Periodic Solutions Of SIR Models And Suspension Bridges Models

Periodic motion exists commonly in the real world, the study on period-ic solutions of ordinary differential equation has been an important issue in the fields of dynamics for a long time. The periodic solutions of ordinary d-ifferential equations which have extensive biological and physical background, can not only represent some cyclical motion, but also depict some aperiodic movement approximatively. In this dissertation, we mainly discuss periodic solutions of SIR models and suspension bridges models.This dissertation is divided into four chapters.In the first chapter, we introduced the origin and development of SIR epi-demic model. Furthermore, we introduce the periodic forced SIR model, SIR model with pulse vaccination and SIR model with media coverage. Moreover, we introduce the suspension bridge model with periodic damping term. In this chapter, we list the basic concepts and some important theorems that we will use in this dissertation and presnet the main result of this dissertation in detail.In the second chapter, we study the existence of periodic solutions of SIR model with both periodic transmission rate and pulse vaccination: where The susceptible population will be vaccinated for k times largely because the susceptible population can be divided into many groups and all the groups can not be vaccinated at the same time. Our vaccination strategies concern the impact of infected population, which can be formulated as where0≤pi<1,α>0large enough.Denote the basic reproduction number with β=1/T∫0Tβ(t)dt. Recently, G. Katriel [58] get the existence of period-ic positive solutions for the periodically forced SIR model by Leray-Schauder degree theory provided R0>1. In this chapter, by Gaines-Mawhin’s contin-uation theorem we get the existence of positive periodic solutions of system (0.0.7). Some numerical simulation are presented to illustrate the effectiveness of such pulse vaccination strategy.In the third chapter, we study the following periodic forced SIR model: The media coverage is an important factor responsible for the transmission of the infectious disease.When a type of contagious disease appears and starts to spread,people’s response to the threat of disease is dependent on their per-ception risk,which is affected by public and private information disseminated widely by the media.We use a decreasing piecewise smooth function f(Ⅰ) to describe the impact of media coverage on the transmission coefficient,given by where a is the factor of influences,σ is a small parameter and Ic is a critical level.Denote R0=β/(γ+μ) with β=1/T∫0Tβ(t)dt.Recently,Wang Wang and Xiao use piecewise continuous transmission rate to describe that the media coverage exhibits its efect once the number of infected individuals exceeds a certain critical level[102].We think that,(I)in(0.0.9)is a good approximation to the discontinuous factor in[102]provided σ is small enough.When f (I)=1in(0.0.9),G.Katriel[58]get the existence of periodic positive solutions for the periodically forced SIR model by Leray-Schauder degree theory provided R0>1.But the methods we mentioned above can not deal with the non-smooth righthand sides directly In this paper,we use an integrall version of Leray-Schauder degree theory under G.Katriel’s frame to prove the existence of periodic solutions of system(0.0.8)-(0.0.9),whenever R0>eα.Some numerical simulations are presented to illustrate the effectiveness of such media coverage. Our main results are as follows.In the fourth chapter,we study the following problem: Utt+p(t)Ut+cUxxxx+du+=h(t,x),(0.0.10) with the boundary conditions U(0,t)=U(L,t)=Uxx(0,t)=Uxx(L,t)=0,(0.0.11) where p(t) is a2π-periodic damping term and h(t,x)=(sin πx/L)f(t) is the2π-periodic external force as same as the assumption in [108]. Looking for a standing-wave solution of (0.0.10) and (0.0.11), we have U(t,x)=(sin πx/L)u(t), which leads to an equivalent ordinary differential equation u"+p(t)u’+bu+-αu-=f(t),(0.0.12) in which α=c(π/L)4and b=d+c(π/L)4.The existence of periodic solutions for ordinary differential equations with variable coefficients damping term are very difficult to handle. In early1990s, Li [123] obtained an ingenious method to discuss the existence and uniqueness of nonlinear two-point boundary value problems with variable coefficient. Re-cently, Zu [124] extended this method to the periodic situation. Under the Dolph-type condition and a small periodic damping term condition, we get existence and uniqueness of periodic solution for the problem (0.0.12) by the above constructive method. Our constructive method is very adaptable to this kind of non-smooth problem. We take some numerical simulations to illustrate the effect of periodic damping term. By the numerical experiment, we know that the effect of the small periodic damping term is limited.