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In this paper, biodiversity and a saturation treatment rate are introduced into a type of mosquito-borne infectious disease model with an incubation time delay. Through the dynamical analysis of the model, conditions for the existence and stability of disease-free and endemic equilibria are determined. The basic reproduction number of the model is calculated, and the condition for the existence of backward bifurcation is outlined. The study finds that under the dual influence of biodiversity and saturation treatment, the threshold characteristic of the basic reproduction number becomes invalidated. When R 0 is less than 1, the model may exhibit four equilibrium states, with both the disease-free equilibrium and the endemic equilibrium being locally stable. In this scenario, whether the virus will become extinct depends on the initial conditions. The study also finds that when the basic reproduction number R 0 is greater than 1, the stability of the model is influenced by the time delay, with Hopf bifurcation occurring at a specific time delay. In addition, another novel contribution of this paper is the formulation of an optimal control problem that takes into account the minimization of damage caused by humans to biodiversity. Based on the Pontryagin's maximum principle, the specific characteristics of the optimal control measures are given, and the optimal strategy is derived by comparing five groups of control strategies. The optimal control results highlight the synergistic effect of multiple control measures with biodiversity. Under optimal control, a significant complementary effect between medical inputs and the dilution effect of biodiversity is evident. The findings imply that maintaining high biodiversity levels can decrease the demand for medical resources in mosquito-borne disease control efforts. [ABSTRACT FROM AUTHOR]

This paper investigates rumor propagation in a multilingual environment, taking into account language usage variations. Firstly, a 2I2S2R model is proposed within a heterogeneous network framework that incorporates both immunologic and cross-transmitted mechanisms. Secondly, the paper calculates the basic reproduction number R0 by the next-generation matrix method. Thirdly, the local asymptotic stability and the global asymptotic stability are further explored, which indicate that whether the rumor continuously spreads or becomes extinct is determined by the threshold. Finally, the numerical simulation and sensitivity analysis are given to illustrate the effectiveness of theoretical results and the influence of model parameters on rumor spreading. [ABSTRACT FROM AUTHOR]

This paper studies an impact of media epidemic system with diffusion and linear source. We first derive the uniform bounds of solutions to impact on media reaction diffusion system. Then, the basic reproduction number is calculated and the threshold dynamics of impact media reaction diffusion system is also given and the Kuratowski measure κ of non-compactness is also considered. In addition, assume the spatial environment is homogeneous, it is shown that the unique endemic equilibrium of the system is global stability by constructing suitable Lyapunov function. Finally, we discuss the asymptotic profile of the system when the diffusion rate of the susceptible (infected) individuals for the system tends to zero or infinity. The main results show that the activities of infected individuals can only be at low risk, and then the virus eventually will be extinct, that is, to control the entry of viruses from abroad and increase the detection of domestic viruses. Finally, some numerical simulations are worked out to confirm the results obtained in this paper. [ABSTRACT FROM AUTHOR]

Unidentified diseases are becoming more prevalent among humans due to various climatic factors, and some of these diseases originate in animals before spreading to humans. One virus that has been of particular concern is the avian influenza virus, which primarily infects bird and can subsequently transmit to humans. This article presents a mathematical model describing the spatio-temporal reaction-diffusion process involved in the transmission of avian flu in human population. The paper begins by studying the proposed model's well-posedness and the calculated basic reproduction number, which provides valuable insights into the dynamics of virus transmission. The paper also provides stability analysis for the disease-free steady state of the model. All theoretical studies are validated using computational results. [ABSTRACT FROM AUTHOR]

This document discusses the challenges faced in mathematical modeling and controlling COVID-19 epidemics. It highlights the global impact of the pandemic on the economy and health, as well as the uneven effects on different age groups. The availability of vaccines led to mass vaccination campaigns, prompting researchers to implement new strategies and innovative approaches. The document also mentions various journals that have published special issues on COVID-19, covering topics such as forecasting, vaccination strategies, and control measures. The special issue presented in this document consists of eleven papers that analyze clinical characteristics, variant spread, economic recovery, mortality rates, non-pharmaceutical interventions, and coordinated actions. The papers propose mathematical models and control strategies to minimize disease spread and prioritize vaccination. The document emphasizes the multidisciplinary nature of the research, involving fields such as applied mathematics, data science, engineering, biology, and economics. Overall, this special issue provides valuable insights into the mathematical modeling and control of COVID-19 epidemics. [Extracted from the article]

In this paper, we formulate and analyze a compartmental model of visceral leishmaniasis (VL). We validate our model by calibrating it to the yearly VL incidence data for India and Bangladesh. The proposed model's basic reproduction number ( R 0 ) has been derived and estimated. We have proved the existence of backward bifurcation in our system. The phenomenon of backward bifurcation has public health implications because the classical requirement of R 0 < 1 , while necessary, is no longer sufficient for effective disease control (or elimination). In such a (backward bifurcation) situation, the initial sizes of the model's subpopulations (state variables) would determine the effectiveness of disease control or elimination. As a result, it is the first attempt to represent and study VL disease dynamics using active case detection (ACD) as a control strategy, although a few experimental studies have been conducted to evaluate ACD in disease transmission. We use sensitivity analysis to investigate the effects of the model's controllable parameters for the basic reproduction number. We found that σ 1 , σ 2 , σ 3 (monitoring rate to infected individuals) have negative impacts on R 0 . The numerical result suggests that the ACD strategy can be useful for the VL elimination program. Due to a small increment in the monitoring rate, the dynamic behavior of infected individuals dramatically decreased. Successful employment of ACD strategy may reduce more than 60–80% symptomatic and post kala-azar dermal leishmaniasis (PKDL) individuals. We found that healthcare organizations should prioritize ACD in symptomatic and PKDL individuals over asymptomatic individuals. We also observed that the use of only culling effect to the reservoirs is not beneficial to society in the control of VL, but spraying of insecticides and the use of treated bednets can be effective control strategies to curtail the outbreak of the disease. [ABSTRACT FROM AUTHOR]

