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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.

This paper presents a fractional mathematical model for analyzing the dynamics of Ebola virus diseases. The model consists of five categories: individuals with low susceptibility to the virus S1(t), individuals with high susceptibility S2(t), infected individuals I(t), exposed individuals E(t), and recovered R(t) individuals. In this study, we have proven the positive bounded solutions for the model under consideration. Additionally, it assesses the durability of a state without disease by utilizing the fundamental reproduction number. Moreover, an examination is conducted to assess the stability of the model. In order to validate and exemplify the research, a computational model is employed, and its results correspond with the analysis outlined in the paper.

The prevention and control of the spread of Cystic Echinococcosis is an important public health issue. Health education has been supported by many governments because it can increase public awareness of echinococcosis, promote the development of personal hygiene habits, and subsequently reduce the transmission of echinococcosis. In this paper, a dynamic model of echinococcosis is used to integrate all aspects of health education. Theoretical analysis and numerical model fitting were used to quantitatively analysed by the impact of health education on the spread of echinococcosis. Theoretical findings indicate that the basic reproduction number is crucial in determining the prevalence of echinococcosis within a given geographical area. The parameters of the model were estimated and fitted by using data from the Ningxia Hui Autonomous Region in China, and the sensitivity of the basic reproduction number was analysed by using the partial rank correlation coefficient method. These findings illustrate that all aspects of health education demonstrate a negative correlation with the basic reproduction number, suggesting the effectiveness of health education in reducing the basic reproduction number and mitigating the transmission of echinococcosis, which is consistent with reality. Particularly, the basic reproduction number showed a strong negative correlation with the burial rate of infected livestock ($ b $) and the incidence of infected livestock viscera that is not fed to dogs ($ q $). This paper further analyzes the implementation plan for canine deworming rates and sheep immunity rates, as well as the transmission of infected hosts over time under different parameters $ b $ and $ q $. According to the findings, emphasizing the management of infected livestock in health education has the potential to significantly reduce the risk of echinococcosis transmission. This study will provide scientific support for the creation of higher quality health education initiatives.

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 R¯0 and R̲0 that bound the effective reproduction number R0, from below and above, that is R̲0≤R0≤R¯0. We prove that cholera tends to disappear when R¯0≤1, while when R̲0>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.

In this paper, we focus on exploring a mathematical model that captures the dynamics of Corona Virus Disease (COVID-19), particularly emphasizing the influence of quarantine measures and the efficiency of detection and diagnosis protocols. Our study aims to comprehensively analyze the model, offering valuable insights and suggestions for effectively addressing the ongoing pandemic situation in various countries. The mathematical model under scrutiny is deterministic, and our approach involves a detailed examination of its equilibria. We rigorously employ methods to identify these points of equilibrium and subsequently undertake a thorough analysis of their stability. This analytical process is crucial in gaining a deeper understanding of the system’s behavior under various conditions, laying the foundation for informed recommendations to manage and mitigate the effects of the pandemic. To enhance the robustness of our findings, we complement our analytic results with numerical simulations. This multi-faceted approach allows us not only to understand the theoretical implications of the model but also to observe and validate its behavior in a simulated environment. The combination of mathematical analysis and numerical simulations strengthens the reliability and applicability of our results, contributing to a more comprehensive understanding of the dynamics at play in the context of COVID-19. Finally, we also discuss the optimal control approach in this paper to bound the epidemic.

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

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 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 $${\mathfrak R}_0$$ 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 $${\mathfrak R}_0$$ 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.

In this paper, an age-space structured disease model with age-dependent relapse rate is investigated. We first prove the well-posedness of the model including the existence and uniqueness of the solution, positivity, and boundedness. By performing the Laplace transformation to renewal equation, we derive the next generation operator, whose spectral radius is defined as the basic reproduction number. By checking the distribution of the roots of the characteristic equation, exploring the strong persistence property of the solution and designing the Lyapunov functionals, we establish the local and global dynamics of the model.

Measles, a highly contagious airborne disease, remains endemic in many developing countries with low vaccination coverage. In this paper, we present a deterministic mathematical compartmental model to analyze the dynamics of measles. We establish global stability conditions for both disease-free and endemic equilibria using the Lyapunov functional stability method. By using arbitrary parameters, we find that the proposed model exhibits forward bifurcation. To simulate the solution of the model for the forward problem, we perform numerical integration using MATLAB software. Moreover, we calibrate the model with real data from Ethiopia and estimate the parameters along with a 95 percent confidence interval (CI) by formulating an inverse problem. It is noteworthy that our model fits well with the actual data from Ethiopia. The estimated basic reproduction number (R0) is determined to be R0=1.3973, demonstrating the endemic status of the disease. Additionally, our local sensitivity analysis indicates that reducing the transmission rate and increasing vaccination coverage can effectively minimize R0.

Abstract 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.

In this paper, a fractional order HIV/HTLV co-infection model with HIV-specific antibody immune response is established. Two cases are considered: constant control and optimal control. For the constant control system, the existence and uniqueness of the positive solutions are proved, and then the sufficient conditions for the existence and stability of five equilibriums are obtained. For the second case, the Pontryagin's Maximum Principle is used to analyze the optimal control, and the formula of the optimal solution are derived. After that, some numerical simulations are performed to validate the theoretical prediction. Numerical simulations indicate that in the case of HIV/HTLV co-infection, the concentration of $CD4^{+}T$ cells is no longer suitable as an effective reference data for understanding the development process of the disease. On the contrary, the number of HIV virus particles should be used as an important indicator for reference.

