| In this thesis, we discuss the dynamics of a stochastic SIQS epidemic model.One intervention procedure to control the spread of infectious diseases is to isolate some infectives, in order to reduce transmissions of the infection to susceptibles. Isolation may have been the first infection control method. Over the centuries quarantine has been used to reduce the transmission of human diseases such as leprosy, plague, cholera, typhus, yellow fever, smallpox, diph-theria, tuberculosis, measles, mumps, ebola and lassa fever. Quarantine has also been used for animal diseases such as rinderpest,foot and mouth, psittaco-sis, Newcastle disease and rabies. Studies of epidemic models with quarantine have become an important area in the mathematical theory of epidemiology, and they have largely been inspired by the works [12,40,55]. Hethcote et al. in [20] discuss an SIQS epidemic model. In fact, epidemic models are in-evitably affected by environmental white noise that is an important component in realism. In Chapter 2, we discuss the following stochastic SIQS model: where parameters A, μ and β are positive constants, and α,γ, ε and δ are non-negative constants, B(t) is a Brownian motion. Here S(t) denotes the number of members who are susceptible to an infection at time t. I(t) denotes the number of members who are infective at time t. Q(t) denotes the num-ber of members who are removed and isolated either voluntarily or coercively from the infectious class. Here A denotes the recruitment rate of susceptibles corresponding to births and immigration;β denotes transmission coefficient between compartments S and I;μ denotes the per capita natural mortality rate;δ denotes the rate for individuals leaving the infective compartment I for the quarantine compartment Q; γ denotes recovery rate of infectious individu-als; ε denotes the rates at which individuals return to susceptible compartment S from compartments Q; α denotes disease-caused death rate of infectious in-dividuals.In this Chapter, we show there exists a unique positive solution of the stochastic SIQS system. The solution of the stochastic SIQS system is expo-nential stability when the volatility of noise is large. In this case, the infective decays exponentially to zero. When the volatility of noise is small, we deduce the condition R0< 1 which will enable the disease to die out exponentially and the condition R0> 1 for the disease being persistent is given. Some outcomes of numerical simulations are reported to illustrate exists the analytical results.Theorem 0.1. There is a unique solution (S(t),I(t),Q(t)) of system (0.0.3) on t≥0 for any initial value (S(0),I(0), Q(0)) ∈ R+, and the solution will remain in R+3 with probability 1, namely, (S(t),I(t),Q(t)) ∈R+3 for all t≥0 almost surely.Theorem 0.2. Let (S(t),I(t),Q(t)) be the solution of system (0.0.3) with initial value (S(0),7(0),Q(0)) ∈Γ*. Assume that Then namely,I(t) tends to zero exponentially a.s.i.e.,the disease will die out withTheorem 0.3 If R0>1 and σ2<βμ/A,then for any initial Value (S(0),I(0),Q(0))∈Γ* ,the solution (S(t),IO(t),Q(t))of system(1.3) has the following properties: where Moreover. andEpidemic models may suffer sudden environmental perturbations, that is, some jump type stochastic perturbations, e.g., earthquakes, hurricanes. In Chapter 3, we discuss the following stochastic SIQS model driven by Levy process: where B(t) is a standard Brownian motion, Nis a Possion measure with com-pensator N, which is in dependent of the Brownian motion. F, F1, G, G1, H, H1 are constants.In this Chapter, we study the stochastic SIQS system driven by Levy noise. We show the existence and uniqueness of the positive solutions of the stochastic SIQS system driven by Levy noise. We obtain the asymptotic be-havior of the stochastic system around the disease-free equilibrium point and endemic disease equilibrium point by Lyapunov function.Theorem 0.4.Let (S(0),I(0),Q(0)) ∈ D={(S,I,Q) ∈R3:S>0,I> 0,Q>0,S+I+Q≤A/n}, and (S(0),/(0),Q(0)) is independent of σ(B). Then the systems (0.0.4) admits a unique continuous time, global solution (S(t),I(t), Q(t)) on t≥0 and this solution is invariant with respect to D.Theorem 0.5.If Then where K= min{2μ-ε-F2-∫Y(F12(u)+F1(u)G1(u))γ(du),2(μ+δ+γ+α)- ε-G2-∫G12(u)+F1(u)+G1(u)+γ(du),2(μ+α)-ε-H2-∫YH12(u)γ(du)}.Theorem 0.6.If Then where k2=2(μ+δ+α)(μ+ε+α)-G2(μ+ε+α)-c2δ-(μ+ε+α)∫YG12(u)γ(du), K3=μ+ε+α-H2-c2/2∫YH2(u)γ(du),K=min{k1/2,k2/(2(μ+ε+α)),k3/2},M is a constant. |
