Steady State Solution and Stability of an Age-Structured MSIQR Epidemic Model

Meihong Qiao, Huan Qi, Tianhai Tian

.

DOI: 10.4236/iim.2010.25037                

Abstract

The importance of epidemiology in our life has stimulated researchers to extend the classic Susceptibles-Infectives-Removed (SIR) model to sophisticated models by including more factors in order to give detailed transmission dynamics of epidemic diseases. However, the integration of the quarantine policy and age-structure is less addressed. In this work we propose an age-structured MSIQR (temporarily immune-susceptibles-infectives-quarantined-removed) model to study the impact of quarantine policies on the spread of epidemic diseases. Specifically, we investigate the existence of steady state solutions and stability property of the proposed model. The derived explicit expression of the basic reproductive number shows that the disease-free equilibrium is globally asymptotically stable if, and that the unique endemic equilibrium exists if. In addition, the stability conditions of the endemic equilibrium are derived.

Keywords

Epidemic Model, Quarantine, Age-Structure, Basic Reproductive Number, Endemic Equilibrium, Stability

Conflicts of Interest

The authors declare no conflicts of interest.

References

[1]	N. M. Ferguson, C. A. Donnelly and R. M. Anderson, “Transmission Intensity and Impact of Control Policies on the Foot and Mouth Epidemic in Great Britain,” Nature, Vol. 413, No. 6855, 2001, pp. 542-548.
[2]	M. Lipsitch, T. Cohen, B. Cooper, J. M. Robins, S. Ma, L. James, G. Gopalakrishna, S. K. Chew, C. C. Tan, M. H. Samore, D. Fisman and M. Murray, “Transmission Dynamics and Control of Severe Acute Respiratory Syndrome,” Science, Vol. 300, No. 5627, 2003, pp. 1966- 1970.
[3]	S. Hsu and L. W. Roeger, “The Final Size of a SARS epidemic Model without Quarantine,” Journal of Mathe- matical Analysis and Applications, Vol. 333, No. 2, 2007, pp. 557-566.
[4]	W. O. Kermack and A. G. Mckendrick, “A Contribution to the Mathematical Theory of Epidemics,” Proceedings of the Royal Society of London – Series A, Vol. 115, No. 772, 1927, pp. 700-721.
[5]	S. Riley, “Large-Scale Spatial-Transmission Models of Infectious Disease,” Science, Vol. 316, No. 5829, 2007, pp. 1298-1301.
[6]	P. E. Parham, B. K. Singh and N. M. Ferguson, “Analytic Approximation of Spatial Epidemic Models of Foot and Mouth Disease,” Theoretical Population Biology, Vol. 73, No. 3, 2008, pp. 349-368.
[7]	M. J. Keeling, M. E. J. Woolhouse, D. J. Shaw, L. Matthews, M. Chase-Topping, D. T. Haydon, S. J. Cornell, J. Kappey, J. Wilesmith and B. T. Grenfell, “Dynamics of the 2001 UK Foot and Mouth Epidemic: Stochastic Dispersal in a Heterogeneous Landscape,” Science, Vol. 294, No. 5543, 2001, pp. 813-817.
[8]	H. C. Tuckwell and R. J. Williams, “Some Properties of a Simple Stochastic Epidemic Model of SIR Type,” Mathe- matical Biosciences, Vol. 208, No. 1, 2007, pp. 76-97.
[9]	S. Gao, L. Chen and Z. Teng, “Impulsive Vaccination of an SEIRS Model with Time Delay and Varying Total Population Size,” Bulletin of Mathematical Biology, Vol. 69, No. 2, 2007, pp. 731-745.
