Dynamical Analysis And Control Of Infectious Diseases Model With Age-Structure

Infectious diseases have been an important factor affecting human’s life and health and social stability from long times ago.The study on the spread of infectious diseases can abstract the evolution process of diseases into differential equation models by establishing ordinary differential equation(group),partial differential equation(group)or differential-integral equation(group).Through the theoretical analysis on the model,including the well-posed and stability of the solution,to reveal the transmission mechanism of disease spread and further predict its activity.For different infectious diseases,there are great differences in the transmission mechanism,transmission route,susceptible population and infectivity;for the same infectious disease,the infectivity of individual with different physiological age is also different;even within the same individual,the infectivity still varies at different stages of the disease.Studies have shown that the strength of infectivity is related to the physiological age of the individual,also related to how long a person has been infected(the age of infection)and how long the antibody is produced after vaccination(the age of immunity).Infectious disease dynamical modeling needs to reflect the infectivity heterogeneity of individuals with different physiological age and different infection age.Heterogeneity of infectious disease transmission can be characterized by discrete age multigroup ODE models or continuous age PDE models.In recent years,the indepth theoretical research on infinite dimensional dynamic systems has greatly promoted the theoretical research on PDE models of infectious diseases with continuous age structure.In the process of dynamic modeling,besides the heterogeneous transmission caused by various ages,it is also necessary to consider that the spread of infectious diseases will be affected and restricted by human interventions such as environmental and vaccination on the whole.In the context of cholera and COVID-19,this paper developed three kinds of infectious disease transmission dynamics model which based on the heterogeneity of susceptibility and infectivity caused by different types of individuals’ physiological age,infection age,immune age,the diversity of contact and environmental transmission routes,as well as the universality of vaccination.We researched the dynamic behavior of the model,combined with the realistic data to estimate the model parameters,analyzed the sensitivity of parameters,as well as the optimal control problem under various control measures.The main works of this paper are as follows:(1)We proposed an age-structure SIR cholera epidemic model with incomplete immunity and vaccine immunity decay,which took into account the age of infection of the infected and the biological age of the pathogen,as well as infection due to vaccination failure and infection due to antibody levels falling below the protective threshold over a long period of vaccination.By analyzing the distribution of the roots of the characteristic equation,the local stability of the equilibria of the model was studied.We calculated the basic reproduction number,and proved that the disease-free equilibrium was globally stable when the basic reproduction number was less than 1,while the endemic equilibrium was globally stable when the basic reproduction number was greater than 1.In addition,the model was successfully applied to the assessment of cholera epidemic in Somalia.We also examined the problem of optimal control with vaccination,treatment and environment as control strategies and finally derove the optimal stratefies with the lowest cost.(2)We developed an age-structure SIRS cholera epidemic model with saturation incidence.We took into account the immune age of the vaccinator,the infective age of the infected and the biological age of the pathogen,and also considered that the infection rate does not always increase linearly as the increase of Vibrio cholerae concentration and the number of the infected,but gradually tends to saturation,thus the saturation rate is used in both environment-person and person-person transmission to capture the crowding effect during the transmission process.In addition,natural immunity acquired after infected is not lifelong,so those who have recovered may recover to suspectible and be reinfected.In theory,the basic reproduction number of the model is firstly calculated,and it is proved that the disease-free equilibrium is locally stable when the basic reproduction number is less than 1,Secondly,the existence of the backward bifurcation of the model is obtained.Numerically,we compared the impact of basic control measures influenced by transmission rate and other various control measures,such as vaccination,on the number of the infected.(3)A discrete age-structured COVID-19 dynamic model with vaccination and environmental transmission is established.We considered some factors such as breakthrough infection caused by the new mutant strain breaks through the immune defense,indirect infection through inhalating aerosols coated with pathogenic microorganisms suspended in the air,weakened immunity due to the decrease of the concentration of antibodies produced after vaccination with time.We analyzed the threshold dynamics about local stability and global stability of the model equilibria.In addition,we fitted the epidemic data of Hong Kong2022,estimated the model parameters,and studied the sensitivity of the parameters.At last we compared the effects of different control measures on different age groups.