Exact analytical solutions of the Susceptible-Infected-Recovered (SIR) epidemic model and of the SIR model with equal death and birth rates

Abstract In this paper, the exact analytical solution of the Susceptible-Infected-Recovered (SIR) epidemic model is obtained in a parametric form. By using the exact solution we investigate some explicit models corresponding to fixed values of the parameters, and show that the numerical solution reproduces exactly the analytical solution. We also show that the generalization of the SIR model, including births and deaths, described by a nonlinear system of differential equations, can be reduced to an Abel type equation. The reduction of the complex SIR model with vital dynamics to an Abel type equation can greatly simplify the analysis of its properties. The general solution of the Abel equation is obtained by using a perturbative approach, in a power series form, and it is shown that the general solution of the SIR model with vital dynamics can be represented in an exact parametric form.

[1]  Graeme C. Wake,et al.  Lyapunov functions and global stability for SIR, SIRS, and SIS epidemiological models , 2002, Appl. Math. Lett..

[2]  Andrei Korobeinikov,et al.  Lyapunov functions for SIR and SIRS epidemic models , 2010, Appl. Math. Lett..

[3]  Rongyan Zhang,et al.  Bifurcation analysis for T system with delayed feedback and its application to control of chaos , 2013 .

[4]  Ganna Rozhnova,et al.  SIRS Dynamics on Random Networks: Simulations and Analytical Models , 2009, Complex.

[5]  F. Brauer,et al.  Mathematical Models in Population Biology and Epidemiology , 2001 .

[6]  Stefan C. Mancas,et al.  Periodic and Chaotic Traveling Wave Patterns in Reaction-Diffusion/ Predator-Prey Models with General Nonlinearities , 2012, 1302.2499.

[7]  P ? ? ? ? ? ? ? % ? ? ? ? , 1991 .

[8]  M. A. Sadiq,et al.  A note on Exact solution of SIR and SIS epidemic models , 2010, 1012.5035.

[9]  James D. Murray.  Mathematical Biology: I. An Introduction , 2007 .

[10]  J. Biazar,et al.  Solution of the epidemic model by Adomian decomposition method , 2006, Appl. Math. Comput..

[11]  Robert A. Van Gorder,et al.  Analytical Hopf Bifurcation and Stability Analysis of T System , 2011 .

[12]  G. Tigan,et al.  Analysis of a 3D chaotic system , 2006, math/0608568.

[13]  N. Rashevsky,et al.  Mathematical biology , 1961, Connecticut medicine.

[14]  Andrei Korobeinikov,et al.  Lyapunov Functions and Global Stability for SIR and SIRS Epidemiological Models with Non-Linear Transmission , 2006, Bulletin of mathematical biology.

[15]  Fengqin Zhang,et al.  Stability analysis for an epidemic model with stage structure , 2008 .

[16]  Tiberiu Harko,et al.  New method for generating general solution of Abel differential equation , 2002 .

[17]  Irina Mazilu,et al.  Stochastic epidemic-type model with enhanced connectivity: exact solution , 2012 .

[18]  Tiberiu Harko,et al.  Solutions generating technique for Abel-type nonlinear ordinary differential equations , 2001 .

[19]  B. Batiha,et al.  A New Method For Solving Epidemic Model , 2011 .

[20]  Rui Xu,et al.  Global stability of a SIR epidemic model with nonlinear incidence rate and time delay , 2009 .

[21]  Andrew J Black,et al.  Stochastic fluctuations in the susceptible-infective-recovered model with distributed infectious periods. , 2009, Physical review. E, Statistical, nonlinear, and soft matter physics.

[22]  W. O. Kermack,et al.  A contribution to the mathematical theory of epidemics , 1927 .

[23]  Tiberiu Harko,et al.  Relativistic dissipative cosmological models and abel differential equation , 2003 .

[24]  J. Graunt,et al.  Natural and political observations made upon the bills of mortality , 1939 .

[25]  Stefan C. Mancas,et al.  Ermakov-Pinney equations with Abel-induced dissipation , 2013 .

[26]  Peter Kulchyski.  and , 2015 .

[27]  Hamid Reza Mohammadi Daniali,et al.  Variational iteration method for solving the epidemic model and the prey and predator problem , 2007, Appl. Math. Comput..

[28]  Hamid Reza Mohammadi Daniali,et al.  Solution of the epidemic model by homotopy perturbation method , 2007, Appl. Math. Comput..

[29]  Toshikazu Kuniya,et al.  Existence of a nontrivial periodic solution in an age-structured SIR epidemic model with time periodic coefficients , 2014, Appl. Math. Lett..