Stability And Optimal Control Of Reaction-diffusion Models For Two Types Of Infectious Diseases

Infectious diseases have been threatening the survival and development of mankind,such as SARS in 2003,respiratory syndrome in the Middle East in 2015 and the recent epidemic of new coronal pneumonia In the struggle against infectious diseases,it is found that the transmission mechanism of infectious diseases is very complicated,for example,HIV and COVID-19 are infectious during incubation period.From a mathematical point of view,analyzing the spread of infectious diseases through modeling can play a certain role in disease prevention and control.We will start with the reaction-diffusion model to study the spread of infectious diseases The details are as follows:1.Firstly,a SEIR model with diffusion,spatial heterogeneity and latent infection is established under Neumann boundary conditions.The threshold dynamic behavior of the model is proved by using the operator semigroup method,and the well posedness and basic regeneration numberR₀are given.Based onR₀we establishes the global threshold dynam-ics.It is proved that whenR₀<1,the disease-free equilibrium is globally asymptotically stable and the disease will be extinct;WhenR₀>1,the epidemic equilibrium is global-ly asymptotically stable,and the disease will develop into an endemic disease and persist.Then,a special caseR₀=1 is given.Finally,the Lyapunov function method is used to prove that the steady state of the unique epidemic equilibrium point is stable under the condition of spatial homogeneity.The corresponding results are given in the numerical simulation2.Next,a SIVR model with incomplete immunity was established by introducing vac-cine intervention.The semi group method is used to discuss the well posedness of the model,deduce the basic reproduction numberR₀of the model,and prove the threshold dynamics of the model:whenR₀<1,the disease-free equilibrium is globally asymptotically stable,and the disease will be extinct;forR₀>1,the epidemic equilibrium point is globally asymp-totically stable,and the disease will continue with a probability of 1.Then,the targeted treatment of patients is introduced into the system as a control parameter,and the optimal control of the system is discussed by using Hamiltonian function and adjoint equation Final-ly,the theoretical results are verified by numerical simulation.