Understanding population migration is essential for controlling highly infectious diseases. This paper studies the global dynamics of an infectious disease epidemic model incorporating population migration and a stochastic transmission rate. Our findings demonstrate that in deterministic and stochastic environments, the models exhibit global Lyapunov stability near the disease-free equilibrium point, determined by a threshold parameter. Furthermore, we analyze the effect of migration on infectious diseases. We discover that the number of infections and the peak value of the infection curve increase with a higher level of population migration. These results are supported by numerical illustrations that hold epidemiological relevance. Additionally, the disease-free equilibrium of the associated time delay model is linearly asymptotically stable, and the endemic equilibrium exhibits more bifurcation for larger time delay values.

In this paper, a deterministic SIR plant mathematical model is proposed and analysed with the application of pesticides as a control measure. The primary purpose of this model is to study the role of pesticides in controlling disease prevalence in plant populations. The total plant population is subdivided into three categories: susceptible, infected, and recovered. Pesticides are considered to be applied to both susceptible and infected populations to prevent the spread of infection to unaffected plant populations. It is considered that plant populations can be recovered only through the use of pesticides. To ensure the biological validity and well-defined nature of the model, the positivity, boundedness, uniqueness and existence of solutions are analysed. The basic reproduction number (R0) of the infection is determined and observed that the disease-free equilibrium state is locally asymptotically stable whenever (R0) is less than unity and unstable otherwise. The sensitivity analysis of the basic reproduction number is carried out, and it is observed that the value of R0 decreases as the value of the death rate and the recovery rate of plants increases. Moreover, it is revealed that above a critical parameter value of the infective induce rate, the population starts oscillating periodically, and the endemic equilibrium state becomes unstable. Finally, numerical simulations are conducted in MATLAB software to compare the analytical findings. Overall, the results obtained from this analysis are both novel and significant, making them an intriguing and potentially valuable contribution to the field of theoretical ecology.

Anthrax is a bacterial infection caused by Bacillus anthracis, primarily affecting animals and occasionally affecting humans. This paper presents two compartmental deterministic models of anthrax transmission having vaccination compartments. In both models, a nonlinear ratio-dependent disease transmission function is employed, and the latter model distinguishes itself by incorporating fractional order derivatives, which adds a novel aspect to the study. The basic reproduction number $ \mathcal{R}_0 $ of the epidemic is determined, below which the disease is eradicated. It is observed that among the various parameters, the contact rate, disease-induced mortality rate, and rate of animal recovery have the potential to influence this basic reproduction number. The endemic equilibrium becomes disease-free via transcritical bifurcations for different threshold parameters of animal recovery rate, disease-induced mortality rate and disease transmission rate, which is validated by utilizing Sotomayor's theorem. Numerical simulations have revealed that a higher vaccination rate contributes to eradicating the disease within the ecosystem. This can be achieved by effectively controlling the disease-induced death rate and promoting animal recovery. The extended fractional model is analyzed numerically using the Adams-Bashforth-Moulton type predictor-corrector scheme. Finally, it is observed that an increase in the fractional order parameter has the potential to reduce the time duration required to eradicate the disease from the ecosystem.

In this paper, we investigate a time-delayed vector-borne disease model with impulsive culling of the vector. The basic reproduction number $ \mathcal{R}_0 $ of our model is first introduced by the theory recently established in [1]. Then the threshold dynamics in terms of $ \mathcal{R}_0 $ are further developed. In particular, we show that if $ \mathcal{R}_0 < 1 $, then the disease will go extinct; if $ \mathcal{R}_0 > 1 $, then the disease will persist. The main mathematical approach is based on the uniform persistent theory for discrete-time semiflows on some appropriate Banach space. Finally, we carry out simulations to illustrate the analytic results and test the parametric sensitivity on $ \mathcal{R}_0 $.

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.

Measles is the most infectious disease with a high basic reproduction number (R0). For measles, it is reported that R0 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 R0=1.

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.

In this paper, we introduce a novel model that simulates the spread of the monkeypox virus. The new model takes into account the effect of the interaction between the human and rodent population along with some realistic factors that have not been introduced before such as imperfect vaccination and nonlinear incidence rates. Moreover, the human population is further divided into low-risk and high-risk groups to better reflect recent observations. To understand the dynamics of the new model, the existence, uniqueness, and continuous dependence on initial conditions, as well as its positivity and boundedness of the model are investigated in detail. The obtained solution is proved to be positive and has bounded behavior. Furthermore, the reproduction number R0 for the proposed system is computed and detailed bifurcation analysis is performed to reveal major qualitative changes in the model's dynamics. In addition, we investigate the influence of key parameters on the basic reproduction number. We also explore and analyze the stability of equilibrium points in the space of parameters. The new findings reveal that the virus is more frequently witnessed among the high-risk group, which may provide some restrictions to stop the spread through social limitations. To validate our theoretical results, we conduct numerical simulations, which provide insights into the behavior of the model under different conditions. The new findings aim in developing preventive control measures to suppress the spread of the virus and to develop effective strategies for controlling and preventing outbreaks, ultimately protecting public health and minimizing the impact of the disease.

