Kermack-McKendrick epidemic model revisited. (English) Zbl 1137.37338

Štěpán, Josef; Hlubinka, Daniel

Kybernetika 43, No. 4, 395-414 (2007).

Summary: An epidemy of a highly infectious disease with a fast recovery (or fatality) in a homogeneous population is considered, the influenza being an example of such epidemics. This classical Kermack-McKendrick model assumes a fixed sized population of \(n\) individuals, the population being divided into three subpopulations which change their respective sizes in the running time of the epidemic: Susceptibles (the individuals exposed to the infection), infectives (the infected individuais that are able to spread the disease) and removals (the individuals restored to health not able further to spread the infection or get themselves to be infected again) numbering by \(x(t)\), \(y(t)\) and \(z(t)\) the individuais that are susceptible, infected and removed at some time \(t\geq 0\), respectively. Hence, \(x(t) + y(t) + z(t) = n\), \(x(t)\) and \(z(t)\) being generally a nonincreasing and nondecreasing function, respectively such that \(z(0) = 0\).
The model assumes the dynamics given by the following three dimensional differential equation
\[ \begin{alignedat}{2} & \dot x(t)=-\beta x(t)y(t),&\quad & x(0)=x_0>0,\\ & \dot y(t)= \beta x(t)y(t) - \gamma y(t),&\quad & y(0)=y_0=n-x_0>0,\tag{M1}\\ &\dot z(t)=\gamma y(t), &\quad & z(0)=0,\end{alignedat} \]
where the intensity \(\beta > 0\) is higher for more infectious diseases and the parameter \(\gamma^{-1} > 0\) is proportional to the average duration of the “being infected” status, i. e. to the average time for which an individual is infected.
Our aim is to propose and justify a diffusion version of (M1) model that allows both more general intensities \(\beta\) and to model the spread of epidemic in a population that changes its size \(N_t\) due to a diffusion type emigration and immigration processes defined for example by the Engelbert-Schmidt stochastic differential equation
\[ dN_t = N_t\sigma(N_t) dW_t,\quad N_0=n_0:=x_0+y_0. \]

MSC:

37N25	Dynamical systems in biology
60H10	Stochastic ordinary differential equations (aspects of stochastic analysis)
60H35	Computational methods for stochastic equations (aspects of stochastic analysis)
92D25	Population dynamics (general)
37H10	Generation, random and stochastic difference and differential equations

Keywords:

SIR epidemic models; stochastic differential equations; weak solution; simulation

References:

[1]	Allen E. J.: Stochastic differential equations and persistence time for two interacting populations. Dynamics of Continuous, Discrete and Impulsive Systems 5 (1999), 271-281 · Zbl 0946.60058
[2]	Allen L. J. S., Kirupaharan N.: Asymptotic dynamics of deterministic and stochastic epidemic models with multiple pathogens. Internat. J. Num. Anal. Model. 2 (2005), 329-344 · Zbl 1080.34033
[3]	Andersson H., Britton T.: Stochastic Epidemic Models and Their Statistical Analysis. (Lecture Notes in Statistics 151.) Springer-Verlag, New York 2000 · Zbl 0951.92021
[4]	Bailey N. T. J.: The Mathematical Theory of Epidemics. Hafner Publishing Comp., New York 1957
[5]	Ball F., O’Neill P.: A modification of the general stochastic epidemic motivated by AIDS modelling. Adv. in Appl. Prob. 25 (1993), 39-62 · Zbl 0777.92018
[6]	Becker N. G.: Analysis of infectious disease data. Chapman and Hall, London 1989 · Zbl 0782.92015
[7]	Daley D. J., Gani J.: Epidemic Modelling; An Introduction. Cambridge University Press, Cambridge 1999 · Zbl 0964.92035
[8]	Greenhalgh D.: Stochastic Processes in Epidemic Modelling and Simulation. Handbook of Statistics 21 (D. N. Shanbhag and C. R. Rao, North-Holland, Amsterdam 2003, pp. 285-335 · Zbl 1017.92030
[9]	Hurt J.: Mathematica\(^{\textregistered }\) program for Kermack-McKendrick model. Department of Probability and Statistics, Charles University in Prague 2005
[10]	Kallenberg O.: Foundations of Modern Probability. Second edition. Springer-Verlag, New York 2002 · Zbl 0996.60001
[11]	Kendall D. G.: Deterministic and stochastic epidemics in closed population. Proc. Third Berkeley Symp. Math. Statist. Probab. 4, Univ. of California Press, Berkeley, Calif. 1956, pp. 149-165
[12]	Kermack W. O., McKendrick A. G.: A contribution to the mathematical theory of epidemics. Proc. Roy. Soc. London Ser. A 115 (1927), 700-721 · JFM 53.0517.01
[13]	Kirupaharan N.: Deterministic and Stochastic Epidemic Models with Multiple Pathogens. PhD Thesis, Texas Tech. Univ., Lubbock 2003 · Zbl 1080.34033
[14]	Kirupaharan N., Allen L. J. S.: Coexistence of multiple pathogen strains in stochastic epidemic models with density-dependent mortality. Bull. Math. Biol. 66 (2004), 841-864 · Zbl 1334.92408
[15]	Rogers L. C. G., Williams D.: Diffusions, Markov Processes and Martingales. Vol. 1: Foundations. Cambridge University Press, Cambridge 2000 · Zbl 0977.60005
[16]	Rogers L. C. G., Williams D.: Diffusions, Markov Processes and Martingales. Vol. 2: Itô Calculus. Cambridge University Press, Cambridge 2000 · Zbl 0977.60005
[17]	Štěpán J., Dostál P.: The \(dX(t)=Xb(X)dt + X\sigma (X)\,dW\) equation and financial mathematics I. Kybernetika 39 (2003), 653-680 · Zbl 1249.91128
[18]	Štěpán J., Dostál P.: The \(dX(t)=Xb(X)dt + X\sigma (X)dW\) equation and financial mathematics II. Kybernetika 39 (2003), 681-701 · Zbl 1249.60128
[19]	Subramaniam R., Balachandran, K., Kim J. K.: Existence of solution of a stochastic integral equation with an application from the theory of epidemics. Nonlinear Funct. Anal. Appl. 5 (2000), 23-29 · Zbl 0967.60065