| Infectious diseases are caused by pathogens,including microorganisms and parasites.In order to analyze the transmission characteristics of the two types of pathogens from a mathematical perspective,the two infectious diseases studied in this paper respectively contain the two types of pathogens,namely,the coronavirus disease 2019(COVID-19)and malaria.Specifically,four dynamic models are established,namely,an infectious disease model with quarantine and standard incidence rate,a short-term COVID-19 dynamic model,a COVID-19dynamic model with time delays,and a malaria transmission dynamic model with vaccination.The next generation matrix method is used to calculate the control reproduction number.The local stability of the equilibria with respect to the control reproduction number is proved by using the Routh-Hurwitz criterion or proof by contradiction.The global dynamic properties of the equilibria of the model with respect to the control reproduction number are studied by constructing the Lyapunov functions/functionals,combining the generalized Lyapunov-La Salle theorem and persistence analysis method.Numerical simulations are carried out by using the COVID-19 dynamical model combined with actual data from India and Nanjing.The specific research contents are as follows:In Chapter 2,inspired by the transmission characteristics of COVID-19,an infectious disease model with quarantine and standard incidence rate is considered,then a novel persistence analysis approach for this type of model is proposed for finding the ultimate lower bound of the number of infected individuals,which means that the infectious disease is uniformly persistent if the control reproduction number(49)_c(29)1.This approach can be applied to related biomathematical models,and some existing works can be improved by using that.In addition,the infection-free equilibriumV~0 of the model is locally stable if(49)_c(27)1 and linearly stable if(49)_c(28)1;while V~0 is unstable if(49)_c(29)1.In Chapter 3,the infectious disease model in Chapter 2 is applied to study the transmission of COVID-19 as a long-term dynamic model for COVID-19.Considering the propagation characteristics of COVID-19 in different regions,the dynamics analysis and numerical demonstration of long-term and short-term models of COVID-19 are carried out,respectively.The long-term model is devoted to investigate the global stability of COVID-19 model with asymptomatic infections and quarantine measures.By adopting the limit system and the generalized Lyapunov-La Salle theorem,it is shown that the COVID-19-free equilibrium V~0is globally stable if the control reproduction number(49)_c(27)1 and globally attractive if(49)_c(28)1,which mean that COVID-19 will die out;the COVID-19 equilibrium V~*is globally stable if(49)_c(29)1,which means that COVID-19 will be persistent.In particular,to obtain the local stability of V~*,we use proof by contradiction and the properties of complex modulus,and to obtain the global attractivity of V~*,we prove the weak persistence of the system instead of its uniform persistence by adopting an analysis approach.Moreover,the final size of the corresponding short-term model is calculated and the stability of its multiple equilibria is analyzed.Numerical simulations of COVID-19 cases show that quarantine measures and asymptomatic infections have a non-negligible impact on the transmission of COVID-19.In Chapter 4,considering the transmission characteristics of COVID-19,there are a certain time delays in the transition from susceptible individuals to exposed individuals after contacting with exposed,symptomatic infected and asymptomatic infected individuals.Based on this,a COVID-19 model with time delays and exposed infection is developed and then the global dynamics of this model is investigated by an improved dynamic method.It is shown that the COVID-19-free equilibrium T~0 is globally stable if and only if the control reproduction number(49)_c?1;while the COVID-19 equilibrium T~*is globally stable if and only if(49)_c(29)1.In Chapter 5,in view of the practical problems of malaria vaccine in malaria transmission,a dynamical model of malaria with vaccination and vaccine failure is constructed,and the control reproduction number(49)_c is calculated.The existence conditions of malaria-free and malaria equilibria in terms of(49)_c are given.By using the Lyapunov function method and the generalized Lyapunov-Lasalle theorem,the sufficient and necessary conditions for the global asymptotic stability of malaria-free and malaria equilibria with respect to(49)_c are established. |