The co-infection of HIV and Zika virus (ZIKV) poses a complex and understudied health challenge, requiring a comprehensive investigation into the synergistic effects, potential complications, and the impact on affected individuals. Consequently, This paper introduces a novel deterministic mathematical model that examines the transmission dynamics of HIV and Zika virus co-infection, considering both vertical and horizontal transmission. The analysis begins with two sub-models: one for HIV-only and another for ZIKV-only. Qualitative examination indicates that the HIV only sub-model achieves a globally asymptotically stable disease-free equilibrium when the associated reproduction number is below unity. In contrast, the ZIKV only sub-model exhibits a backward bifurcation phenomenon, where both stable disease-free and stable endemic equilibria co-exist when the associated reproduction number of the ZIKV only sub-model is less than unity. Thus, the backward bifurcation property makes effective control of ZIKV infection in the population difficulty when the associated reproduction number is less than unity. It is shown, using the center manifold theory that the full HIV-ZIKV co-infection model undergoes the phenomenon of backward bifurcation. We carried out the sensitivity analysis of the HIV and ZIKV basic reproduction numbers to determine the parameters that positively influence the spread of the two diseases. It is also revealed that an increase in HIV infection in the population will positively influence the transmission of ZIKV. We validated the ZIKV only sub-model by fitting the ZIKV only sub-model to the confirmed cases of ZIKV data in Brazil. The outcome of the numerical simulations of HIV-ZIKV co-infection model reveals that the two diseases co-exist when their basic reproduction number surpasses one. Furthermore, increasing HIV treatment rate significantly reduces the burden of co-infection with the Zika virus.

Social networking platforms give people an opportunity to get in contact with each other irrespective of boundaries. Through these platforms, any individual can easily follow the ideas of other individuals like friends, relatives, politicians, actors, etc, and get in contact with them. In this paper, we examine rumor propagation with two types of infected people: highly influential people and ordinary people. Some ordinary people, blindly follow the messages shared by highly influential people. By using this concept we analyze the rumor-spreading model in networks (homogeneous and heterogeneous). We applied control strategies in homogeneous networks and observed that government policies to control rumor propagation can reduce its spread. Immunization in heterogeneous networks by giving proper information and education to people about facts can also reduce the spread and help people make the right decision when they come across a rumor.

The host population in epidemiology may actually be at risk of more than two infectious diseases with stochastic complicated interaction, e.g., HIV and HBV. In this paper, we propose a class of stochastic epidemic model that applies the double epidemic hypothesis and Crowley-Martin incidence rate in order to explore how stochastic disturbances affect the spread of diseases. While disregarding stochastic disturbances, we examine the dynamic features of the system in which the local stability of equilibria are totally determined by the basic reproduction numbers. We focus particularly on the threshold dynamics of the corresponding stochastic system, and we obtain the extinction and permanency conditions for a pair of infectious diseases. We find that the threshold dynamics of the deterministic and stochastic systems vary significantly: (ⅰ) disease outbreaks can be controlled by appropriate stochastic disturbances; (ⅱ) diseases die out when the intensity of environmental perturbations is higher. The effects of certain important parameters on deterministic and stochastic disease transmission were obtained through numerical simulations. Our observations indicate that controlling epidemics should improve the effectiveness of prevention measures for susceptible individuals while improving the effectiveness of treatment for infected individuals.

Strangles is one of the most prevalent horse diseases globally. The infected horses may be asymptomatic and can still carry the infectious pathogen after it recovers, which are named asymptomatic infected horses and long-term subclinical carriers, respectively. Based on these horses, this paper establishes a dynamical model to screen, measure, and model the spread of strangles. The basic reproduction number $ \mathcal{R}_0 $ is computed through a next generation matrix method. By constructing Lyapunov functions, we concluded that the disease-free equilibrium is globally asymptotically stable if $ \mathcal{R}_0 < 1 $, and the endemic equilibrium exits uniquely and is globally asymptotically stable if $ \mathcal{R}_0 > 1 $. For example, while studying a strangles outbreak of a horse farm in England in 2012, we computed an $ \mathcal{R}_0 = 0.8416 $ of this outbreak by data fitting. We further conducted a parameter sensitivity analysis of $ \mathcal{R}_0 $ and the final size by numerical simulations. The results show that the asymptomatic horses mainly influence the final size of this outbreak and that long-term carriers are connected to an increased recurrence of strangles. Moreover, in terms of the three control measures implemented to control strangles(i.e., vaccination, implementing screening regularly and isolating symptomatic horses), the result shows that screening is the most effective measurement, followed by vaccination and isolation, which can provide effective guidance for horse management.

In this paper, we propose a multi-patch SVEIR epidemic model that incorporates vaccination of both newborns and susceptible populations. We determine the basic reproduction number $ R_{0} $ and prove that the disease-free equilibrium $ P_{0} $ is locally and globally asymptotically stable if $ R_{0} < 1, $ and it is unstable if $ R_{0} > 1. $ Moreover, we show that the disease is uniformly persistent in the population when $ R_{0} > 1. $ Numerical simulations indicate that vaccination strategies can effectively control disease spread in all patches while population migration can either intensify or prevent disease transmission within a patch.

