{"title":"Susceptible-infectious-susceptible (SIS) model with virus mutation in a variable population size","authors":"Ayse Peker Dobie","doi":"10.1016/j.ecocom.2022.101004","DOIUrl":null,"url":null,"abstract":"<div><p>The complex dynamics of a contagious disease in which populations experience horizontal and vertical transmissions, size variation, and virus mutations are of considerable practical and theoretical interest. We model such a system by dividing a population into three distinct groups: susceptibles (<span><math><mi>S</mi></math></span>), <span><math><mi>C</mi></math></span>-infected (<span><math><mi>C</mi></math></span>) and <span><math><mi>F</mi></math></span>-infected (<span><math><mi>F</mi></math></span>), based on the Susceptible-Infectious-Susceptible (<span><math><mrow><mi>S</mi><mi>I</mi><mi>S</mi></mrow></math></span>) model. Once the individuals in the <span><math><mi>C</mi></math></span>-infected group recover from the disease, they gain no permanent immunity. The virus can mutate in the group <span><math><mi>C</mi></math></span>. When it does, the individuals become members of the <span><math><mi>F</mi></math></span>-infected group. The mutated virus causes a lethal and incurable disease with a high mortality rate. We discuss the model for two cases. For the first case, all the newborns from infected mothers develop the disease shortly after their birth. For the second case, there exist equal transmission rates and the <span><math><mi>C</mi></math></span>-infected population is lifelong infectious. Our analysis shows that both systems have positive solutions, and the first model possesses four equilibrium points, the trivial one (extinction of the species), <span><math><mi>C</mi></math></span>-free equilibrium (extinction of the ancestor virus) and two endemic equilibria of different properties. We identify the net population growth rates of the susceptible and <span><math><mi>C</mi></math></span>-infected groups for the existence of the equilibria of the first model. We define the conditions of parameters for which species extinction and endemic equilibria are locally asymptotically stable. We observe that bifurcation occurs at the <span><math><mi>C</mi></math></span>-free equilibrium. For the second model, we find that there is only one endemic equilibrium and it is always locally asymptotically stable. We also determine the region for the net population growth rates of the susceptible and <span><math><mi>F</mi></math></span>-infected groups for the existence of the endemic equilibrium.</p></div>","PeriodicalId":50559,"journal":{"name":"Ecological Complexity","volume":null,"pages":null},"PeriodicalIF":3.1000,"publicationDate":"2022-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"5","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Ecological Complexity","FirstCategoryId":"93","ListUrlMain":"https://www.sciencedirect.com/science/article/pii/S1476945X22000265","RegionNum":3,"RegionCategory":"环境科学与生态学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q2","JCRName":"ECOLOGY","Score":null,"Total":0}
引用次数: 5
Abstract
The complex dynamics of a contagious disease in which populations experience horizontal and vertical transmissions, size variation, and virus mutations are of considerable practical and theoretical interest. We model such a system by dividing a population into three distinct groups: susceptibles (), -infected () and -infected (), based on the Susceptible-Infectious-Susceptible () model. Once the individuals in the -infected group recover from the disease, they gain no permanent immunity. The virus can mutate in the group . When it does, the individuals become members of the -infected group. The mutated virus causes a lethal and incurable disease with a high mortality rate. We discuss the model for two cases. For the first case, all the newborns from infected mothers develop the disease shortly after their birth. For the second case, there exist equal transmission rates and the -infected population is lifelong infectious. Our analysis shows that both systems have positive solutions, and the first model possesses four equilibrium points, the trivial one (extinction of the species), -free equilibrium (extinction of the ancestor virus) and two endemic equilibria of different properties. We identify the net population growth rates of the susceptible and -infected groups for the existence of the equilibria of the first model. We define the conditions of parameters for which species extinction and endemic equilibria are locally asymptotically stable. We observe that bifurcation occurs at the -free equilibrium. For the second model, we find that there is only one endemic equilibrium and it is always locally asymptotically stable. We also determine the region for the net population growth rates of the susceptible and -infected groups for the existence of the endemic equilibrium.
期刊介绍:
Ecological Complexity is an international journal devoted to the publication of high quality, peer-reviewed articles on all aspects of biocomplexity in the environment, theoretical ecology, and special issues on topics of current interest. The scope of the journal is wide and interdisciplinary with an integrated and quantitative approach. The journal particularly encourages submission of papers that integrate natural and social processes at appropriately broad spatio-temporal scales.
Ecological Complexity will publish research into the following areas:
• All aspects of biocomplexity in the environment and theoretical ecology
• Ecosystems and biospheres as complex adaptive systems
• Self-organization of spatially extended ecosystems
• Emergent properties and structures of complex ecosystems
• Ecological pattern formation in space and time
• The role of biophysical constraints and evolutionary attractors on species assemblages
• Ecological scaling (scale invariance, scale covariance and across scale dynamics), allometry, and hierarchy theory
• Ecological topology and networks
• Studies towards an ecology of complex systems
• Complex systems approaches for the study of dynamic human-environment interactions
• Using knowledge of nonlinear phenomena to better guide policy development for adaptation strategies and mitigation to environmental change
• New tools and methods for studying ecological complexity