In this paper, a nonlinear mathematical model for tuberculosis transmission, which incorporates multiple saturated exogenous reinfections, is proposed and explored. The existence of disease-free and endemic steady states is investigated. Disease-free equilibrium (DFE) is locally asymptotically stable (LAS) but not globally asymptotically stable (GAS) when the basic reproduction number, ℛ 0 < 1. However, it is GAS only when there is no exogenous reinfection. The local asymptotic stability and global asymptotic stability of the unique endemic equilibrium point (EEP) are established under certain conditions when ℛ 0 > 1. Further, the EEP is GAS when ℛ 0 > 1 , provided there is no exogenous reinfection. When ℛ 0 is below unity, the presence of multiple endemic equilibria is found which leads to backward bifurcation. It is demonstrated that the system encounters a Hopf-bifurcation when the transmission rate β crosses a critical value, resulting in the formation of limit cycles, i.e. periodic solutions bifurcate around the EEP when β passes a critical value. The stability and direction of Hopf-bifurcation are also studied. The results of the analytical work are validated through numerical simulations. A numerical simulation illustrates that EEP losses its stability via Hopf-bifurcation for specific parameters. However, when the bifurcation parameter β is increased further, the EEP regains its stability. In addition, Hopf-bifurcation occurs due to exogenous reinfection rates p and . Thus, our model shows some important nonlinear dynamical behaviors. [ABSTRACT FROM AUTHOR]

In this paper, a three-dimensional (3D) Filippov SEIS epidemic model is proposed to characterize the impact of limited medical resources on disease transmission with discontinuous treatment functions. Qualitative analysis of non-smooth dynamical behaviors are performed on two subsystems and sliding modes. Criteria on the stability of various kinds of feasible equilibria and bifurcations, e.g., saddle-node bifurcation, transcritical bifurcation, and boundary equilibrium bifurcation, are established. The theoretical results are illustrated by numerical simulation, from which we find there could exist bistable phenomena, e.g., endemic and pseudo-equilibria, endemic equilibria of the two subsystems, or endemic and disease-free equilibria, even the basic reproduction numbers of two subsystems are less than 1. The disease spread is dependent both on the limited medical resources and latent compartment, which are more beneficial to effective disease control than planar Filippov and smooth models. [ABSTRACT FROM AUTHOR]

The Zika virus (ZIKV) is a serious global public health crisis. A major control challenge is its multiple transmission modes. This paper aims to simulate the transmission patterns of ZIKV using a dynamic process-based epidemiological model written in ordinary differential equations, which incorporates the human-to-mosquito infection by bites and sewage, mosquito-to-human infection by bites, and human-to-human infection by sex. Mathematical analyses are carried out to calculate the basic reproduction number and backward bifurcation, and prove the existence and stability of the equilibria. The model is validated with infection data by applying it to the 2015–2016 ZIKV epidemic in Brazil. The results indicate that the reproduction number is estimated to be 2.13, in which the contributions by mosquito bite, sex and sewage account for 85.7%, 3.5% and 10.8%, respectively. This number and the morbidity rate are most sensitive to parameters related to mosquito ecology, rather than asymptomatic or human-to-human transmission. Multiple transmission routes and suitable temperature exacerbate ZIKV infection in Brazil, and the vast majority of human infection cases were prevented by the intervention implemented. These findings may provide new insights to improve the risk assessment of ZIKV infection. [ABSTRACT FROM AUTHOR]

In this paper, we consider a multigroup stochastic SIQR epidemic model with varying total population size. After proving the existence and uniqueness of the global solution to the system, we developed sufficient conditions for the existence of an stationary ergodic distribution of the positive solutions. Then we gave sufficient conditions for extinction of the diseases which is based on the basic reproduction number in its corresponding deterministic system. [ABSTRACT FROM AUTHOR]

In this paper, following previous works of ours, we deal with a mathematical model of hepatitis B virus (HBV) infection. We assume spatial diffusion of free HBV particles, logistic growth for both healthy and infected hepatocytes, and use the standard incidence function for viral infection. Moreover, one time delay is introduced to account for actual virus production. Another time delay is used to account for virus maturation. The existence, uniqueness, positivity and boundedness of solutions are established. Analyzing the model qualitatively and using a Lyapunov functional, we establish the existence of a threshold $ \mathcal {T}_0 $ T 0 such that, if the basic reproduction number $ \mathcal {R}_0 $ R 0 verifies $ \mathcal {R}_0\leq \mathcal {T}_0{ \lt }1 $ R 0 ≤ T 0 < 1 , the infection-free equilibrium is globally asymptotically stable. When $ \mathcal {R}_0 $ R 0 is greater than one, we discuss the local asymptotic stability of the unique endemic equilibrium and the occurrence of a Hopf bifurcation. Also, when $ \mathcal {R}_0 $ R 0 is greater than one, the system is uniformly persistent, which means that the HBV infection is endemic. Finally, we carry out some relevant numerical simulations to clarify and interpret the theoretical results. [ABSTRACT FROM AUTHOR]

