The Study Of A Tuberculosis Dynamic Model With The Environment Pollution

Tuberculosis is a chronic infectious diseases caused by mycobacterium tuberculosis.At present,China is second only to India in the world's second largest high burden of tuberculosis countries.Tuberculosis patients can spread the droplet spray with mycobacterium tuberculosis in the air when they are coughing or sneezing.Mycobacterium suspended in the air can survive more than 9 hours,and adhere to the dust it can remain infectious 8 to 10 days,and in the dry sputum it can survive 6 to 8 months.Mycobacterium tuberculosis invades susceptibles' body through respiratory tract,digestive tract and skin of mucous membrane damage and other ways.Therefore,the pollution of environment has an important impact on the spread of tuberculosis.In this paper,we set up a model with indirect infection(infected by mycobacterium tuberculosis in the contaminated environment)to study the effects of environmental pollution on the patterns of pulmonary tuberculosis transmission.The model is analyzed in the whole parameter space by using the theory of differential equation qualitative and stability and the theory method of epidemiology dynamics.We obtain the basic reproductive number,the necessary and sufficient conditions for the existence of equilibrium points of disease-free disease and multiple endemic diseases.The local asymptotic stability of each equilibrium point is obtained by Hurwitz criterion and matrix theory.It is proved that the model not only exists posterior branch but also has a forward branch and a branch point for the forward and backward branches is found.The results showed that the release rate of mycobacterium tuberculosis in the environment and other parameters of the model affected the dynamic behavior and the tuberculosis transmission patterns.Finally,the influence of different pollution conditions on the transmission of the disease is presented by numerical simulation.