{"title":"Dynamics and backward bifurcations of SEI tuberculosis models in homogeneous and heterogeneous populations","authors":"Wei Li , Yi Wang , Jinde Cao , Mahmoud Abdel-Aty","doi":"10.1016/j.jmaa.2024.128924","DOIUrl":null,"url":null,"abstract":"<div><div>The main difference between tuberculosis (TB) and other infectious diseases is that the transmission of the bacterium should be considered not only as the development of a primary infection, but also as exogenous reinfection or endogenous reactivation. Moreover, individuals in the population may have heterogeneous contact rates, which can be described by complex networks. To this end, we present two differential equation-based TB models in homogeneous and heterogeneous populations. The first model assumes that the number of contacts per unit time is constant for the whole population, whereas the second model considers the heterogeneous number of contacts per unit time for each individual. We derive the basic reproduction number of each model using the next-generation matrix method, and analyze the dynamical properties of each model in detail. We find that the two models undergo backward bifurcations and have the same threshold condition for backward bifurcation. From this threshold condition, we see that the reduced rate of exogenous reinfection of individuals plays an important role in causing the backward bifurcation. Interestingly, the second model allows the threshold condition for backward bifurcation to be independent of network parameters. Thus, unlike other infectious disease models on complex networks, in controlling the spread of tuberculosis among populations with different numbers of contacts, we only need to focus on disease parameters during treatment. Finally, numerical simulations more intuitively demonstrate the impact of parameter changes on the prevalence of tuberculosis and reveal the model's richer and more interesting dynamical properties, such as bistability. Sensitivity analysis indicates that the basic reproduction number is highly correlated with both the relapse rate of latent individuals progressing to active infection and the probability of healthy individuals becoming infected after contact with the pathogen. Therefore, enhancing the detection and treatment of latent cases and reducing contact between infected and uninfected individuals are the most crucial public health interventions.</div></div>","PeriodicalId":50147,"journal":{"name":"Journal of Mathematical Analysis and Applications","volume":"543 2","pages":"Article 128924"},"PeriodicalIF":1.2000,"publicationDate":"2024-10-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Journal of Mathematical Analysis and Applications","FirstCategoryId":"100","ListUrlMain":"https://www.sciencedirect.com/science/article/pii/S0022247X24008461","RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"MATHEMATICS","Score":null,"Total":0}
引用次数: 0
Abstract
The main difference between tuberculosis (TB) and other infectious diseases is that the transmission of the bacterium should be considered not only as the development of a primary infection, but also as exogenous reinfection or endogenous reactivation. Moreover, individuals in the population may have heterogeneous contact rates, which can be described by complex networks. To this end, we present two differential equation-based TB models in homogeneous and heterogeneous populations. The first model assumes that the number of contacts per unit time is constant for the whole population, whereas the second model considers the heterogeneous number of contacts per unit time for each individual. We derive the basic reproduction number of each model using the next-generation matrix method, and analyze the dynamical properties of each model in detail. We find that the two models undergo backward bifurcations and have the same threshold condition for backward bifurcation. From this threshold condition, we see that the reduced rate of exogenous reinfection of individuals plays an important role in causing the backward bifurcation. Interestingly, the second model allows the threshold condition for backward bifurcation to be independent of network parameters. Thus, unlike other infectious disease models on complex networks, in controlling the spread of tuberculosis among populations with different numbers of contacts, we only need to focus on disease parameters during treatment. Finally, numerical simulations more intuitively demonstrate the impact of parameter changes on the prevalence of tuberculosis and reveal the model's richer and more interesting dynamical properties, such as bistability. Sensitivity analysis indicates that the basic reproduction number is highly correlated with both the relapse rate of latent individuals progressing to active infection and the probability of healthy individuals becoming infected after contact with the pathogen. Therefore, enhancing the detection and treatment of latent cases and reducing contact between infected and uninfected individuals are the most crucial public health interventions.
期刊介绍:
The Journal of Mathematical Analysis and Applications presents papers that treat mathematical analysis and its numerous applications. The journal emphasizes articles devoted to the mathematical treatment of questions arising in physics, chemistry, biology, and engineering, particularly those that stress analytical aspects and novel problems and their solutions.
Papers are sought which employ one or more of the following areas of classical analysis:
• Analytic number theory
• Functional analysis and operator theory
• Real and harmonic analysis
• Complex analysis
• Numerical analysis
• Applied mathematics
• Partial differential equations
• Dynamical systems
• Control and Optimization
• Probability
• Mathematical biology
• Combinatorics
• Mathematical physics.