Traveling Wave Solutions In A Two-group SIR Epidemic Model With Nonlocal Dispersal

In this paper,we study the existence and nonexistence of traveling wave solutions in a two-group epidemic model with nonlocal dispersal,and the continuous dependence of minimum wave speed on parameters.Firstly,the Fourier transformation method is used to derive the specific model studied in this paper.That is,a class of a two-group SIR epidemic model with nonlocal dispersal.Secondly,the existence and nonexistence of traveling wave solutions in a two-group SIR epidemic model with nonlocal dispersal are studied under the consideration of latent period.Due to the introduction of nonlocal operators,the solution of the system itself does not have enough regularity,so the existing methods of the existence of the traveling wave solutions can not be applied directly.In order to overcome this difficulty,we use truncation method to study the existence of wave solutions when the basic reproduction numberR0?S₁⁰,S₂⁰?>1 and c>c*,where c*is the critical wave velocity.The existence of traveling wave solutions is obtained by constructing closed cones on bounded regions and using Schauder's fixed point theorem and approximation method.Furthermore,the boundary asymptotic behavior of the traveling wave solution is discussed.Because the solution does not have sufficient regularity,the boundedness of the affected individual and the asymptotic behavior of the susceptible individual at +? are difficult to obtain.Nevertheless,this paper overcomes this difficulty by careful analysis under the condition of high wave velocity.On the other hand,for the case of Ro?S10^,S20?>1 and c??a,c*?,the nonexistence of traveling wave solutions is studied by using the bilateral laplace transform.At the same time,the nonexistence of traveling wave solutions is discussed by Perron-Frobenius theorem when R0?S10:S20?? 1.Finally,we consider the continuous dependence of the minimum wave velocity c*on the parameters.The diffusion rate Di of the infected individuals,the diffusion rate DLi of latent individual,and the infection rate ?ij can accelerate the spread of the disease.Emigration rate ri of the infected individuals,the migration rate Mi of latent individuals and latent period T can slow the spread of the disease.