Study On Dynamical Models Of Epidemic Diseases

It is well known that infectious diseases are the archenemy of mankind. Over the past decades, human made permanent struggle to all kinds of infectious and have obtained many brilliant achievements. However, the reports from the World Health Organization (WTO) show that infectious diseases are still the first killer for human until now. Hence, it is long way to go for the infectious diseases.As we do not study infectious diseases using the experiments from humman, theoretical analysis and computer simulations have become the main schems for the various kinds of infectious disease study; while the constructions of proper mathematical model is the basis for these theoretical analysis and simulations.Recently, a rapid progress has been made for studies on the mathematical model of many kinds of diseases. These mathematical models are used to study the prevalent law. And the theoretical tools used are Lyapunov's stability approach, limited equation theory, matrix theory, bifurcation theory, K sequence monotone system theory and the center manifold theory.For the classic models of infectious diseases, the basic reproduction number R₀ is a very important factor. It is defined as the average number of secondary infections produced when a infected individual is introduced into a host populationwhere everyone is susceptible. It is clear that R₀ = 1 is a threshold which determines whether an infection can invade or persist in a new host population. when R₀ < 1, the model has only a global stable disease-free equilibrium and the disease will naturally disappearance; when R₀ > 1, the model still has only global stable endemic equilibrium point and the disease will always exist in the polulations.Recently, there are many researchers are focused on how to find the basic reproductive number for a infectious diseases model. Two effective methods, the quarantine and vaccination for the epidemic individual, have become main schemes for the diseases control.In this paper,we construct and study a SIR epidemic model with constant vertical transmission, vaccinationa. we first give the equilibria of this model and develop the basis reproduction number determining whether the diseases occur and die out or not.First, we based on the following assumptions:1. Infectious diseases through mother-to-child transmission channels can be disseminated through the effective engagement;2. b and b' are the birth rate coefficient of non-infected (S+R) and infected I, dand d' are the mortality rate coefficient;3.γ> 0 is the recovery rate coefficient;4.m is the ratio of vaccination of the newborn from susceptible S and removed R (or understanding of the innate immunity);5.Set removed R are lifelong immunity;6.The freshmen who are not vaccinated are susceptible;7.q is the probability of vertical infection(p + q = 1);And let b = b',d = d',The model to be studied takes the following form:Based on the analysis ,we know the model (1) always has a disease-free equilibrium E₀(1 - m, 0) Define the basic reproduction number as followsWe can see that if R₀ > 1 ,then the model has a unique positive equilibriumWe have the following results on the stability of the equilibriumstheorem 1 If R₀≤l,then there is no endemic equilibrium,and the diseasefree equilibrium E₀ of (1) is globally asymptotically stable ;if R₀ > l,the diseasefree equilibrium E₀ is unstable,the endemic equilibrium E₊ is globally asymptotically stable.It shows that the following schemes can be used to control the spread of infectious disease:(1) Increase the vaccination rate of the newborns ,who is born by the susceptible individuals and the removed individuals.(2) Improve the recovery rateγ.If the illness is a bird or animal ,we could kill them: If the illness are human beings ,we can only speed up the study of drugs, improve the cure rate.(3) Reduce the probability of vertical transmission of infectious diseases, which can be taken artificial means to fewer sick or second generation of Health: increasing the research efforts on blocking vertical transmissionIn addition ,we construct a SIQS epidemic model as follows:where A is the recruitment rate of the population, a > 0,b≥0, c≥0, {a+b+c = 1) represent the inputing rate of the susceptible individuals ,infective individuals, isolation individuals , respectively.d > 0 represents the natural mortality rate andα> 0 represents the death rate due to disease respectively.k(k < d) is the birth rate and we assumed the newborns are susceptibles .εis the rate that the isolation returns to the susceptible.βis the transmission rate rely on the the total population N.We can see thatWhen 6 = 0 ,the model always has a disease-free equilibrium P₀(0,((cA)/(d+ε)),(A/(d-k))),and the basic reproduction numberIf R₀ > l,the model has a unique endemic equilibrium P*(I*, Q*, N*).WhereN* is unique solution of the following equation in the interval (0, (A/(d-k))If R₀≤1,the disease-free equilibrium P₀ is globally asymptotically stable,if R₀ > 1, the disease-free equilibrium P₀ is unstable ,the endemic equilibrium P* is local asymptotically stable.When 6 > 0 ,there are infected persons from the outside world population moves to the population,so the model has no disease-free equilibrium,but has aunique endemic equilibrium P*(I*,Q*,N*). Where I*,Q* satisfies (3).N* is unique solution of the following equation in the interval (0, (A(D-K))We have the theorem theorem 2 The unique endemic equilibrium P*(I*, Q*, N*) is globally asymptotically stable.The conclusions are:(1) increasing the death rate of diseases individual;(2) increasing isolated rate for the diseases invdividual are the effective method for the diseases control.