The chikungunya virus (CHIKV) infects macrophages and adherent cells and it can be transmitted via a direct contact with the virus or with an already infected cell. Thus, the CHIKV infection can have two routes. Furthermore, it can exhibit seasonal peak periods. Thus, in this paper, we consider a dynamical system model of the CHIKV dynamics under the conditions of a seasonal environment with a general incidence rate and two routes of infection. In the first step, we studied the autonomous system by investigating the global stability of the steady states with respect to the basic reproduction number. In the second step, we establish the existence, uniqueness, positivity and boundedness of a periodic orbit for the non-autonomous system. We show that the global dynamics are determined by using the basic reproduction number denoted by $ \mathcal{R}_0 $ and they are calculated using the spectral radius of an integral operator. We show the global stability of the disease-free periodic solution if $ \mathcal{R}_0 < 1 $ and we also show the persistence of the disease if $ \mathcal{R}_0 > 1 $ where the trajectories converge to a limit cycle. Finally, we display some numerical investigations supporting the theoretical findings.

In this paper, we establish a spatial heterogeneous SEIRS patch model with asymmetric mobility kernel. The basic reproduction ratio $ \mathcal{R}_{0} $ is defined, and threshold-type results on global dynamics are investigated in terms of $ \mathcal{R}_{0} $. In certain cases, the monotonicity of $ \mathcal{R}_{0} $ with respect to the heterogeneous diffusion coefficients is established, but this is not true in all cases. Finally, when the diffusion rate of susceptible individuals approaches zero, the long-term behavior of the endemic equilibrium is explored. In contrast to most prior studies, which focused primarily on the mobility of susceptible and symptomatic infected individuals, our findings indicate the significance of the mobility of exposed and recovered persons in disease dynamics.

In this paper, we examine the stability of an endemic equilibrium in a chronological age-structured SIR (susceptible, infectious, removed) epidemic model with age-dependent infectivity. Under the assumption that the transmission rate is a shifted exponential function, we perform a Hopf bifurcation analysis for the endemic equilibrium, which uniquely exists if the basic reproduction number is greater than $ 1 $. We show that if the force of infection in the endemic equilibrium is equal to the removal rate, then there always exists a critical value such that a Hopf bifurcation occurs when the bifurcation parameter reaches the critical value. Moreover, even in the case where the force of infection in the endemic equilibrium is not equal to the removal rate, we show that if the distance between them is sufficiently small, then a similar Hopf bifurcation can occur. By numerical simulation, we confirm a special case where the stability switch of the endemic equilibrium occurs more than once.

In this paper, based on the statistical data, we investigate the effects of long-term exposure to ambient particulate air pollution on the transmission dynamics of tuberculosis (TB) in Jiangsu, China by studying the threshold dynamics of the TB epidemic model via the statistical data analytically and numerically. The basic reproduction number R0>1 reveals that TB in Jiangsu, China is an endemic disease and will persist for a long time. And the numerical results show that, in order to control the TB in Jiangsu effectively, we must decrease the depuration coefficient of PM10 in the body, the proportion of TB symptomatic infectious by direct transmission, the reactivation rate of the pre-symptomatic infectious and the effect coefficient of PM10 and MTB inhaled of TB transmission, and increase the uptake coefficient, the recovery rate of the symptomatic/pre-symptomatic infectious and the influence coefficient of PM10 on the body of mortality. Our study shows that PM10 is closely related to the incidence of TB, and the effective control efforts are suggested to focus on increasing close-contact distance and wearing protective mask to decrease the influence of PM10 on the TB transmission, which may shed a new light on understanding the environmental drivers to TB.

The impact of the emergence of Corona virus which affected all parts of the world cannot be over emphasized. COVID-19 has cost hundreds of thousands of human lives globally, presenting healthcare professionals with pressing challenges, and exposed the weaknesses of national health systems worldwide. Hence, there is a need for more vaccination of individuals which will in turn leads to the eradication of the deadly disease. In this paper, an investigation is carried out for the convergent solution of the model by making use of a reliable Homotopy Perturbation Method (HPM) in exploring the possible solution. Basic Reproduction number was computed using Next Generation Method, The Equilibriums points are determined and the model also explored the sensitivities aspect when the parameters are varied. Fractional model numerically using the Homotopy Perturbation Method (HPM) to obtain the iterative solution of the epidemic model scheme and presenting different forms of graphical results that can be useful to analyze the model. The numerical results show that the spread of the corona – virus disease is reduced by taking adequate and effective vaccines with time

In this paper, a stochastic SIQR epidemic model with non-monotone incidence is investigated. First of all, we consider the disease-free equilibrium of the deterministic model is globally asymptotically stable by using the Lyapunov method. Secondly, the existence and uniqueness of positive solution to the stochastic model is obtained. Then, the sufficient condition for extinction of the stochastic model is established. Furthermore, a unique stationary distribution to stochastic model will exist by constructing proper Lyapunov function. Finally, numerical examples are carried out to illustrate the theoretical results, with the help of numerical simulations, we can see that the higher intensities of the white noise or the bigger of the quarantine rate can accelerate the extinction of the disease. This theoretically explains the significance of quarantine strength (or isolation measures) when an epidemic erupts.