Bovine tuberculosis (bTB) is a disease of major public health concern and economic importance. The disease affects the livelihood of rural livestock farmers, economies of developing countries, human and animal health. In this paper, both deterministic and continuous time Markov chain (CTMC) models are formulated and analyzed to study the dynamics of bovine tuberculosis in humans and cattle. The next generation matrix method is applied to compute the basic reproduction number R 0 for the deterministic model while the multitype branching process is applied to compute the stochastic threshold for CTMC model. The normalized forward sensitivity index method is used to compute sensitivity indices of model parameters with respect to basic reproduction number R 0 . The results reveal that infection rate among cattle is the most positive sensitive parameter whereas cattle's natural death is the most negative sensitive parameter. This suggests that more efforts should be directed at reducing number of infected cattle for possible control of the spread of bTB in human and cattle populations. Numerical simulations for the CTMC stochastic model using 10,000 sample paths is carried out to compute probabilities for disease extinction by varying initial values for infected classes. The results show that, the solutions of CTMC stochastic model are relatively close to the deterministic model solutions. The possibility of bTB extinction is high when it emanates from either humans or infectious dairy products or from a small number of M. bovis bacteria from the environment. However, the disease outbreak occurs when it emerges from infected cattle population. Moreover, the results show that if bTB emerges from infected humans, cattle, dairy product and environment at the same time, then there is disease outbreak. Thus, any intervention measure that targets on reducing number of infected cattle will have significant impact to contain bTB in human and cattle populations. [ABSTRACT FROM AUTHOR]

This paper presents and examines a COVID-19 model that takes comorbidities and up to three vaccine doses into account. We analyze the stability of the equilibria, examine herd immunity, and conduct a sensitivity analysis validated by data on COVID-19 in Indonesia. The disease-free equilibrium is locally and globally asymptotically stable whenever the basic reproduction number is less than one, while an endemic equilibrium exists and is globally asymptotically stable when the number is greater than one. Subsequently, the model incorporates two effective measures, namely public education and enhanced medical care, to determine the most advantageous approach for mitigating the transmission of the disease. The optimal control model is then determined using Pontryagin’s maximum principle. The integrated control strategy is the best method for reliably safeguarding the general population against COVID-19 infection. Cost evaluations and numerical simulations corroborate this conclusion. [ABSTRACT FROM AUTHOR]

This paper deals with the problem of the prediction and control of cholera outbreak using real data of Cameroon. We first develop and analyze a deterministic model with seasonality for the cholera, the novelty of which lies in the incorporation of undetected cases. We present the basic properties of the model and compute two explicit threshold parameters R0 and R0 that bound the effective reproduction number R0, from below and above, that is R0 ≤ R0 ≤ R0. We prove that cholera tends to disappear when R0 ≤ 1, while when R0 >1, cholera persists uniformly within the population. After, assuming that the cholera transmission rates and the proportions of newly symptomatic are unknown, we develop the EnKf approach to estimate unmeasurable state variables and these unknown parameters using real data of cholera from 2014 to 2022 in Cameroon. We use this result to estimate the upper and lower bound of the effective reproduction number and reconstructed active asymptomatic and symptomatic cholera cases in Cameroon, and give a short-term forecasts of cholera in Cameroon until 2024. Numerical simulations show that (i) the transmission rate from free Vibrio cholerae in the environment is more important than the human transmission and begin to be high few week after May and in October, (ii) 90% of newly cholera infected cases that present the symptoms of cholera are not diagnosed and (iii) 60.36% of asymptomatic are detected at 14% and 86% of them recover naturally. The future trends reveals that an outbreak appeared from July to November 2023 with the number of cases reported monthly peaked in October 2023. An impulsive control strategy is incorporated in the model with the aim to avoid or prevent the cholera outbreak. In the first year of monitoring, we observed a reduction of more than 75% of incidences and the disappearance of the peaks when no control are available in Cameroon. A second monitoring of control led to a further reduction of around 60% of incidences the following year, showing how impulse control could be an effective means of eradicating cholera. [ABSTRACT FROM AUTHOR]

The Zika virus has been shown to infect glioblastoma stem cells via the membrane receptor α v β 5 , which is activated by the stem-specific transcription factor SOX2. Since the expression level of SOX2 is an important predictive marker for successful virotherapy, it is important to understand the fundamental mechanisms of the role of SOX2 in the dynamics of cancer stem cells and Zika viruses. In this paper, we develop a mathematical ODE model to investigate the effects of SOX2 expression levels on Zika virotherapy against glioblastoma stem cells. Our study aimed to identify the conditions under which SOX2 expression level, viral infection, and replication can reduce or eradicate the glioblastoma stem cells. Analytic work on the existence and stability conditions of equilibrium points with respect to the basic reproduction number are provided. Numerical results were in good agreement with analytic solutions. Our results show that critical threshold levels of both SOX2 and viral replication, which change the stability of equilibrium points through population dynamics such as transcritical and Hopf bifurcations, were observed. These critical thresholds provide the optimal conditions for SOX2 expression levels and viral bursting sizes to enhance therapeutic efficacy of Zika virotherapy against glioblastoma stem cells. This study provides critical insights into optimizing Zika virus-based treatment for glioblastoma by highlighting the essential role of SOX2 in viral infection and replication. [ABSTRACT FROM AUTHOR]

This paper presents a novel approach to studying divorce dynamics and elimination strategies using nonlinear differential equations. A mathematical model is formulated to capture the key factors influencing divorce rates. The model undergoes a rigorous theoretical analysis, including parameter estimation, solution existence/uniqueness, positivity, boundedness, and invariant regions. A qualitative analysis explores equilibria, stability conditions, and a sensitivity analysis. Numerical simulations and discussions are presented to validate the model and shed light on divorce dynamics. Finally, conclusions and future research directions are outlined. This work offers valuable insights for understanding and potentially mitigating divorce rates through targeted interventions. [ABSTRACT FROM AUTHOR]

Measles is the most infectious disease with a high basic reproduction number ( R 0 ). For measles, it is reported that R 0 lies between 12 and 18 in an endemic situation. In this paper, a fractional order mathematical model for measles disease is proposed to identify the dynamics of disease transmission following a declining memory process. In the proposed model, a fractional order differential operator is used to justify the effect and success rate of vaccination. The total population of the model is subdivided into five sub-compartments: susceptible (S), exposed (E), infected (I), vaccinated (V), and recovered (R). Here, we consider the first dose of measles vaccination and convert the model to a controlled system. Finally, we transform the control-induced model to an optimal control model using control theory. Both models are analyzed to find the stability of the system, the basic reproduction number, the optimal control input, and the adjoint equations with the boundary conditions. Also, the numerical simulation of the model is presented along with using the analytical findings. We also verify the effective role of the fractional order parameter alpha on the model dynamics and changes in the dynamical behavior of the model with R 0 = 1 . [ABSTRACT FROM AUTHOR]

