Kermack-McKendrick epidemic model revisited

Josef Štěpán; Daniel Hlubinka

Kybernetika (2007)

  • Volume: 43, Issue: 4, page 395-414
  • ISSN: 0023-5954

Abstract

top
This paper proposes a stochastic diffusion model for the spread of a susceptible-infective-removed Kermack–McKendric epidemic (M1) in a population which size is a martingale N t that solves the Engelbert–Schmidt stochastic differential equation (). The model is given by the stochastic differential equation (M2) or equivalently by the ordinary differential equation (M3) whose coefficients depend on the size N t . Theorems on a unique strong and weak existence of the solution to (M2) are proved and computer simulations performed.

How to cite

top

Štěpán, Josef, and Hlubinka, Daniel. "Kermack-McKendrick epidemic model revisited." Kybernetika 43.4 (2007): 395-414. <http://eudml.org/doc/33866>.

@article{Štěpán2007,
abstract = {This paper proposes a stochastic diffusion model for the spread of a susceptible-infective-removed Kermack–McKendric epidemic (M1) in a population which size is a martingale $N_t$ that solves the Engelbert–Schmidt stochastic differential equation (). The model is given by the stochastic differential equation (M2) or equivalently by the ordinary differential equation (M3) whose coefficients depend on the size $N_t$. Theorems on a unique strong and weak existence of the solution to (M2) are proved and computer simulations performed.},
author = {Štěpán, Josef, Hlubinka, Daniel},
journal = {Kybernetika},
keywords = {SIR epidemic models; stochastic differential equations; weak solution; simulation; SIR epidemic models; stochastic differential equations; weak solution; simulation},
language = {eng},
number = {4},
pages = {395-414},
publisher = {Institute of Information Theory and Automation AS CR},
title = {Kermack-McKendrick epidemic model revisited},
url = {http://eudml.org/doc/33866},
volume = {43},
year = {2007},
}

TY - JOUR
AU - Štěpán, Josef
AU - Hlubinka, Daniel
TI - Kermack-McKendrick epidemic model revisited
JO - Kybernetika
PY - 2007
PB - Institute of Information Theory and Automation AS CR
VL - 43
IS - 4
SP - 395
EP - 414
AB - This paper proposes a stochastic diffusion model for the spread of a susceptible-infective-removed Kermack–McKendric epidemic (M1) in a population which size is a martingale $N_t$ that solves the Engelbert–Schmidt stochastic differential equation (). The model is given by the stochastic differential equation (M2) or equivalently by the ordinary differential equation (M3) whose coefficients depend on the size $N_t$. Theorems on a unique strong and weak existence of the solution to (M2) are proved and computer simulations performed.
LA - eng
KW - SIR epidemic models; stochastic differential equations; weak solution; simulation; SIR epidemic models; stochastic differential equations; weak solution; simulation
UR - http://eudml.org/doc/33866
ER -

References

top
  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 (1999) Zbl0946.60058MR1678255
  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 Zbl1080.34033MR2112651
  3. Andersson H., Britton T., Stochastic Epidemic Models and Their Statistical Analysis, (Lecture Notes in Statistics 151.) Springer–Verlag, New York 2000 Zbl0951.92021MR1784822
  4. Bailey N. T. J., The Mathematical Theory of Epidemics, Hafner Publishing Comp., New York 1957 MR0095085
  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 (1993) Zbl0777.92018MR1206532
  6. Becker N. G., Analysis of infectious disease data, Chapman and Hall, London 1989 Zbl0782.92015MR1014889
  7. Daley D. J., Gani J., Epidemic Modelling; An Introduction, Cambridge University Press, Cambridge 1999 Zbl0964.92035MR1688203
  8. Greenhalgh D., Stochastic Processes in Epidemic Modelling and Simulation, In: Handbook of Statistics 21 (D. N. Shanbhag and C. R. Rao, eds.), North–Holland, Amsterdam 2003, pp. 285–335 Zbl1017.92030MR1973547
  9. Hurt J., Mathematica ® 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 Zbl0996.60001MR1876169
  11. Kendall D. G., Deterministic and stochastic epidemics in closed population, In: Proc. Third Berkeley Symp. Math. Statist. Probab. 4, Univ. of California Press, Berkeley, Calif. 1956, pp. 149–165 (1956) MR0084936
  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 (1927) 
  13. Kirupaharan N., Deterministic and Stochastic Epidemic Models with Multiple Pathogens, PhD Thesis, Texas Tech. Univ., Lubbock 2003 Zbl1080.34033MR2704799
  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 MR2255779
  15. Rogers L. C. G., Williams D., Diffusions, Markov Processes and Martingales, Vol. 1: Foundations. Cambridge University Press, Cambridge 2000 Zbl0977.60005MR1796539
  16. Rogers L. C. G., Williams D., Diffusions, Markov Processes and Martingales, Vol. 2: Itô Calculus. Cambridge University Press, Cambridge 2000 Zbl0977.60005MR1780932
  17. Štěpán J., Dostál P., The d X ( t ) = X b ( X ) d t + X σ ( X ) d W equation and financial mathematics I, Kybernetika 39 (2003), 653–680 MR2035643
  18. Štěpán J., Dostál P., The d X ( t ) = X b ( X ) d t + X σ ( X ) d W equation and financial mathematics II, Kybernetika 39 (2003), 681–701 MR2035644
  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 MR1795707

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.