In this paper, an optimal control model for the transmission dynamics of COVID-19 is investigated. We established important model properties like nonnegativity and boundedness of solutions, and also the region of invariance. Further, an expression for the basic reproduction number is computed and its sensitivity w.r.t model parameters is carried out to identify the most sensitive parameter. Based on sensitivity analysis, optimal control strategies were presented to reduce the disease burden and related costs. It is demonstrated that optimal control does exist and is unique. The characterization of optimal trajectories is analytically studied via Pontryagin's Minimum Principle. Moreover, various simulations were performed to support analytical results. The simulation results showed that the proposed controls significantly influence the disease burden compared to the absence of control cases. Further, it reveals that the applied control strategies are effective throughout the intervention period in reducing COVID-19 diseases in the community. Besides, the simulation results of the optimal control suggested that concurrently applying all controlling strategies outperform in mitigating the spread of COVID-19 compared to any other preventive measures.

In this paper, we investigate the effects of ambient air pollution (AAP) on the spread of influenza in an AAP-dependent dynamic influenza model. The value of this study lies in two aspects. Mathematically, we establish the threshold dynamics in the term of the basic reproduction number $ \mathcal{R}_0 $: If $ \mathcal{R}_0 < 1 $, the disease will go to extinction, while if $ \mathcal{R}_0 > 1 $, the disease will persist. Epidemiologically, based on the statistical data in Huaian, China, we find that, in order to control the prevalence of influenza, we must increase the vaccination rate, the recovery rate and the depletion rate, and decrease the rate of the vaccine wearing off, the uptake coefficient, the effect coefficient of AAP on transmission rate and the baseline rate. To put it simply, we must change our traveling plan and stay at home to reduce the contact rate or increase the close-contact distance and wear protective masks to reduce the influence of the AAP on the influenza transmission.

Dengue is one of the most infectious diseases in the world. In Bangladesh, dengue occurs nationally and has been endemic for more than a decade. Therefore, it is crucial that we model dengue transmission in order to better understand how the illness behaves. This paper presents and analyzes a novel fractional model for the dengue transmission utilizing the non-integer Caputo derivative (CD) and are analysed using q-homotopy analysis transform method (q-HATM). By using the next generation method, we derive the fundamental reproduction number $ R_0 $ and show the findings based on it. The global stability of the endemic equilibrium (EE) and the disease-free equilibrium (DFE) is calculated using the Lyapunov function. For the proposed fractional model, numerical simulations and dynamical attitude are seen. Moreover, A sensitivity analysis of the model is performed to determine the relative importance of the model parameters to the transmission.

Since the beginning of March 2022, the epidemic due to the Omicron variant has developed rapidly in Jilin Province. To figure out the key controlling factors and validate the model to show the success of the Zero-COVID policy in the province, we constructed a Recursive Zero-COVID Model quantifying the strength of the control measures, and defined the control reproduction number as an index for describing the intensity of interventions. Parameter estimation and sensitivity analysis were employed to estimate and validate the impact of changes in the strength of different measures on the intensity of public health preventions qualitatively and quantitatively. The recursive Zero-COVID model predicted that the dates of elimination of cases at the community level of Changchun and Jilin Cities to be on April 8 and April 17, respectively, which are consistent with the real situation. Our results showed that the strict implementation of control measures and adherence of the public are crucial for controlling the epidemic. It is also essential to strengthen the control intensity even at the final stage to avoid the rebound of the epidemic. In addition, the control reproduction number we defined in the paper is a novel index to measure the intensity of the prevention and control measures of public health.

This paper uses a fractional-order epidemic model to describe the transmission dynamics of the Ebola virus. The proposed model uses the fractional-order derivative in Caputo-Fabrizio’s sense. It calculates the time-independent solutions of the proposed model, and the next-generation matrix method is used to calculate the basic reproduction number. It provides the conditions for the existence and uniqueness of solutions to the model. Further, the conditions for generalized Ulam-Hyers-Rassias stability of the proposed model are obtained. Numerical simulations show how the proposed model’s approximate solution varies for integer and fractional orders. They also show the behavior of the Ebola in terms of infections, deceased, and susceptible counts, for various contact rates. To demonstrate efficiency while using less time, CPU times are given in tabular form.

The malware spreading in Wireless Sensor Network (WSN) has lately attracted the attention of many researchers as a hot problem in nonlinear systems. WSN is a collection of sensor nodes that communicate with each other wirelessly. These nodes are linked in a decentralised and distributed structure, allowing for efficient data collection and communication. Due to their decentralised architecture and limited resources, WSN is vulnerable to security risks, including malware attacks. Malware can attack sensor nodes, causing them to malfunction and consume more energy. These attacks can spread from one infected node to others in the network, making it essential to protect WSN against malware attacks. In this paper, we focus on the analysis of a novel fractional epidemiology model, specifically the fractional order SEIVR epidemic model in the sense of Caputo’s fractional derivative of order $0{ < {\alpha }{\leq }}1$ with the goal of examining the efficacy of vaccination strategies and the heterogeneity of a scale-free network on epidemic spreading. First, using the next-generation technique and obtain the basic reproduction number of the proposed epidemic model, which is essential for determining both the locally asymptotically stable equilibrium point of the worm-free system and the unique existence of the endemic equilibrium point. To numerically solve the model, the Adam-Bashforth-Moulton predictor-corrector (ABM) method is applied. The fractional calculus enables us to deal directly with the “memory effect” of numerous phenomena, taking into account the system’s dependence on previous stages. This method provides the results of a complex system. Additionally, research demonstrates that vaccine treatments are quite effective at preventing the spread of malware. The outcome of the study reveals that the applied ABM predictor-corrector method is computationally strong and effective to analyse fractional order dynamical systems in the SEIVR epidemic model for malware propagation in WSN. The results show that the order of the fractional derivative has a significant effect on the dynamic process. Also, from the result, it is obvious that the memory effect is zero for ${\alpha }$ = 1. When the fractional order ${\alpha }$ is decreased from 1, the memory effect appears, and its dynamics vary according to the time. This memory effect points out the difference between derivatives of fractional and integer orders. The theorems and their proofs are presented to validate the validity of the proposed model. To validate the proposed model, extensive theoretical study and computational analysis have also been applied.