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]

Knowledge diffusion in regional innovation ecosystems is an important factor that influences the regional innovation efficiency. In regional innovation ecosystems under digital empowerment, the knowledge diffusion enables the optimal allocation of innovation resources and promotes the sustainable development and ecological evolution of regional innovation ecosystems. In this paper, a SEIR (Susceptible–Exposed–Infected–Recovered) model is proposed for knowledge diffusion in regional innovation ecosystems under digitization. The basic reproduction number of the proposed model is calculated and its stability is validated. Finally, the expressions for the optimal control system and the optimal control parameters are presented. According to the research conclusions of this paper, the knowledge-diffusion ability of innovation agents in an innovation ecosystem affects the knowledge diffusion in a system; the contact rate between innovation agents affects the efficiency of knowledge diffusion in the system and the structure of the system; the digital transmission ability of innovation agents affects the breadth of knowledge diffusion in the system; and the self-learning ability of innovation agents affects the efficiency of knowledge diffusion in the system.The digital technologies help heterogeneous innovation agents in regional innovation ecosystems to break down the knowledge silos. At the same time, the digital technologies enhance the ability of innovation agents to absorb and learn knowledge in regional innovation ecosystems under digitization, thereby increasing the infection rate of knowledge diffusion in such systems.These conclusions extend the theoretical boundaries of innovation ecosystems and knowledge diffusion and offer management implications for enterprises and governments in decision-making. [ABSTRACT FROM AUTHOR]

Today, because infectious diseases have a large impact on people, control methods for all infectious diseases are crucial. In this paper, we propose a SIQR model with Crowley-Martin type incidence rate and saturated treatment. We have incorporated precautionary measures taken by the population through the saturated incidence rate. Screening of individuals who have come into contact with infected individuals is modelled. Thus, we explore a four-dimensional mathematical model to study the dynamics of the infectious diseases and their management. Stability and bifurcation analysis ofthe system is done. The Mensitivity analysis of all parameters in relation to the basic reproduction number has been performed and the theoretical results have been validated using numerical simulation. Mathematical results in this paper suggest that both government interventions and individual efforts must be incorporated to achieve better control of emerging infectious diseases in the population. [ABSTRACT FROM AUTHOR]

The basic reproduction, or reproductive number, is a useful index that indicates whether or not there will be an epidemic. However, it is also very important to determine whether an epidemic will eventually decrease and disappear or persist as an endemic. Different infectious diseases have different behaviors and mathematical models used to simulated them should capture the most important processes; however, the models also involve simplifications. Influenza epidemics are usually short-lived and can be modeled with ordinary differential equations without considering demographics. Delays such as the infection time can change the behavior of the solutions. The same is true if there is permanent or temporary immunity, or complete or partial immunity. Vaccination, isolation and the use of antivirals can also change the outcome. In this paper, we introduce several new models and use them to find the effects of all the above factors paying special attention to whether the model can represent an infectious process that eventually disappears. We determine the equilibrium solutions and establish the stability of the disease-free equilibrium using various methods. We also show that many models of influenza or other epidemics with a short duration do not have solutions with a disappearing epidemic. The main objective of the paper is to introduce different ways of modeling immunity in epidemic models. Several scenarios with different immunities are studied since a person may not be re-infected because he/she has total or partial immunity or because there were no close contacts. We show that some relatively small changes, such as in the vaccination rate, can significantly change the dynamics; for example, the existence and number of the disease-free equilibria. We also illustrate that while introducing delays makes the models more realistic, the dynamics have the same qualitative behavior. [ABSTRACT FROM AUTHOR]

In this paper, a mathematical model for the transmission dynamics of Fasciola hepatica in cattle and snail populations is formulated and analyzed. The snail mortality rate (μs) is the most important factor that indirectly impacts the basic reproduction number (R0). A 50% change, either an increase or decrease, in the snail mortality rate will result in an approximate 50% change in the opposite direction in the value of R0. The model shows a forward bifurcation at R0 = 1, indicating that the disease dynamics undergo a critical transition at this threshold. This change signifies a transition from a disease‐free state to a persistent infection, highlighting the possibility of a continuous disease presence given specific epidemiological conditions. Simulations show that reducing miracidia, metacercariae, and snail populations, improving treatment, and lowering pathogen transfer between cattle and snails significantly decrease disease prevalence in cattle. To control the disease, transmission rates for cattle and snails must be reduced below γc = 1.4338 × 10−7 and γs = 1.1473 × 10−8, respectively. Current treatments are insufficient, and a combination of improved treatments reduced transmission rates, and increased snail mortality is recommended for better disease control. [ABSTRACT FROM AUTHOR]

