Cannon, John R.; Galiffa, Daniel J. An epidemiology model suggested by yellow fever. (English) Zbl 1232.92056 Math. Methods Appl. Sci. 35, No. 2, 196-206 (2012). Summary: We construct and analyze a nonlinear reaction-diffusion epidemiology model consisting of two integral-differential equations and an ordinary differential equation, which is suggested by various insect borne diseases, for example yellow fever. We begin by defining a nonlinear auxiliary problem and establishing the existence and uniqueness of its solution via a priori estimates and a fixed point argument, from which we prove the existence and uniqueness of the classical solution to the analytic problem. Next, we develop a finite-difference method to approximate our model and perform some numerical experiments. We conclude with a brief discussion of some subsequent extensions. Cited in 5 Documents MSC: 92C60 Medical epidemiology 45K05 Integro-partial differential equations 35K57 Reaction-diffusion equations 65M06 Finite difference methods for initial value and initial-boundary value problems involving PDEs 65R20 Numerical methods for integral equations 65D30 Numerical integration 92D30 Epidemiology 37N25 Dynamical systems in biology Keywords:existence; uniqueness; fixed point; nonlinear; nonlocal; integral equations PDF BibTeX XML Cite \textit{J. R. Cannon} and \textit{D. J. Galiffa}, Math. Methods Appl. Sci. 35, No. 2, 196--206 (2012; Zbl 1232.92056) Full Text: DOI OpenURL References: [1] Capasso, Convergence to equilibrium states for a reaction-diffusion system modeling the spatial spread of a class of bacterial and viral diseases, Journal of Mathematical Biology 13 pp 173– (1981) · Zbl 0468.92016 [2] Anita, Note on the stabilization of a reaction-diffusion model in epidemiology, Nonlinear Analysis: Real World Applications 6 pp 537– (2005) · Zbl 1138.93399 [3] Capasso, Asymptotic stability for an integro-differential reaction-diffusion system, Journal of Mathematical Analysis and Applications 103 pp 575– (1984) · Zbl 0595.45020 [4] Capasso, Analysis of a reaction-diffusion system modeling man-environment-man epidemics, SIAM Journal on Applied Mathematics 57 pp 327– (1997) · Zbl 0872.35053 [5] Anita, A stabilization strategy for a reaction-diffusion system modeling a class of spatially structured epidemic systems (think globally, act locally), Nonlinear Analysis: Real World Applications 10 pp 2026– (2009) · Zbl 1163.91510 [6] Thieme, Asymptotic speeds of spread and traveling waves for integral equations and delayed reaction-diffusion models, Journal of Differential Equations 195 pp 430– (2003) · Zbl 1045.45009 [7] Wu, Asymptotic speed of spread and traveling fronts for a nonlocal reaction-diffusion model with distributed delay, Applied Mathematical Modeling 33 pp 2757– (2009) · Zbl 1205.35150 [8] Crosby, The American Plague: The Untold Story of Yellow Fever, the Epidemic that Shaped Our History (2006) [9] Cannon, A numerical method for a nonlocal elliptic boundary value problem, Journal of Integral Equations and Applications 20 pp 243– (2008) · Zbl 1149.65099 [10] Cannon, On a numerical method for a homogeneous, nonlinear, nonlocal, elliptic boundary value problem, Nonlinear Analysis 74 pp 1702– (2010) · Zbl 1236.34028 [11] Cannon, The One-Dimensional Heat Equation, Encyclopedia of Mathematics and Its Applications (1984) · Zbl 0567.35001 [12] Courant, Methods of Mathematical Physics II (1962) [13] Diekmann, Thresholds and traveling waves for the geographical spread of infection, Journal of Mathematical Biology 6 pp 109– (1978) · Zbl 0415.92020 [14] Diekmann, Run for your life. A note on the asymptotic speed of propagation of an epidemic, Journal of Differential Equations 33 pp 58– (1979) · Zbl 0377.45007 [15] Thieme, Asymptotic estimates of the solutions of nonlinear integral equations and asymptotic speeds for the spread of populations, Journal für die reine und angewandte Mathematik 306 pp 94– (1979) · Zbl 0395.45010 [16] Thieme, Density-dependent regulation of spatially distributed populations and their asymptotic speed of spread, Journal of Mathematical Biology 8 pp 173– (1979) · Zbl 0417.92022 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.