Due to the fast advancement of Internet technology, the popularity of Online Social Networks (OSN) over the Internet is increasing day by day. In the modern world, people are using OSN to communicate with others around the world who may or may not know each other. OSN has become the most convenient means to transmit media (news/content) and gather or spread information in the world. The posts (contents) on OSN affect and impact people, and minds at least for some time. These contents are important because they play a crucial role in taking the decision. The posts which are available on the OSN may be information or just misinformation. The misinformation may be a type of fake news or rumour. This is very difficult for people to differentiate whether the posts are information or rumour. Therefore, the development of techniques that can prevent the transmission of false information or rumours that might harm society in any way is critical. In this paper, a model is developed based on the epidemic approach, for examining and controlling fake information dissemination in OSN. The proposed model illustrates how different misinformation debunking measures impact and how misinformation spreads among different groups. In this article, we explain that the proposed model will be able to recognize and eradicate fake news from OSN. The model is written as a system of differential equations. Its equilibrium and stability are also carefully examined. The basic reproduction number $(R_{0})$ is calculated, which is an important parameter in the study of message propagation in OSN. If $R_{0} < 1$ , the propagation of rumor in the OSN will be minimal; nevertheless, if $R_{0} > 1$ , the fake information/rumor will continue in OSN. The effects of disinformation of rumours in OSN in the real world are explored. In addition, the model covers the fake information/rumour dissemination control mechanism. The comparative study shows that the proposed model provides a better mechanism to prevent the dissemination of fake information in OSN in comparison to other previous models Extensive theoretical study and computation analysis have also been used to validate the proposed model

In this paper, we introduce a novel multi-scale network model of two epidemics: HIV infection and opioid addiction. The HIV infection dynamics is modeled on a complex network. We determine the basic reproduction number of HIV infection, $ \mathcal{R}_{v} $, and the basic reproduction number of opioid addiction, $ \mathcal{R}_{u} $. We show that the model has a unique disease-free equilibrium which is locally asymptotically stable when both $ \mathcal{R}_{u} $ and $ \mathcal{R}_{v} $ are less than one. If $ \mathcal{R}_{u} > 1 $ or $ \mathcal{R}_{v} > 1 $, then the disease-free equilibrium is unstable and there exists a unique semi-trivial equilibrium corresponding to each disease. The unique opioid only equilibrium exist when the basic reproduction number of opioid addiction is greater than one and it is locally asymptotically stable when the invasion number of HIV infection, $ \mathcal{R}^{1}_{v_i} $ is less than one. Similarly, the unique HIV only equilibrium exist when the basic reproduction number of HIV is greater than one and it is locally asymptotically stable when the invasion number of opioid addiction, $ \mathcal{R}^{2}_{u_i} $ is less than one. Existence and stability of co-existence equilibria remains an open problem. We performed numerical simulations to better understand the impact of three epidemiologically important parameters that are at the intersection of two epidemics: $ q_v $ the likelihood of an opioid user being infected with HIV, $ q_u $ the likelihood of an HIV-infected individual becoming addicted to opioids, and $ \delta $ recovery from opioid addiction. Simulations suggest that as the recovery from opioid use increases, the prevalence of co-affected individuals, those who are addicted to opioids and are infected with HIV, increase significantly. We demonstrate that the dependence of the co-affected population on $ q_u $ and $ q_v $ are not monotone.

Corona-virus disease 2019 (COVID-19) is an infectious disease that has affected different groups of humankind such as farmers, soldiers, drivers, educators, students, healthcare workers and many others. The transmission rate of the disease varies from one group to another depending on the contact rate. Healthcare workers are at a high risk of contracting the disease due to the high contact rate with patients. So far, there exists no mathematical model which combines both public control measures (as a parameter) and healthcare workers (as an independent compartment). Combining these two in a given mathematical model is very important because healthcare workers are protected through effective use of personal protective equipment, and control measures help to minimize the spread of COVID-19 in the community. This paper presents a mathematical model named SWE IsIaHR; susceptible individuals (S), healthcare workers (W), exposed (E), symptomatic infectious (Is), asymptomatic infectious (Ia), hospitalized (H), recovered (R). The value of basic reproduction number R0for all parameters in this study is 2.8540. In the absence of personal protective equipment ξ and control measure in the public θ, the value of R0≈4.6047which implies the presence of the disease. When θ and ξ were introduced in the model, basic reproduction number is reduced to 0.4606, indicating the absence of disease in the community. Numerical solutions are simulated by using Runge–Kutta fourth-order method. Sensitivity analysis is performed to presents the most significant parameter. Furthermore, identifiability of model parameters is done using the least square method. The results indicated that protection of healthcare workers can be achieved through effective use of personal protective equipment by healthcare workers and minimization of transmission of COVID-19 in the general public by the implementation of control measures. Generally, this paper emphasizes the importance of using protective measures.