In this paper, we suggest a mathematical model of COVID-19 with multiple variants of the virus under optimal control. Mathematical modeling has been used to gain deeper insights into the transmission of COVID-19, and various prevention and control strategies have been implemented to mitigate its spread. Our model is a SEIR-based model for multi-strains of COVID-19 with 7 compartments. We also consider the circulatory structure to account for the termination of immunity for COVID-19. The model is established in terms of the positivity and boundedness of the solution and the existence of equilibrium points, and the local stability of the solution. As a result of fitting data of COVID-19 in Ghana to the model, the basic reproduction number of the original virus and Delta variant was estimated to be 1.9396, and the basic reproduction number of the Omicron variant was estimated to be 3.4905, which is 1.8 times larger than that. We observe that even small differences in the incubation and recovery periods of two strains with the same initial transmission rate resulted in large differences in the number of infected individuals. In the case of COVID-19, infections caused by the Omicron variant occur 1.5 to 10 times more than those caused by the original virus. In terms of the optimal control strategy, we formulate three control strategies focusing on social distancing, vaccination, and testing-treatment. We have developed an optimal control model for the three strategies outlined above for the multi-strain model using the Pontryagin's Maximum Principle. Through numerical simulations, we analyze three optimal control strategies for each strain and also consider combinations of the two control strategies. As a result of the simulation, all control strategies are effective in reducing disease spread, in particular, vaccination strategies are more effective than the other two control strategies. In addition the combination of the two strategies also reduces the number of infected individuals by 1/10 compared to implementing one strategy, even when mild levels are implemented. Finally, we show that if the testing-treatment strategy is not properly implemented, the number of asymptomatic and unidentified infections may surge. These results could help guide the level of government intervention and prevention strategy formulation. [ABSTRACT FROM AUTHOR]

This paper investigates a nonlocal dispersal epidemic model under the multiple nonlocal distributed delays and nonlinear incidence effects. First, the minimal wave speed c ∗ and the basic reproduction number R 0 are defined, which determine the existence of traveling wave solutions. Second, with the help of the upper and lower solutions, Schauder's fixed point theorem, and limiting techniques, the traveling waves satisfying some asymptotic boundary conditions are discussed. Specifically, when ℛ 0 > 1 , for every speed c > c ∗ there exists a traveling wave solution satisfying the boundary conditions, and there is no such traveling wave solution for any 0 < c < c ∗ when ℛ 0 > 1 or c > 0 when ℛ 0 < 1. Finally, we analyze the effects of nonlocal time delay on the minimum wave speed. [ABSTRACT FROM AUTHOR]

In this paper, an SIRS epidemic model using Grunwald–Letnikov fractional-order derivative is formulated with the help of a nonlinear system of fractional differential equations to analyze the effects of fear in the population during the outbreak of deadly infectious diseases. The criteria for the spread or extinction of the disease are derived and discussed on the basis of the basic reproduction number. The condition for the existence of Hopf bifurcation is discussed considering fractional order as a bifurcation parameter. Additionally, using the Grunwald–Letnikov approximation, the simulation is carried out to confirm the validity of analytic results graphically. Using the real data of COVID-19 in India recorded during the second wave from 15 May 2021 to 15 December 2021, we estimate the model parameters and find that the fractional-order model gives the closer forecast of the disease than the classical one. Both the analytical results and numerical simulations presented in this study suggest different policies for controlling or eradicating many infectious diseases. [ABSTRACT FROM AUTHOR]

In this paper, we have delved into the intricate dynamics of a discrete-time Hepatitis B virus (HBV) model, shedding light on its local dynamics, topological classifications at equilibrium states, and pivotal epidemiological parameters such as the basic reproduction number. Our analysis extended to exploring convergence rates, control strategies, and bifurcation phenomena crucial for understanding the behavior of the HBV system. Employing linear stability theory, we meticulously examined the local dynamics of the HBV model, discerning various equilibrium states and their topological classifications. Subsequently, we identified bifurcation sets at these equilibrium points, providing insights into the system’s stability and potential transitions. Further, through the lens of bifurcation theory, we conducted a comprehensive bifurcation analysis, unraveling the intricate interplay of parameters that govern the HBV model’s behavior. Our investigation extended beyond traditional stability analysis to explore chaos and convergence rates, enriching our understanding of the dynamics of the understudied HBV model. Finally, we validated our theoretical findings through numerical simulations, confirming the robustness and applicability of our analysis in real-world scenarios. By integrating biological and epidemiological insights into our mathematical framework, we offered a holistic understanding of the dynamics of HBV transmission dynamics, with implications for public health interventions and disease control strategies. [ABSTRACT FROM AUTHOR]

In this paper, we consider a reaction–diffusion dengue model on a varying domain that monotonically increases in time and gradually approaches saturation arising from environmental change. By the upper and lower solutions, comparison principle, asymptotic autonomous semiflows and the technique of Lyapunov function, we investigate the stabilities of equilibria in terms of the basic reproduction number $ \Re _0^{\rho } $ ℜ 0 ρ . The results show that (i) if $ \Re _0^{\rho } \gt 1 $ ℜ 0 ρ > 1 , the nontrivial solutions starting from the upper and lower solutions of the model approach to the set formulated by the maximal and minimal solutions of its related elliptic problem; (ii) the disease-free equilibrium is globally asymptotically stable when $ \Re _0^{\rho } \lt 1 $ ℜ 0 ρ < 1. Comparing our problem in different settings including growing domain, fixed domain and without spatial structure, our results demonstrate that the disease can spread in the growing domain, while vanish in the fixed domain; and the spatial model decreases the transmission risk compared with the system without spatial structure. [ABSTRACT FROM AUTHOR]

This paper is devoted to the study of the global dynamics for a large class of reaction-diffusion systems with a time-varying domain. By appealing to the theories of asymptotically autonomous and periodic semiflows, we establish the threshold-type results on the long-time behavior of solutions for such a system in the cases of asymptotically bounded and periodic domains, respectively. To investigate the model system in the case of asymptotically unbounded domain, we first prove the global attractivity for nonautonomous reaction-diffusion systems with asymptotically vanishing diffusion coefficients via the method of sub- and supersolutions and then use the comparison arguments to obtain the threshold dynamics. We also apply these analytical results to a reaction-diffusion model of dengue fever transmission to investigate the effect of time-varying domain on the basic reproduction number. It turns out that the basic reproduction numbers with dengue fever transmission for the asymptotically bounded and unbounded domains are always less than that for the spatially homogeneous case, and under appropriate conditions, the basic reproduction numbers for asymptotically bounded and periodic domains are larger than or equal to that for the stationary bounded domain. [ABSTRACT FROM AUTHOR]

