一类考虑垂直传染、接种及人均病床数的SIVS传染病模型分析

Analysis of an SIVS Epidemic Model with Vertical Transmission,Vaccination and Number of Beds Per Capita

基于疫苗接种不完全有效的实际情况,研究了一类考虑垂直传染、接种及人均病床数的SIVS传染病模型,同时采用与人均病床数有关的治疗函数以研究医疗资源数量对传染病防治工作的影响.首先,利用下一代生成矩阵法得到疾病基本再生数的表达式,并且分别采用几何法和特征值法得到各平衡点稳定性的一些结论;其次,通过对平衡点的讨论发现此类模型会发生后向分支,同时结合理论证明和数值模拟对其进行验证;最后,基于已有文献对加入垂直传染和新生儿接种因素后的理论结果异同点进行比较,得出了通过大幅减少有效接触和提供丰富的医疗资源可以避免发生后向分支从而消灭疾病,通过增加疫苗的接种比例和接种效率可以控制疾病传播的结论.

In this paper,based on the fact that vaccination is not fully effective,a SIVS infectious disease model considering vertical transmission,vaccination and per capita bed number is studied.Meanwhile,the treatment function related to per capita bed number has been used to study the influence of medical resources on the prevention and treatment of infectious diseases.Firstly,the expression of the basic reproduction has been obtained with the next generation matrix and the existence and stability of every points are discussed by means of geometric method and eigenvalue method respectively.Secondly,through the discussion of the equilibrium points,it has been found that it will appear backward bifurcation in such model and the condition of the backward bifurcation is verified by theoretical proof and numerical simulation.And finally,the similarities and differences of the theoretical results after adding the factors of vertical infection and neonatal vaccination are compared based on the existing literature.It has been shown that minimal effective contact and abundant medical resources could avoid the appearance of backward bifurcation,leading to disease extinction finally.Meanwhile the epidemic diseases can also be controlled by increasing vaccination rates and improving the effectiveness of vaccination.