Deciphering Dynamics of Epidemic Spread: The Case of Influenza Virus

In this paper, we have proposed and analyzed a simple model of Influenza spread with an asymptotic transmission rate. Existence and uniqueness of solutions are established and shown to be uniformly bounded for all non-negative initial values. We have also found a sufficient condition which ensures the persistence of the model system. This implies that both susceptible and infected will always coexist at any location of the inhabited domain. This coexistence is independent of values of the diffusivity constants for two subpopulations. The global stability of the endemic equilibrium is established by constructing a Lyapunov function. By linearizing the system at the positive constant steady-state solution and analyzing the associated characteristic equation, conditions for Hopf and Turing bifurcations are obtained. We have also studied the criteria for diffusion-driven instability caused by local random movements of both susceptible and infective subpopulations. Turing patterns selected by the reaction–diffusion system under zero flux boundary conditions have been explored. Numerical simulations show that contact rate, β which is related to the reproduction number (R0 = β α ), plays an important role in spatial pattern formation. It was found that diffusion has appreciable influence on spatial spread of epidemics. The wave of chaos appears to be a dominant mode of disease dispersal. This suggests a bidirectional spread for influenza epidemics. The epidemic propagates in the form of nonchaotic and chaotic waves as observed in H1N1 incidence data of positive tests in 2009 in the United States. We have conducted numerical simulations to confirm the analytic work and observed interesting behaviors. This suggests that influenza has a complex dynamics of spatial spread which evolves with time.