[10]	H. W. Hethcode, “The Mathematics of Infectious Diseases,” SIAM Review, Vol. 42, No. 4, 2000, pp. 599-653.
[11]	J. Arino, J. R. Davis, D. Hartley, R. Jordan, J. M. Miller and P. van den Driessche, “A Multi-Species Epidemic Model with Spatial Dynamics,” Mathematical Medicine and Biology, Vol. 22, No. 2, 2005, pp. 129-142.
[12]	F. Hoppensteadt, “An Age Dependent Epidemic Model,” Journal of the Franklin Institute, Vol. 297, No. 5, 1974, pp. 325-333.
[13]	F. Hoppensteadt, “Mathematical Theories of Populations: Demographics,” Genetics and Epidemics, SIAM, Philadelphia, 1975.
[14]	Z. Zhang and J. G. Peng, “A SIRS Epidemic Model with Infection-Age Dependence,” Journal of Mathematical Analysis and Applications, Vol. 331, No. 2, 2007, pp. 1396-1414.
[15]	H. R. Thieme and C. Castillo-Chavez, “How may Infection Age-Dependent Infectivity Affect the Dynamics of HIV/AIDS?” SIAM Journal on Applied Mathematics, Vol. 53, No. 5, 1993, pp. 1447-1479.
[16]	C. M. Kribs-Zaleta and M. Martcheva, “Vaccination Stra- tegies and backward Bifurcation in an Age-since-Infection Structured Model,” Mathematical Biosciences, Vol. 177- 178, 2002, pp. 317-332.
[17]	H. Inaba and H. Sekine, “A Mathematical Model for Chagas Disease with Infection-Age-Dependent Infectivity,” Mathematical Biosciences, Vol. 190, No. 1, 2004, pp. 39- 69.
[18]	B. Fang and X. Li, “Stability of an Age-Structured MSEIS Epidemic Model with Infectivity in Latent Period,” Acta Mathematicae Applicatae Sinica, Vol. 31, No. 1, 2008, pp. 110-125.
[19]	G. B. Webb, “Theory of Nonlinear Age-Dependent Population Dynamics,” CRC Press, Boca Raton, 1985.
[20]	J. Li, Y. C. Zhou, Z. Z. Ma and M. Hyman, “Epidemiological Models for Mutating Pathogens,” SIAM Journal on Applied Mathematics, Vol. 65, No. 1, 2004, pp. 1-23.
[21]	H. W. Hethcotevan and P. den Driessche, “Two SIS Epidemiologic Models with Delays,” Journal of Mathematical Biology, Vol. 40, No. 1, 2000, pp. 3-26.
[22]	W. H. McNeill, “Plagues and Peoples,” Updated Edition, Anchor, Garden City, 1976.
[23]	A. Mandavilli, “SARS Epidemic Unmasks Age-Old Quarantine Conundrum,” Nature Medicine, Vol. 9, No. 5, 2003, p. 487.
[24]	B. Diamond, “SARS Spreads New Outlook on Quarantine Models,” Nature Medicine, Vol. 9, No. 12, 2003, p. 1441.
[25]	Y. H. Hsieh, C. C. King, C. W. Chen, M. S. Ho, S. B. Hsu and Y. C. Wu, “Impact of Quarantine on the 2003 SARS Outbreak: A Retrospective Modeling Study,” Journal of Theoretical Biology, Vol. 244, No. 4, 2007, pp. 729-736.
[26]	L. Sattenspiel and D. A. Herring, “Simulating the Effect of Quarantine on the Spread of the 1918-19 Flu in Central Canada,” Bulletin of Mathematical Biology, Vol. 65, No. 1, 2003, pp. 1-26.
[27]	H. W. Hethcote, Z. E. Ma and S. B. Liao, “Effects of Quarantine in Six Endemic Models for Infectious Diseases,” Mathematical Biosciences, Vol. 180, No. 1, 2002, pp. 141-160.
[28]	C. T. Bauch, J. O. Lloyd-Smith, M. P. Coffee and A. P. Galvani, “Dynamically Modelling SARS and Other Newly Emerging Respiratory Illnesses: Past, Present and Future,” Epidemiology, Vol. 16, No. 6, 2005, pp. 791-801.