In this paper, a mathematical analysis of the HIV/AIDS deterministic model studied in the paper called Mathematical Model of HIV/AIDS Considering Sexual Preferences Under Antiretroviral Therapy, a case study in the previous works preformed by Espitia is performed. The objective is to gain insight into the qualitative dynamics of the model determining the conditions for the persistence or effective control of the disease in the community through the study of basic properties such as positiveness and boundedness; the calculus of the basic reproduction number; stationary points such as disease-free equilibrium (DFE), boundary equilibrium (BE) and endemic equilibrium (EE); and the local stability (LAS) of disease-free equilibrium. The findings allow us to conclude that the best way to reduce contagion and consequently reach a DFE is thought to be the reduction in the rate of homosexual partners, as they are the most affected population by the virus and are therefore the most likely to become infected and spread it. Increasing the departure rate of infected individuals leads to a decrease in untreated infected heterosexual men and untreated infected women.

Taking into account the effects of the immune response and delay, and complexity on HIV-1 transmission, a multiscale AIDS/HIV-1 model is formulated in this paper. The multiscale model is described by a within-host fast time model with intracellular delay and immune delay, and a between-host slow time model with latency delay. The dynamics of the fast time model is analyzed, and includes the stability of equilibria and properties of Hopf bifurcation. Further, for the coupled slow time model without an immune response, the basic reproduction number R0h is defined, which determines whether the model may have zero, one, or two positive equilibria under different conditions. This implies that the slow time model demonstrates more complex dynamic behaviors, including saddle-node bifurcation, backward bifurcation, and Hopf bifurcation. For the other case, that is, the coupled slow time model with an immune response, the threshold dynamics, based on the basic reproduction number R˜0h, is rigorously investigated. More specifically, if R˜0h<1, the disease-free equilibrium is globally asymptotically stable; if R˜0h>1, the model exhibits a unique endemic equilibrium that is globally asymptotically stable. With regard to the coupled slow time model with an immune response and stable periodic solution, the basic reproduction number R0 is derived, which serves as a threshold value determining whether the disease will die out or lead to periodic oscillations in its prevalence. The research results suggest that the disease is more easily controlled when hosts have an extensive immune response and the time required for new immune particles to emerge in response to antigenic stimulation is within a certain range. Finally, numerical simulations are presented to validate the main results and provide some recommendations for controlling the spread of HIV-1.

In this paper, the Caputo-based fractional derivative optimal control model is looked at to learn more about how the human respiratory syncytial virus (RSV) spreads. Model solution properties such as boundedness and non-negativity are checked and found to be true. The fundamental reproduction number is calculated by using the next-generation matrix’s spectral radius. The fractional optimal control model includes the control functions of vaccination and treatment to illustrate the impact of these interventions on the dynamics of virus transmission. In addition, the order of the derivative in the fractional optimal control problem indicates that encouraging vaccination and treatment early on can slow the spread of RSV. The overall analysis and the simulated behavior of the fractional optimum control model are in good agreement, and this is due in large part to the use of the MATLAB platform.

The purpose of this paper is to investigate the transmission dynamics of COVID-19 with Dengue coinfection using a mathematical model. The human population was divided into six compartments, while the mosquito population was divided into two sections. The model considers that COVID-19 infection might be symptomatic or asymptomatic. First, we analyzed the dengue infection model. The basic reproduction number of the COVID-19 infection system and the Dengue infection system are used to forecast illness mitigation and persistence (denoted by ℛ0C and ℛ0D respectively). The qualitative examination of the sub-systems indicated that the disease-free equilibrium (DFE) is locally asymptotically stable provided the corresponding reproduction numbers are less than one. The coinfection model is then analyzed to yield the basic reproduction number, designated by ℛ0. The DFE and stability of the coinfection model are dependent on ℛ0 = max {ℛ0D, ℛ0C}. The numerical simulation of the coinfection model showed the existence of the endemic equilibrium of the coinfection model. Furthermore, we studied the dynamic solutions of the coinfection model by establishing the equilibrium points and evaluated the stability requirements.

Communicable diseases are one of the challenges to people’s health and fitness. Consuming contaminated water or food and inhaling contaminated water droplets are the main causes of water-borne pathogens spread. This paper studies the spreading dynamics of water-borne diseases with vaccination and preventive measures over a multilayer network. SVIR-process for disease spread over humans and SIS-process for disease spread over water supply mediums and UAU-process for awareness diffusion over humans via social media are taken to study the spreading dynamics. The Discrete-time Markov chain approach is used to frame and analyse the model. The optimal value corresponding to the vaccination parameter is found to control the disease spread. The impact of awareness and vaccination over disease spread and the characteristics of basic reproduction number are found via numerical simulation.