In this paper, we developed a mathematical model for the dynamics of coronavirus disease (COVID‐19) transmission. The model embraces the notion of contact tracing and contaminated surfaces which are vital for disease control and contribute to disease transmission, respectively. We analyzed the model properties such as the positivity of the solution, invariant region, existence, and stability nature of equilibria. Besides, we computed the basic reproduction number R0. The local stability and the global stability of disease‐free equilibrium (DFE) points are proved by using the Routh–Hurwitz criteria and the Castillo‐Chavez and Song approach, respectively. LaSalle's invariant principle is applied to prove the stability of an endemic equilibrium (EE) point. The possibility of bifurcation is discussed using the center manifold theory. We used real data on the spread and control of COVID‐19 disease in Ethiopia. Based on the data reported, we estimated the values of the parameters using the least squares method together with the fmin function in the MATLAB optimization toolbox. The sensitivity analysis of the model is explored numerically to illustrate the impact of the parameters on disease transmission. The study addressed that contact tracing is especially important because COVID‐19 often has asymptomatic carriers, and there are many asymptomatic individuals unaware in Ethiopia. The new infections would decrease in the communities by detecting and isolating COVID‐19 cases before they could spread the virus to others. Moreover, the study endorsed that the contaminated surface has contributed to disease transmission. The sensitivity analysis shows that if the rate of disinfected contaminated objects (ϕ) rises, then the transmission of the disease is reduced. Consequently, this study will aid in the fight against COVID‐19 policymakers and NGOs. It can also be used as a policy input for different countries under this crisis. Because of the mathematical modeling of this global pandemic, there is another point of view rather than public health research outputs. Additionally, with the concept of contact tracing and contaminated surfaces incorporated into the model, the result provides insight for disease prevention. [ABSTRACT FROM AUTHOR]

Background: To control resurging infectious diseases like mumps, it is necessary to resort to effective control and preventive measures. These measures include increasing vaccine coverage, providing the community with advice on how to reduce exposure, and closing schools. To justify such intervention, it is important to understand how well each of these measures helps to limit transmission. Methods: In this paper, we propose a simple SEILR (susceptible-exposed-symptomatically infectious-asymptomatically infectious-recovered) model by using a novel transmission rate function to incorporate temperature, humidity, and closing school factors. This new transmission rate function allows us to verify the impact of each factor either separately or combined. Using reported mumps cases from 2004 to 2018 in the mainland of China, we perform data fitting and parameter estimation to evaluate the basic reproduction number R 0 . As a wide range of one-dose measles, mumps, and rubella (MMR) vaccine programs in China started only in 2008, we use different vaccination proportions for the first Stage I period (from 2004 to 2008) and the second Stage II period (from 2009 to 2018). This allows us to verify the importance of higher vaccine coverage with a possible second dose of MMR vaccine. Results: We find that the basic reproduction number R 0 is generally between 1 and 3. We then use the Akaike Information Criteria to assess the extent to which each of the three factors contributed to the spread of mumps. The findings suggest that the impact of all three factors is substantial, with temperature having the most significant impact, followed by school opening and closing, and finally humidity. Conclusion: We conclude that the strategy of increasing vaccine coverage, changing micro-climate (temperature and humidity), and closing schools can greatly reduce mumps transmission. [ABSTRACT FROM AUTHOR]

In this paper, we construct an infectious disease model with a testing compartment and analyze the existence and stability of its endemic states. We obtain the basic reproduction number, R 0 , and demonstrate the existence of one endemic equilibrium without testing and one endemic equilibrium with testing and prove their local and global stabilities based on the value of the basic reproduction number, R 0 . We then apply our model to the US COVID-19 pandemic and find that, for a large parameter set, including those relevant to the SARS-CoV-2 virus, our analytic and numerical results suggest that the trajectories will be trapped to the testing-free state when the testing number is small enough. This indicates that the pandemic may end with a testing-free endemic state through a novel and surprising mechanism called stochastic trapping. [ABSTRACT FROM AUTHOR]

Mathematical models play a crucial role in controlling and preventing the spread of diseases. Based on the communication characteristics of diseases, it is necessary to take into account some essential epidemiological factors such as the time delay that takes an individual to progress from being latent to become infectious, the infectious age which refers to the duration since the initial infection and the occurrence of reinfection after a period of improvement known as relapse, etc. Moreover, age-structured models serve as a powerful tool that allows us to incorporate age variables into the modeling process to better understand the effect of these factors on the transmission mechanism of diseases. In this paper, motivated by the above fact, we reformulate an SEIR model with relapse and age structure in both latent and infected classes. Then, we investigate the asymptotic behavior of the model by using the stability theory of differential equations. For this purpose, we introduce the basic reproduction number R 0 of the model and show that this threshold parameter completely governs the stability of each equilibrium of the model. Our approach to show global attractivity is based on the fluctuation lemma and Lyapunov functionals method with some results on the persistence theory. The conclusion is that the system has a disease-free equilibrium which is globally asymptotically stable if R 0 < 1 , while it has only a unique positive endemic equilibrium which is globally asymptotically stable whenever R 0 > 1 . Our results imply that early diagnosis of latent infection with decrease in both transmission and relapse rates may lead to control and restrict the spread of disease. The theoretical results are illustrated with numerical simulations, which indicate that the age variable is an essential factor affecting the spread of the epidemic. [ABSTRACT FROM AUTHOR]

In this paper, a dynamic HIV model with cell-to-cell transmission, two immune responses, and induced apoptosis is proposed and studied. First, the non-negativity and boundedness of the solutions of the model are given, and then the exact expression of the basic reproduction number R0 is obtained by using the next generation matrix method. Second, criteria are obtained for the local stability of the disease-free equilibrium, immune response-free equilibrium, and the infected equilibrium with both humoral and cellular immune responses. Furthermore, the threshold conditions are also derived for the global asymptotic stability of the disease-free equilibrium, immune responsefree equilibrium, and the infected equilibrium with both humoral and cellular immune responses by constructing the suitable Lyapunov function. Finally, some numerical simulations are conducted to verify the theoretical results; the numerical simulation results show that the increase of apoptosis rate had a positive role in the control of viral infection. [ABSTRACT FROM AUTHOR]

