people receiving health education. [ABSTRACT FROM AUTHOR]

Quarantine of those individuals who are suspected of being infected is one of the intervention measures to contain the spread of an infectious disease. We propose an age structured SEQIR (S—Susceptible, E—Exposed, Q—Quarantine, I—Infected, R—Recovered) model with vaccination of susceptible and exposed individuals. Firstly without vaccination, basic reproduction number is derived by using the presence of endemic steady state. With an appropriate Lyapunov function, stability of disease free equilibrium point is checked. Vaccinating a population with reduction in number of infected individuals at minimal cost is considered as an optimization problem. We show that the vaccination strategy is concentrated on atmost three age classes. We also solve our model numerically to study the effect of quarantine and for better illustrations of our analytical findings. [ABSTRACT FROM AUTHOR]

The existing research models of SIQRS are often defined by linear incidence, ignoring the impact of lagging public awareness during outbreaks of infectious diseases. Their research results often deviate from the actual situation. For decision‐makers, the delay of quarantine measures according to its formulation is liable to bring losses. Given this, we establish a new class of SIQRS models based on the nonlinear incidence and focus on the effect of time delay on the propagation characteristics of the system. Firstly, we calculate the basic reproduction number of the transmission dynamics model to obtain the expression of the transmission threshold. Secondly, by constructing suitable Lyapunov functions, we prove the stability of the disease‐free equilibrium point and endemic equilibrium point of the system. Thirdly, we add model immune parameters to study three different immunization strategies, namely, uniform immunization, targeted immunization, and acquaintance immunization. Studies show that compared with uniform immunization, targeted and acquaintance immunization have better control effects. When the immunization ratio increases, the number of infected people decreases, and immune factors can control the spread of the disease. Meanwhile, we calculate the optimal quarantine level value u*≈0.56 to solve the problem of financial costs in quarantine measures to control infectious diseases. Finally, we verify conclusions of the theoretical research by numerical simulation and further discover that when an infection is in the high‐density outbreak, the time delay will lead to a slower convergence rate of the disease. And with the increase of time delay, the control cost Ju* will also increase gradually. [ABSTRACT FROM AUTHOR]

In ecosystem, the most critical issue is the extinction and persistence of population. For this reason, various deterministic and stochastic models have been studied in the past decades. However, the effect of transitions which is caused by either internal or external environmental noise has given less attention. In this work, we introduce a robust and an efficient numerical scheme based on Legendre spectral collocation method (LSCM) to explore the asymptomatic behavior of stochastic influenza avian model. We consider the model in which both birds and human population are exposed class, while for the human population, the asymptomatic class is also considered, taking into account that the asymptomatic human beings may get re‐infected and migrate to symptomatic class. A fragmented treatment gives more consideration as they relocate symptomatic individuals towards the asymptomatic class. A generation approach is utilized to figure out the essential propagation number (called reproduction number) denoted by R0. For R0 < 1, the system is asymptotically stable locally and has an ailment disease free equilibrium (DFE) and may have up to five endemic equilibria. Our simulations results recommend that the pace of complete predominance of influenza will be high, if all the individuals show symptoms upon infection and experience a deficient treatment. Further, the predominance pace of influenza is low, if all the individuals first move to symptomatic class and then treated effectively. An average rate of total prevalence is secured when more individuals upon disease move towards asymptomatic class. Our numerical simulations justified the theoretical results. [ABSTRACT FROM AUTHOR]

As the pandemic of Coronavirus Disease 2019 (COVID‐19) rages worldwide, accurate modeling of the dynamics thereof is essential. However, since the availability and quality of data vary dramatically from region to region, accurate modeling directly from a global perspective is difficult. Nevertheless, via local data collected by certain regions, it is possible to develop accurate local prediction tools, which may be coupled to develop global models. In this study, we analyze the dynamics of local outbreaks of COVID‐19 via a system of ordinary differential equations (ODEs). Utilizing a large amount of data available from the ebbing outbreak in Hubei, China, as a testbed, we predict the trajectory of daily cases, daily deaths, and other features of the Hubei outbreak. Through numerical experiments, we observe the effects of social distancing on the dynamics of the outbreak. Using knowledge gleaned from the Hubei outbreak, we apply our model to analyze the dynamics of the outbreak in Turkey. We provide forecasts for the peak of the outbreak and the daily number of cases and deaths in Turkey, by varying levels of social distancing and the transition rate which is from infected class to confirmed infected class. [ABSTRACT FROM AUTHOR]

We extend an ordinary differential equation (ODE) model for the meningococcal meningitis disease to a partial differential equation (PDE). This extension istwofold: (i) consideration of two modes of contamination, namely through a direct transmission (human to human) and an indirect transmission (via free‐living bacteria). (ii) consideration of the human movement and the dispersal of bacteria in a heterogeneous environment. Furthermore we show the existence of the solutions and define the basic reproduction number R0 for the model. Additionally, the existence and the stability of both disease‐free and endemic equilibria were investigated. The sensitivity of R0 and the endemic state with respect of the model parameters were studied. Numerical computations of the spreading bacteria are carried out to illustrate our mathematical results. [ABSTRACT FROM AUTHOR]

To reflect the impacts of vertical transmission in human and mosquitoes on the transmission of dengue fever, we propose an SIS‐SI dengue fever model with time periodic in spatially heterogeneous environments. Firstly, we obtain the properties of the basic reproduction number R0 for the dengue model by the eigenvalue problem and the spectral radius of next infection operator. Secondly, we investigate the existence and attractivity of the positive ω‐periodic solution for the dengue fever model. We also investigate the stability of ω‐periodic solution. The theoretical results indicate that vertical transmission can enhance the persistence of dengue virus in spatially heterogeneous environments with time periodic when R0 > 1. We perform numerical simulations to illustrate theoretical results of dengue fever model and give some epidemiological explanations. Finally, we combine medical advice to seek some effective prevention and control measures for vertical transmission. [ABSTRACT FROM AUTHOR]