BackgroundSevere acute respiratory syndrome (SARS) is a form of atypical pneumonia which took hundreds of lives when it swept the world two decades ago. The pathogen of SARS was identified as SARS-coronavirus (SARS-CoV) and it was mainly transmitted in China during the SARS epidemic in 2002-2003. SARS-CoV and SARS-CoV-2 have emerged from the SARS metapopulation of viruses. However, they gave rise to two different disease dynamics, a limited epidemic, and an uncontrolled pandemic, respectively. The characteristics of its spread in China are particularly noteworthy. In this paper, the unique characteristics of time, space, population distribution and transmissibility of SARS for the epidemic were discussed in detail.MethodsWe adopted sliding average method to process the number of reported cases per day. An SEIAR transmission dynamics model, which was the first to take asymptomatic group into consideration and applied indicators of R0, Reff, Rt to evaluate the transmissibility of SARS, and further illustrated the control effectiveness of interventions for SARS in 8 Chinese cities.ResultsThe R0 for SARS in descending order was: Tianjin city (R0 = 8.249), Inner Mongolia Autonomous Region, Shanxi Province, Hebei Province, Beijing City, Guangdong Province, Taiwan Province, and Hong Kong. R0 of the SARS epidemic was generally higher in Mainland China than in Hong Kong and Taiwan Province (Mainland China: R0 = 6.058 ± 1.703, Hong Kong: R0 = 2.159, Taiwan: R0 = 3.223). All cities included in this study controlled the epidemic successfully (Reff

In this paper, we formulate a mathematical model of vector-borne disease dynamics. The model is constructed by considering two models : a baseline model of vector population dynamics due to Lutambi et al. that takes into account the development of the aquatic stages and the female mosquitoes gonotrophic cycle and an SI-SIR model describing the interaction between mosquitoes and human hosts. We briefly study the baseline model of vectors dynamics and, for the transmission model, we explicitly compute the equilibrium points, and by using the method of Van den Driesshe and J. Watmough, we derive the basic reproduction number ℛ0. Otherwise, thanks to Lyapunov’s principle, Routh-Hurwitz criteria and a favorable result due to Vidyasagar, we establish the local and global stability results of the equilibrium points. Furthermore, we establish an interesting relationship between the mosquito reproduction number ℛv and the basic reproduction number ℛ0. It then follows that aquatic stages and behavior of adult mosquitoes have a significant impact on disease transmission dynamics. Finally, some numerical simulations are carried out to support the theoretical findings of the study.

Anthrax is an acute infectious zonootic disease caused by Bacillus anthracis, a gram-positive, rod-shaped non-motile bacterium. It is a disease that mainly affects herbivorous animals of both domestic and wildlife, and causes devastating spillover infections into the human population. Anthrax epidemic results in serious and fatal infections in both animals and humans globally. In this paper, a non-linear differential equation model is proposed to study the transmission dynamics of anthrax in both animal and human populations taking into accounts saturation effect within the animal population and behavioural change of the general public towards the outbreak of the disease. The model is shown to have two unique equilibrium points, namely; the anthrax-free and endemic equilibrium points. The anthrax-free equilibrium point is globally asymptotically stable whenever the reproduction number is less than unity $ (\mathcal{R}_{0} < 1) $ and the endemic equilibrium point is locally asymptotically stable whenever $ \mathcal{R}_{0} > 1 $. Sensitivity analysis suggests that the most influential factors on the spread of anthrax are the infection force $ \beta_{a} $, pathogen shedding rate $ \xi_{a} $, recruitment rate $ \Lambda_a $, natural death rate in animals $ \mu_{a} $ and recovery rate in animals $ \phi_{a} $. Numerical simulations demonstrate that the saturation effect and behavioural change of the general public towards the outbreak of the disease increase the size of the susceptible population, reduce the size of the infective population and the pathogen levels in the environment. Findings of this research show that anthrax epidemic can be controlled by reducing the rate of anthrax infection and pathogen shedding rate, while increasing the rate of pathogen decay through proper environmental hygiene as well as increasing treatment to ensure higher recovery rate in infected animals. The results also show that positive behavioural change of the general public through mass awareness interventions can help control the spread of the disease.

In this paper, a deterministic ordinary differential equations model for the transmission dynamics of maize streak virus disease (MSVD) in maize plants is proposed and analyzed qualitatively. Using the next generation matrix approach, the basic reproduction number, $ \mathcal{R}_{0} $ with respect to the MSVD free equilibrium is found to be $ 4.7065\approx5 $. The conditions for local stability of the disease-free equilibrium and endemic equilibrium points were established. From the results, the disease-free equilibrium point was found to be unstable whenever $ \mathcal{R}_{0} > 1 $ and the endemic equilibrium point was found to be locally asymptotically stable whenever $ \mathcal{R}_{0} > 1 $. The sensitivity indices of various parameters with respect to the MSVD eradication or spreading were determined. It was found that $ b\; , \beta_{{1}}\; , \beta_{{11}}\; , $ and $ \beta_{{2}} $ are the parameters that are directly related to $ \mathcal{R}_{0} $, and $ H_{2}, \mu, \mu_{{1}} $ and $ \gamma $ are inversely related to $ \mathcal{R}_{0} $. Numerical simulation was performed and displayed graphically to justify the analytical results.

The goal of this paper is to investigate the influence of the waning immunity on the dynamics of Tilapia Lake Virus (TiLV) transmission in wild and farmed tilapia within freshwater. We formulate a model for which susceptible individuals can contract the disease in two ways: (i) direct mode caused by contact with infected individuals; (ii) indirect mode due to the presence of pathogenic agents in the water. We obtain an age-structured model which combines both age since infection and age since recovery. We derive an explicit formula for the reproductive number $ \mathcal {R}_0 $ and show that the disease-free equilibrium is locally asymptotically stable when, $ \mathcal {R}_0 \lt 1 $ . We discuss on the form of the waning immunity parameter and show numerically that a Hopf bifurcation may occur for suitable immunity parameter values, which means that there is a periodic solution around the endemic equilibrium when, $ \mathcal {R}_0 \gt 1 $ .