Traveling waves of the spread of avian influenza

Communications Faculty Of Science University of Ankara Series A1Mathematics and Statistics, 2017

In this paper, we aim to study spread of an epidemic in a spatially strati…ed population with non-overlapping generations. We consider mean …eld equation of an endemic chain-binomial process and allow individuals to disperse in the spatial habitat. To be able to model the spatial movement, we used an averaging kernel. The existence of traveling waves for traveling wave speeds greater than a certain minimum is proved. In addition, an explicit formula for the critical wave speed is given in terms of the moment generating function of the dispersal kernel and the basic reproductive ratio of the infectives.

British Journal of Mathematics & Computer Science, 2017

In this paper, we investigate the global stability and the existence of traveling waves for a delayed diffusive epidemic model. The disease transmission process is modeled by a specific nonlinear function that covers many common types of incidence rates. In addition, the global stability of the disease-free equilibrium and the endemic equilibrium is established by using the direct Lyapunov method. By constructing a pair of upper and lower solutions and applying the Schauder fixed point theorem, the existence of traveling wave solution which connects the two steady states is obtained and characterized by two parameters that are the basic reproduction number and the minimal wave speed. Furthermore, the models and main results studied the existence of traveling waves presented in the literature are extended and generalized.

Journal of Theoretical Biology, 2003

Between major pandemics, the influenza A virus changes its antigenic properties by accumulating point mutations (drift) mainly in the RNA genes that code for the surface proteins hemagglutinin (HA) and neuraminidase (NA). The successful strain (variant) that will cause the next epidemic is selected from a reduced number of progenies that possess relatively high transmissibility and the ability to escape from the immune surveillance of the host. In this paper, we analyse a one-dimensional model of influenza A drift (Z. Angew. Math. Mech. 76 (2) (1996) 421) that generalizes the classical SIR model by including mutation as a diffusion process in a phenotype space of variants. The model exhibits traveling wave solutions with an asymptotic wave speed that matches well those obtained from numerical simulations. As exact solutions for these waves are not available, asymptotic estimates for the amplitudes of infected and recovered classes are provided through an exponential approximation based on the smallness of the diffusion constant. Through this approximation, we find simple scaling properties to several parameters of relevance to the epidemiology of the disease. r

Journal of the Indonesian Mathematical Society, 2010

We propose two disease transmission models: the SI model with di®u-sive terms as a model for transmission of avian in°uenza viruses among °ock withconstant population density, and the host-vector SI model with di®usive terms fortransmission among °ock and human. There is a threshold number that determinewhether the diseases become epidemic or not. It depends on three parameters: theaverage length of the infective period, the contact parameter, and the density ofsusceptible population at the initial time. With di®usive terms in the model, thespatial spread of the infection is obvious. Threshold number greater than one isalso the necessary condition for the spreading of infection in the form of a travelingpulse. Its minimum velocity can be determined. We implement the MacCormackmethod, and simulate the generation of a traveling pulse starting from one infectedbird. Numerically, we obtain the percentage of survival in °ock and human afterthe epidemic. We also simulate the minimal porti...

Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, 2009

We obtain full information about the existence and non-existence of travelling wave solutions for a general class of diffusive Kermack–McKendrick SIR models with non-local and delayed disease transmission. We show that this information is determined by the basic reproduction number of the corresponding ordinary differential model, and the minimal wave speed is explicitly determined by the delay (such as the latent period) and non-locality in disease transmission, and the spatial movement pattern of the infected individuals. The difficulty is the lack of order-preserving property of the general system, and we obtain the threshold dynamics for spatial spread of the disease by constructing an invariant cone and applying Schauder’s fixed point theorem.

Mathematical Modelling of Natural Phenomena, 2008

Mechanisms contributing to the spread of avian influenza seem to be well identified, but how their interplay led to the current worldwide spread pattern of H5N1 influenza is still unknown due to the lack of effective global surveillance and relevant data. Here we develop some deterministic models based on the transmission cycle and modes of H5N1 and focusing on the interaction among poultry, wild birds and environment. Some of the model parameters are obtained from existing literatures, and others are allowed to vary in order to assess the effectiveness of various control strategies involving bird migration, agro-ecological environments, live and dead poultry trading, smuggling of wild birds, mechanical movement of infected materials and specific farming practices. Our simulations are carried out for a set of parameters that leads to the basic reproduction number 3.3. We show that by reducing 95% of the initial susceptible poultry population or by killing all infected poultry birds within one day, one may control the disease outbreak in a local setting. Our simulation shows that cleaning the environment is also a feasible and useful control measure, but culling wild birds and destroying their habitat are ineffective control measures. We use a one dimensional PDE model to examine the contribution to the spatial spread rate by the size of the susceptible poultry birds, the diffusion rates of the wild birds and the virus. We notice the diffusion rate of the wild birds with high mortality has very little impact on the spread speed. But for the wild birds who can survive the infection, depending on the direction of convection, their diffusion rate can substantially increase the spread rate, indicating a significant role of the migration of these type of wild birds in the spread of the disease.

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. Keywords: Influenza virus; epidemics; persistence; pandemic; Hopf bifurcation; Turing instability; wave of chaos. 1.

Journal of elliptic and parabolic equations, 2024

This paper deals with the asymptotic behavior of solutions for the diffusive epidemic model with logistic growth. In the first part, we consider the initial boundary value problem on the bounded domain and derive the stabilization of the solutions of the reaction-diffusion system to a constant equilibrium. In the second part, we consider the initial value problem on R, and derive the stability of forced waves under certain perturbations of a class of initial data.

Proceedings of the Royal Society of Edinburgh: Section A Mathematics, 2005

The existence and uniqueness of travelling-wave solutions is investigated for a system of two reaction-diffusion equations where one diffusion constant vanishes. The system arises in population dynamics and epidemiology. Travelling-wave solutions satisfy a three-dimensional system about (u, u , v), whose equilibria lie on the u-axis. Our main result shows that, given any wave speed c > 0, the unstable manifold at any point (a, 0, 0) on the u-axis, where a ∈ (0, γ) and γ is a positive number, provides a travelling-wave solution connecting another point (b, 0, 0) on the u-axis, where b := b(a) ∈ (γ, ∞), and furthermore, b(·) : (0, γ) → (γ, ∞) is continuous and bijective.

Applied Mathematical Modelling, 2011

A diffusive epidemic model for H1N1 influenza is formulated with a view to gain basic understanding of the virus behavior. All newborns are assumed to be susceptible. Mortality rate for infective individuals in the population is assumed to be greater than natural mortality rate. Latent, infectious and immune periods are assumed to be constants throughout this study. The numerical solutions of this model are carried out under three different initial populations distribution. In order to investigate the effect of the disease transmission coefficient on the spread of disease, b is taken to be constant as well as a function of seasonally varying time t and a function of spatial variable x. The threshold quantity ðR 0 Þ that governs the disease dynamics is derived. Numerical simulation shows that the system supports the existence of sustained and damped oscillations depending on initial populations distribution, the disease transmission rate and diffusion.