Immunotherapy is a targeted therapy that can be applied to cervical cancer patients to prevent DNA damage caused by human papillomavirus (HPV). The HPV infects normal cervical cells withing a specific cell age interval, i.e., between the G1 to S phase of the cell cycle. In this study, we developed a new mathematical model of age-dependent immunotherapy for cervical cancer. The model is a four-dimensional first-order partial differential equation with time- and age-independent variables. The cell population is divided into four sub-populations, i.e., susceptible cells, cells infected by HPV, precancerous cells, and cancer cells. The immunotherapy term has been added to precancerous cells since these cells can experience regression if appointed by proper treatments. The immunotherapy process is closely related to the rate of T-cell division. The treatment works in the same cell cycle that stimulates and inhibits the immune system. In our model, immunotherapy is represented as a periodic function with a small amplitude. It is based on the fluctuating interaction between Tcells and precancerous cells. We have found that there are two types of steady-state conditions, i.e., infection-free and endemic. The local and global stability of an infection-free steady-state has been analyzed based on basic reproduction numbers. We have solved the Riccati differential equation to show the existence of an endemic steady-state. The stability analysis of the endemic steady-state has been determined by using the perturbation approach and solving integral equations. Some numerical simulations are also presented in this paper to illustrate the behavior of the solutions. [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]

Gumboro disease is a viral poultry disease that causes immune suppression on the infected birds leading to poor production, mortality, and exposure to secondary infections, hence a major threat in the poultry industry worldwide. A mathematical model of the transmission dynamics of Gumboro disease is developed in this paper having four compartments of chicken population and one compartment of Gumboro pathogen population. The basic reproduction number R og is derived, and the dynamical behaviors of both the disease-free equilibrium (DFE) and endemic equilibrium are analyzed using the ordinary differential equation theory. From the analysis, we found that the system exhibits an asymptotic stable DFE whenever R og < 1 and an asymptotic stable EE whenever R og > 1. The numerical simulation to verify the theoretical results was carried out using MATLAB ode45 solver, and the results were found to be consistent with the theoretical findings. [ABSTRACT FROM AUTHOR]

In this paper, a new generalized non-monotonic and saturated incidence rate was introduced into a susceptible-infected-susceptible (SIS) epidemic model to account for inhibitory effect and crowding effect. The dynamic properties of the model were studied by qualitative theory and bifurcation theory. It is shown that when the influence of psychological factors is large, the model has only disease-free equilibrium point, and this disease-free equilibrium point is globally asymptotically stable; when the influence of psychological factors is small, for some parameter conditions, the model has a unique endemic equilibrium point, which is a cusp point of co-dimension two, and for other parameter conditions the model has two endemic equilibrium points, one of which could be weak focus or center. In addition, the results of the model undergoing saddle-node bifurcation, Hopf bifurcation and Bogdanov–Takens bifurcation as the parameters vary were also proved. These results shed light on the impact of psychological behavior of susceptible people on the disease transmission. [ABSTRACT FROM AUTHOR]

This paper is concerned with the principal spectral theory of time-periodic cooperative systems with nonlocal dispersal and Neumann boundary condition. First we present a sufficient condition for the existence of principal eigenvalues by using the theory of resolvent positive operators with their perturbations. Then we establish the monotonicity of principal eigenvalues with respect to the frequency and investigate the limiting properties of principal eigenvalues as the frequency tends to zero or infinity. We also study the effects of dispersal rates and dispersal ranges on the principal eigenvalues, and the difficulty is that principal eigenvalues of time-periodic cooperative systems with Neumann boundary conditions are not monotone with respect to the domain. Finally, we apply our theory to a man-environment-man epidemic model and consider the impacts of dispersal rates, frequency, and dispersal ranges on the basic reproduction number and positive time-periodic solutions. [ABSTRACT FROM AUTHOR]

This paper explores the dynamics of COVID-19 transmission, particularly focusing on super-spreaders, through the lens of the SIR model. The model comprises six compartments representing susceptible, exposed, symptomatic infected, super-spreader, asymptomatic infected, and recovered individuals. Utilizing a set of non-linear, interdependent differential equations, we numerically solve for model parameters to examine the influence of super-spreaders on infection spread within the population. We calculate the basic reproduction number (R0) and discuss the stability of disease-free equilibrium. Our findings underscore the significant role played by super-spreaders and asymptomatic individuals in disease dissemination. Drawing on the epidemiological concept of the iceberg phenomenon, we offer insights into super-spreader events (SSEs) in India and their ramifications. [ABSTRACT FROM AUTHOR]

In this paper, an epidemic SI model with n -infectious stages is studied. Lyapunov functions are used to conduct the global stability analysis for equilibrium points. The n -basic reproduction ratios R 1 , R 2 , ..., R n are computed, and the basic reproduction number ( R 0 ) is the max value between this ratios is obtained. For, j = 1 , 2 ,... , n when R j is less than one, all strains die out, and if it is greater than one, then persists. The disease-free and endemic equilibrium points are found, and we studied the global stability for them by using the direct Lyapunov functions. The Maple program is used for carrying a numerical simulations to support the analytically results. [ABSTRACT FROM AUTHOR]

