Evolutionary invasion model of the viral virulence in a symptom-dependent SIS community

The impacts of viral virulence and its evolution on transmission dynamics are important for the public and the policymakers during the spread of infectious diseases. To reveal the impacts of viral virulence evolution on transmission dynamics within the population, we formulate a coupled SIS-v model with viral virulence, in which transmission rate and disease-induced death rate are set as functions of viral virulence. The dynamic analysis for SIS model and SIS-v model are extensively explored within the symptom-dependent community against viral virulence evolution. Precisely, the expressions of basic reproduction numbers for two models are derived by the next generation matrix method, the existence and stabilities of equilibrium points are then derived by Routh-Hurwitz criterion and LaSalle's invariance principle. By the adaptive dynamics method, the results reveal that when the asymptomatic boundary equilibrium point is globally asymptotically stable, the mutant virus could invade the population with resident virus under certain conditions, and the globally asymptotically stable viral virulence is the evolutionary singularity that maximizes both the invasion fitness and the absolute fitness for mutant virus. When the symptomatic boundary equilibrium point is globally asymptotically stable, the conclusion is the same. Furthermore, the numerical simulations with respect to viral virulence are performed using SIS model and surveillance data. The corresponding results show that SARS-CoV-2 virulence is relaxed, when the evolution starts from wild strain to Delta strain and further to Omicron strain. The increase of viral virulence contributes to the enhancement of infection scale. While, the decrease of viral virulence contributes to the decline of basic reproduction number. The invasion fitness of mutant virus obeys the increasing tendency with the enhancement of viral virulence, until the invasion fitness reaches the maximum. The main results provide three significant insights for the public and the policymakers of local government to fight against the spread of infectious diseases: the features of viral virulence evolution are helpful to the public against the newly emerging strains; the non-pharmacological interventions of the local government effectively decline basic reproduction number for the asymptomatic individuals and the symptomatic individuals within the key population; when the key population consists of the asymptomatic individuals, the local government should focus on the prompt nucleic acid screening against the spread of epidemic; when the key population consists of the symptomatic individuals, the local government should strengthen isolation measures for the symptomatic individuals.