| Avian influenza is a zoonotic disease caused by the transmission of the avian influenza A virus.High pathogenic avian influenza has the characteristics of high mortality,and it not only affects the poultry farming industry but also poses a great threat to human health.Therefore,it is of great theoretical value and practical significance to study the transmission rule and control strategies of highly pathogenic avian influenza.By analyzing the characteristics of avian influenza transmission and considering the different factors affecting the transmission of avian influenza,five avian influenza models are established by using the modeling idea of infectious disease dynamics in this paper.The dynamic behavior of these models are analyzed and the optimal control of avian influenza are further studied.The details are as follows:(1)Since poultry slaughtering can control the spread of avian influenza in time,a delayed SI-SIR avian influenza model with poultry slaughtering is established.First of all,the local asymptotic stability of the equilibria of the system are studied by using Routh-Hurwitz criterion and relat-ed theory,and the global asymptotic stability of the equilibria are proved through constructing Lyapunov function and applying LaSalle invariant principle.Secondly,the corresponding opti-mal control problem is formulated by introducing the control variables,namely the slaughtering for poultry and education campaign for the susceptible humans.Meanwhile,the existence and uniqueness of the optimal control is investigated.In the end,the accuracy of the analytical results are verified by numerical simulations.The numerical results show that slaughtering sus-ceptible(infected)poultry and educating susceptible humans can not only reduce the number of infected poultry(humans),but also minimize the cost of implementing these control strategies.(2)On the basis of research content(1),a delayed SI-SIR avian influenza model with Beddington-DeAngelis incidence rate in both poultry and human population is established.Then,three control variables(one with time delay)are introduced to study the optimal control of the mod-el.The conclusions obtained in this part are as follows:implementation of hybrid controls can significantly reduce the number of infected poultry(humans),and as the delay τ1 or τ2 increases,the transmission rate of avian influenza will decline.Therefore,in order to prevent the spread of avian influenza,relevant departments should take multiple control strategies at the same time as soon as possible,which can delay the outbreak of avian influenza and reduce the cost of control the avian influenza.(3)A stochastic delayed SEI-SIR avian influenza model with vaccination and mutation is proposed by considering the avian influenza virus mutates.Firstly,the existence and uniqueness of glob-al positive solutions of the system are studied.Secondly,the thresholds of persistence and extinction for the stochastic avian-only subsystem and the stochastic avian-human system are investigated by using Lyapunov functional method.The research results show that if R01s<1,the avian influenza dose not spread in poultry population.Further,if R02s<1,neither avian influenza nor the mutant avian influenza is transmitted among human population;On the con-trary,if R02s>1,avian influenza does not spread in the human population but the mutant avian influenza spreads in the human population.(4)A stochastic delayed SEI-SIR avian influenza model with saturated incidence rate is given by considering the incubation period of avian influenza virus in poultry.At first,the existence and uniqueness of positive solutions of the system under certain conditions are proved.Then,the asymptotic behavior of the solutions of stochastic system near the equilibrium are inves-tigated by constructing appropriate Lyapunov functionals and using Young inequality,Holder inequality and so on.(5)A stochastic SI-SEIR avian influenza model with psychological effects is presented.Then,the problem of stochastic near-optimal control is proposed by introducing slaughtering for poultry and treatment for infected humans into the model,and the sufficient and necessary conditions for the existence of stochastic near-optimal control are given by using the Ekeland theorem. |