A time-delayed virus dynamic model is proposed with general monotonic incidence, different nonlinear CTL (cytotoxic T lymphocyte) responses [CTL elimination function p y g 1 (z) and CTL stimulation function c y g 2 (z) ], and immune impairment. Indeed, the different CTL responses pose challenges in obtaining the dissipativeness of the model. By constructing appropriate Lyapunov functionals with some detailed analysis techniques, the global stability results of all equilibria of the model are obtained. By the way, we point out that the partial derivative f v (x , 0) is increasing (but not necessarily strictly) in x > 0 for the general monotonic incidence f (x , v). However, some papers defaulted that the partial derivative was strictly increasing. Our main results show that if the basic reproduction number R 0 ≤ 1 , the infection-free equilibrium E 0 is globally asymptotically stable (GAS); if CTL stimulation function c y g 2 (z) = 0 for z = 0 and the CTL threshold parameter R 1 ≤ 1 < R 0 , then the immunity-inactivated infection equilibrium E 1 is GAS; if the immunity-activated infection equilibrium E + exists, then it is GAS. Two specific examples are provided to illustrate the applicability of the main results. The main results acquired in this paper improve or extend some of the existing results. [ABSTRACT FROM AUTHOR]

Non-pharmaceutical interventions (NPIs) are effective in mitigating infections during the early stages of an infectious disease outbreak. However, these measures incur significant economic and livelihood costs. To address this, we developed an optimal control framework aimed at identifying strategies that minimize such costs while ensuring full control of a cross-regional outbreak of emerging infectious diseases. Our approach uses a spatial SEIR model with interventions for the epidemic process, and incorporates population flow in a gravity model dependent on gross domestic product (GDP) and geographical distance. We applied this framework to identify an optimal control strategy for the COVID-19 outbreak caused by the Delta variant in Xi'an City, Shaanxi, China, between December 2021 and January 2022. The model was parameterized by fitting it to daily case data from each district of Xi'an City. Our findings indicate that an increase in the basic reproduction number, the latent period or the infectious period leads to a prolonged outbreak and a larger final size. This indicates that diseases with greater transmissibility are more challenging and costly to control, and so it is important for governments to quickly identify cases and implement control strategies. Indeed, the optimal control strategy we identified suggests that more costly control measures should be implemented as soon as they are deemed necessary. Our results demonstrate that optimal control regimes exhibit spatial, economic, and population heterogeneity. More populated and economically developed regions require a robust regular surveillance mechanism to ensure timely detection and control of imported infections. Regions with higher GDP tend to experience larger-scale epidemics and, consequently, require higher control costs. Notably, our proposed optimal strategy significantly reduced costs compared to the actual expenditures for the Xi'an outbreak. Author summary: In the early stage of the outbreak of an emerging infectious disease, non-pharmaceutical interventions (NPIs) are the most effective way to control the spread of the disease given unavailability of effective medicines or vaccines. However, the implementation of NPIs may have a certain negative impact on the society and economy. There are many challenges to designing an effective and economical control regime on the basis of spatial, populate and economical heterogeneity. We developed a modelling framework to couple transmission dynamics, all possible control measures and the spatial mobility, aiming at controlling the outbreak spreading across regions and reducing the economic losses caused by control measures. We applied this framework to identify the optimal control strategy for the COVID-19 outbreak in Xi'an City, China, between December 2021 and January 2022. We analyzed the key factors related to the outbreak size and control cost, and the results showed that the outbreak size and control cost were significantly correlated with the transmissibility of the virus and the demographic and economic conditions of the epidemic area. Our results demonstrate that optimal control regimes exhibit spatial, economic, and population heterogeneity. [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]

The spatial structure of a host population has a profound effect on the dynamics of infectious diseases. The basic reproduction number, a central quantity in the study of epidemic dynamics, is affected by host clustering as well as host density. Several authors have developed methods to quantify the basic reproduction number in a spatially structured host population. The methods used and the expressions derived are however difficult to apply to real life spatial host structures. In this paper we introduce an explicit expression for the basic reproduction number using the O‐ring statistic, developed in spatial statistics, that quantifies the host density as a function of the distance from a randomly selected host individual. The O‐ring statistic is frequently used in the study of the ecology of spatially structured plant populations, being a convenient summary of the properties of a landscape by way of a single function. The connection we develop between spatial statistics and epidemic dynamics can be used to study the effect of host spatial pattern on the basic reproduction number of infectious diseases. As well as showing how explicit expressions for the basic reproduction number can be derived for landscapes with standard structures, our expression for the basic reproduction number is tested against a simulation model. The model structure in our simulation is motivated by the spread of a plant disease epidemic, although it is applicable more broadly. The agreement between our analytic expression for the basic reproduction number and the corresponding numeric quantity extracted from simulations is close to perfect across a wide range of landscape structures and model parameterisations, and including cases in which more than one species of host is at risk of infection. [ABSTRACT FROM AUTHOR]

In this paper, we introduce an age-structured SIS (susceptible-infectious-susceptible) epidemic model with random diffusion and advection, in which birth and transmission rates depend on individuals. First, the well-posedness of this model was obtained. Next, we established the existence and uniqueness of the nontrivial nonnegative steady state, and derived the basic reproduction number which is also a threshold for the existence of nontrivial nonnegative steady states. Then, we studied the local stability of the nontrivial nonnegative steady state. In particular, we determined the upper and lower bounds of the principal eigenvalue to better obtain the local stability. Finally, we investigated the asymptotic profiles of the principal eigenvalue and the nontrivial nonnegative steady state with respect to diffusion rate and advection rate, respectively. [ABSTRACT FROM AUTHOR]

In this paper, we study an SIS epidemic reaction-diffusion model with mass action infection mechanism and subject to homogeneous Dirichlet boundary conditions. We define a basic reproduction number $ \mathcal{R}_0 $ for the model and prove that there exists a unique endemic equilibrium if $ \mathcal{R}_0> 1 $. We then establish the global attractivity of the disease-free equilibrium and the endemic equilibrium for a special case. Furthermore, we analyze the asymptotic profiles of the endemic equilibrium. We show that limited movement of the individuals cannot eradicate the disease because of a varying total population, while large population mobility can cause the disease extinction due to the hostile exterior environment. [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]