New Method For Basic Reproduction Number And Dynamics Analysis Of Infectious Disease Models

The application of systems thinking in the study of infectious diseases has always been a hot topic.This article mainly analyzes the basic reproduction number and dynamic properties of infectious disease models.Based on the bilinear incidence model,a new method for calculating the basic reproduction number is proposed,and this method is extended to the nonlinear incidence model.A bilinear incidence model with N infection populations is developed based on the infection pathway of the disease.All infected populations undergo an incubation period and then freely transfer from top to bottom.First list all pathways of infection for the disease.The basic reproduction number for the infectious disease model is obtained by using the next generation method.A new method for calculating the basic reproduction number is obtained by combining the basic reproduction number formula and the infection path called path analysis method.The basic reproduction number of the bilinear incidence infectious disease model is equal to the sum of the basic reproduction numbers of the infected population.The basic reproduction number of the infected population is equal to the sum of the basic reproduction numbers of the infected paths.The basic reproduction number of infection pathways is equal to the product of infection rate,susceptible target population,and infection time.The path analysis method is implemented in four steps.The first step is to identify infected populations with the ability to spread through the infection function.There is an infected population without the ability to transmit such as an infected population that is isolated and treated.The second step is to find the infection path of the infected population.There is an infected population with multiple infection paths.The third step is to calculate the susceptible target population for the infection path.The fourth step is to calculate the basic reproduction number of the model.The number of polynomials is the same as the number of infection paths.The path analysis method is less computationally intensive and easier for the average reader to understand than the traditional next-generation method.Note that the path analysis method is only applicable to infectious disease models where the incidence is bilinear.To solve the problem that the path analysis method is not applicable to the nonlinear incidence model,we modeled the bi-nonlinear incidence of N infection populations.Infection population transfer conforms to a nonlinear function.Infection paths are the same as in the bilinear incidence model.The basic reprodution number is calculated by the next generation method.We generalized the path analysis method to a bi-nonlinear incidence model by combining the basic reproduction number formula and infection path.And obtain the generalized path analysis method.The basic reproduction number of the bilinear incidence infectious disease model is equal to the sum of the basic reproduction numbers of the infected population.The basic reproduction number of the infected population is equal to the sum of the basic reproduction numbers of the infected path.The basic reproduction number of infection pathways is equal to the product of infection rate,susceptible target population and infection time.The susceptible target population is equal to the derivative of the infection function at the zero point.The elimination rate is equal to the derivative of the elimination function at the zero point.Compared with the path analysis method,the difference is the calculation of the susceptible target populations and the elimination rate.The generalized path analysis method is not only applicable to the nonlinear incidence model,but also to the bilinear incidence model.Chapters 3,4,and 5 of this paper study the bilinear incidence model,and chapters6 and 7 study the nonlinear incidence model.Chapter 3 develops a SEIR model with isolation to simulate the outbreak development in the context of COVID-19 transmission in Wuhan.Chapter 4 builds a bilinear model of N infected populations with sequential transfer of infected populations to neighboring populations.Chapter 5proposes a new method for calculating the basic reproduction number-the path analysis method.Chapter 6 develops a nonlinear incidence SIR model,focusing on the Hopf bifurcation.Chapter 7 extends the path analysis method to the nonlinear incidence model to obtain the generalized path analysis method.