Stochastic SIR Household Epidemic Model with Misspecification
Keywords:
Final size epidemic, infectious period distribution, maximum likelihood estimates, misclassification probabilities
Abstract
The stochastic SIR household epidemic model is well discussed in [3], [4], [5] and also in [1]
by assuming that the infection period distribution is known. Sometimes this may wrongly
be assumed in the model estimation and hence the adequacy of the model fittness to the
final size data is affected. we examined this problem using simulations with large population size and theoretical parameters in which the final size data is first simulated with exp(4.1) infectious period distribution and estimated with Gamma(2,4.1/2) infectious period distribution and vice versa. The estimates of the two dimensional models are further explored for a range of local and global infection rates with corresponding proportion infected and found to be biased and imprecise.
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References
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20
[8] F. G. BALL AND P. NEAL, A general model for the stochastic SIR epidemic with two
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[10] , R. BARTOSZYNSKI, Branching Processes and the Theory of Epidemic. Fifth Berkeley
symposium on Mathematical Statistics and Probability, University of California Press,
Vol. 4, (1967), pp.259-269.
[11] , R. BARTOSZYNSKI, On a Certain Model of An Epidemic. Journal of Applicatione
Mathematicae, Vol. 13, (1972), pp. 139-151.
[12] N. G. BECKER, A stochastic Model for Interacting Population. Journal of Applied Prob
ability, Vol. 7, No. 3, (1970), pp. 544-564.
[13] N. G. BECKER, Analysis of Infectious Disease Data: Monographs on Statistics and
Applied Probability. Chapman and Hall/CRC, (1989).
[14] F. BRAUER, P. VAN DEN DRIESSCHE AND J. WU (EDS.) Mathematical Epidemiol
ogy, Mathematical Biosciences Subseries, Springer-Verlag, Berlin (2008).
[15] D.J.DALEY AND J.GANI, Epidemic Modelling: An Introduction. Cambridge University
Press, New York, (1999).
[16] O. DIEKMANN, H. HEESTERBEEK AND T. BRITTON, Mathematical Tools for Un
derstanding Infectious Diseases Dynamics: Princeton Series in Theoretical and Computa
tional Biology. Princeton University Press, New Jersey, (2013).
[17] C. LEFEVRE AND P. PICARD, A Non-Standard Family of Polynomial and The Fi
nal Size Distribution of Reed-Frost Epidemic Processes. Advances in Applied Probability,
Vol.22, No.1, (1990), pp. 25-48.
[18] I. M. LONGINI, JR AND J.S. KOOPMAN, Household and Community Transmission
Parameters from Final Distribution of Infections in Households. Biometrics, Vol. 38, No.
1, (1982), pp. 115-126.
21
[19] I. M. LONGINI, JR, J. S. KOOPMAN, A. S. MONTO, AND J. P. FOX, Estimating
Household and Community Transmission Parameters for Influenza. American Journal of
Epidemiology, Vol. 115, No. 5, (1982), pp. 736-750.
[20] P. NEAL, Efficient Likelihood-free Bayesian Computation for Household Epidemics .
Journal of Statistics and computing, Vol. 22, No.6, (2012), pp. 1239-1256.
[21] P. NEAL, A Household SIR Epidemic Model Incorporating Time of Day Effects. Journal
of Applied Probability, Vol. 53, (2016), pp. 489-501.
[22] A. M. UMAR, Effects on minimum epidemic and population sizes on a global epidemic
in simulation of final size epidemic data. Journal of progressive research in Mathematics,
Vol. 16, (2020), pp. 3093-3108.
22
the Analysis of Infectious Disease Final Size Data. Biometrics, Vol. 47, No. 3, (1991), pp.
961-974.
[2] H. ANDERSSON AND T. BRITTON, Lecture Notes in Statistics: Stochastic Epidemic
Models and Their Statistical Analysis. Springer, Verlag (2000).
[3] F. G. BALL, The Threshold Behaviour of Epidemic Models. Journal of Applied Probability,
Vol. 20, No. 2, (1983), pp. 227-241.
[4] F. G. BALL, A Unified Approach to the Distribution of the total size and Total Area under
the Trajectory of Infection in Epidemic Models. Advances in Applied Probability, Vol. 18,
No. 2, (1986), pp. 289-310.
