Dynamic Model Analysis Of Pulmonary Tuberculosis With Linear Population Input And Age Structure

Tuberculosis,also known as "phthisis",is a chronic infectious disease caused by Mycobacterium tuberculosis.In recent years,with the growth of the population,the increase of population mobility and the infection of the novel coronavirus,the incidence of tuberculosis is also increasing,which seriously endangers human life and health.In order to explore the impact of population input on the spread of tuberculosis,several dynamic models of tuberculosis with linear input and age structure are established based on the pathological background and transmission mechanism of tuberculosis with the help of the age structure model.Firstly,a SEIT model with latent age structure and population input as bS is established.And the model is transformed into a class of abstract non-dense Cauchy problem,which verifies the non-negative and boundedness of the model solution.Then the existence of the equilibria is discussed,and the global stability of the extinction equilibrium P0 and disease-free equilibrium P1 is proved by constructing Lyapunov functional.At the same time,the instability of the equilibrium of endemic diseases is explored.Finally,the numerical examples are executed to illustrate the theoretical results.In addition,considering the mobility of the population,a tuberculosis model with latent age structure and population input model as bN is developed,and the existence and stability of the equilibria of the model are also explored.Comparing these two models,we can see that the dynamic behavior of pulmonary tuberculosis is influenced by different population input modes.Secondly,the age structure model of SIRS infection with population input as bS is established,and the influence of time delay and the recurrence of disease on the dynamic state of the model of age structure epidemic model is considered.The non-negative and bounded of the model solutions are discussed,and the existence conditions of the three equilibria points in the system are given.The stability of the equilibria and the conditions of Hopf branching are proved by theoretical derivation.Numerical simulations show that different population sizes can lead to multiple stable switches in the SIRS model of age structure.Finally,an age structure model with two types of latent infection with population input as bS is established.The main consideration is to divide latent tuberculosis infected persons into two categories:those who will develop the disease and those who will not develop the disease for life.Aiming to explore the effect of linear population input on the dynamic behavior of the model,the well-posedness of the model solution is analyzed,and the existence and stability of equilibria are discussed,when b<d,there is a globally asymptotically stable population extinction equilibrium Q0;When b=d,there is a globally asymptotically stable disease-free equilibrium Q1;When b>d,there is an unstable endemic equilibrium Q*.The results are verified by numerical simulation.