[5] F. G. BALL, D. MOLLISON AND G. SCALIA-TOMBA, Epidemics with Two Levels of
Mixing. Annals of Applied Probability, Vol. 7, No. 1, (1997), pp. 46-89.
[6] F. G. BALL AND O. D. LYNE, Epidemics Among A Population of Households. Mathemat
ical Approaches for the Emerging and Reemerging Infectious Disease: Models, Methods
and Theory, (The IMA Volumes in Mathematics and its Applications), Springer, Editor:
Castillo-Chavez, Vol. 126, (2000), pp. 115-125.
[7] F. G. BALL AND O. D LYNE, Stochastic Multitype SIR Epidemic Among A Population
Partitioned into Households. Advances in Applied Probability, Vol. 33, No. 1 (Mar. 2001),
pp. 99-123
20
[8] F. G. BALL AND P. NEAL, A general model for the stochastic SIR epidemic with two
levels of mixing. Journal of Math. Biosciences, Vol. 180, (2002), pp. 73-102.
[9] F. BALL AND P. NEAL, Network Epidemic Models With Two Levels of Mixing. Mathe
matical Biosciences, Vol. 212, (2008), pp. 69-87.
[10] , R. BARTOSZYNSKI, Branching Processes and the Theory of Epidemic. Fifth Berkeley
symposium on Mathematical Statistics and Probability, University of California Press,
Vol. 4, (1967), pp.259-269.
[11] , R. BARTOSZYNSKI, On a Certain Model of An Epidemic. Journal of Applicatione
Mathematicae, Vol. 13, (1972), pp. 139-151.
[12] N. G. BECKER, A stochastic Model for Interacting Population. Journal of Applied Prob
ability, Vol. 7, No. 3, (1970), pp. 544-564.
[13] N. G. BECKER, Analysis of Infectious Disease Data: Monographs on Statistics and
Applied Probability. Chapman and Hall/CRC, (1989).
[14] F. BRAUER, P. VAN DEN DRIESSCHE AND J. WU (EDS.) Mathematical Epidemiol
ogy, Mathematical Biosciences Subseries, Springer-Verlag, Berlin (2008).
[15] D.J.DALEY AND J.GANI, Epidemic Modelling: An Introduction. Cambridge University
Press, New York, (1999).
[16] O. DIEKMANN, H. HEESTERBEEK AND T. BRITTON, Mathematical Tools for Un
derstanding Infectious Diseases Dynamics: Princeton Series in Theoretical and Computa
tional Biology. Princeton University Press, New Jersey, (2013).
[17] C. LEFEVRE AND P. PICARD, A Non-Standard Family of Polynomial and The Fi
nal Size Distribution of Reed-Frost Epidemic Processes. Advances in Applied Probability,
Vol.22, No.1, (1990), pp. 25-48.
[18] I. M. LONGINI, JR AND J.S. KOOPMAN, Household and Community Transmission
Parameters from Final Distribution of Infections in Households. Biometrics, Vol. 38, No.
1, (1982), pp. 115-126.
21
[19] I. M. LONGINI, JR, J. S. KOOPMAN, A. S. MONTO, AND J. P. FOX, Estimating
Household and Community Transmission Parameters for Influenza. American Journal of
Epidemiology, Vol. 115, No. 5, (1982), pp. 736-750.
[20] P. NEAL, Efficient Likelihood-free Bayesian Computation for Household Epidemics .
Journal of Statistics and computing, Vol. 22, No.6, (2012), pp. 1239-1256.
[21] P. NEAL, A Household SIR Epidemic Model Incorporating Time of Day Effects. Journal
of Applied Probability, Vol. 53, (2016), pp. 489-501.
[22] A. M. UMAR, Effects on minimum epidemic and population sizes on a global epidemic
in simulation of final size epidemic data. Journal of progressive research in Mathematics,
Vol. 16, (2020), pp. 3093-3108.
22
Published
2021-08-09
How to Cite
Umar, A. (2021). Stochastic SIR Household Epidemic Model with Misspecification. Journal of Progressive Research in Mathematics, 18(3), 31-46. Retrieved from http://scitecresearch.com/journals/index.php/jprm/article/view/2074
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