# American Institute of Mathematical Sciences

2013, 10(5&6): 1475-1497. doi: 10.3934/mbe.2013.10.1475

## Mathematical models of contact patterns between age groups for predicting the spread of infectious diseases

 1 Los Alamos National Laboratory, Los Alamos, NM 87545, United States 2 Tulane University, New Orleans, LA, 70118 3 Swiss Tropical and Public Health Institute, 4002 Basel, Switzerland

Received  August 2012 Revised  October 2012 Published  August 2013

The spread of an infectious disease is sensitive to the contact patterns in the population and to precautions people take to reduce the transmission of the disease. We investigate the impact that different mixing assumptions have on the spread an infectious disease in an age-structured ordinary differential equation model. We consider the impact of heterogeneity in susceptibility and infectivity within the population on the disease transmission. We apply the analysis to the spread of a smallpox-like disease, derive the formula for the reproduction number, $\Re_{0}$, and based on this threshold parameter, show the level of human behavioral change required to control the epidemic. We analyze how different mixing patterns can affect the disease prevalence, the cumulative number of new infections, and the final epidemic size. Our analysis indicates that the combination of residual immunity and behavioral changes during a smallpox-like disease outbreak can play a key role in halting infectious disease spread; and that realistic mixing patterns must be included in the epidemic model for the predictions to accurately reflect reality.
Citation: Sara Y. Del Valle, J. M. Hyman, Nakul Chitnis. Mathematical models of contact patterns between age groups for predicting the spread of infectious diseases. Mathematical Biosciences & Engineering, 2013, 10 (5&6) : 1475-1497. doi: 10.3934/mbe.2013.10.1475
##### References:
 [1] R. M. Anderson and R. M. May, "Infectious Diseases of Humans: Dynamics and Control," Oxford Science Publications, Oxford University Press, USA, 1992. [2] I. Arita, Duration of immunity after smallpox vaccination: A study on vaccination policy against smallpox bioterrorism in Japan, Japan Journal of Infectious Diseases, 55 (2002), 112-116. [3] C. L. Barrett, S. G. Eubank and J. P. Smith, If smallpox strikes Portland, Scientific American, 292 (2005), 54-61. doi: 10.1038/scientificamerican0305-54. [4] S. P. Blythe and C. Castillo-Chavez, Like-with-like preference and sexual mixing models, Mathematical Biosciences, 96 (1989), 221-238. doi: 10.1016/0025-5564(89)90060-6. [5] S. A. Bozzette, R. Boer, V. Bhatnagar, J. L. Brower, E. B. Keeler, S. C. Morton and M. A. Stoto, A model for a smallpox vaccination policy, The New England Journal of Medicine, 348 (2003), 416-425. doi: 10.1056/NEJMsa025075. [6] S. Busenberg and C. Castillo-Chavez, A general solution of the problem of mixing of sub-populations and its application to risk- and age-structured epidemic models for the spread of AIDS, IMA J. Math. Appl. Med. Biol., 8 (1991), 1-29. doi: 10.1093/imammb/8.1.1. [7] Centers for Disease Control and Prevention (CDC), Clinical evaluation tools for smallpox vaccine adverse reactions, (2004). Available from: http://www.bt.cdc.gov/agent/smallpox/vaccination/clineval/ [Online; accessed 19-July-2011]. [8] J. Chin, "Control of Communicable Diseases Manual," $17^th$ edition, American Public Health Association, Washington, D. C., 2002. [9] N. Chitnis, J. M. Hyman, J. Restrepo and J. Li, DSDISP, (2004). Available from: http://math.lanl.gov/~mac/dsdisp/ [Online; accessed 19-January-2012]. [10] G. Chowell, P. W. Fenimore, M. A. Castillo-Garsow and C. Castillo-Chavez, SARS outbreaks in Ontario, Hong Kong and Singapore: The role of diagnosis and isolation as a control mechanism, Emerging Infectious Diseases, 10 (2004), 1-8. doi: 10.1016/S0022-5193(03)00228-5. [11] S. Del Valle, H. Hethcote, J. M. Hyman and C. Castillo-Chavez, Effects of behavioral changes in a smallpox attack model, Mathematical Biosciences, 195 (2005), 228-251. doi: 10.1016/j.mbs.2005.03.006. [12] S. Y. Del Valle, J. M. Hyman, S. G. Eubank and H. W. Hethcote, Mixing patterns between age groups using social networks, Social Networks, 29 (2007), 539-554. [13] M. Eichner, Analysis of historical data suggests long-lasting protective effects of smallpox vaccination, American Journal of Epidemiology, 158 (2003), 717-723. doi: 10.1093/aje/kwg225. [14] M. Eichner and K. Dietz, Transmission potential of smallpox: Estimates based on detailed data from an outbreak, American Journal of Epidemiology, 158 (2003), 110-117. doi: 10.1093/aje/kwg103. [15] S. Eubank, H. Guclu, V. S. Anil Kumar, M. V. Marathe, A. Srinivasan, Z. Toroczkai and N. Wang, Modeling disease outbreaks in realistic urban social networks, Nature, 429 (2004), 180-184. doi: 10.1038/nature02541. [16] T. Fararo, Biased networks and the strength of weak ties, Social Networks, 5 (1983), 1-11. doi: 10.1016/0378-8733(83)90013-8. [17] E. P. Fenichel, C. Castillo-Chavez, M. G. Ceddia, G. Chowell, P. A. Gonzalez Parra, G. J. Hickling, G. Holloway, R. Horan, B. Morin, C. Perrings, M. Springborn, L. Velazquez and C. Villalobos, Adaptive human behavior in epidemiological models, PNAS, 108 (2011), 6306-6311. doi: 10.1073/pnas.1011250108. [18] F. Fenner, D. A. Henderson, I. Arita, Z. Jezek and I. D. Ladnyi, Smallpox and its eradication, World Health Organization Geneva, Switzerland, (1988). [19] N. M. Ferguson, M. J. Keeling, W. J. Edmunds, R. Gani, B. T. Grenfell, R. M. Anderson and S. Leach, Planning for smallpox outbreaks, Nature, 425 (2003), 681-685. doi: 10.1038/nature02007. [20] R. Gani and S. Leach, Transmission potential of smallpox in contemporary populations, Nature, 414 (2001), 748-751. [21] J. Glasser, Z. Feng, A. Moylan, R. Germundsson, S. Del Valle and C. Castillo-Chavez, Mixing in age-structured population models of infectious diseases, Mathematical Biosciences, 235 (2012), 1-7. doi: 10.1016/j.mbs.2011.10.001. [22] K. P. Hadeler and C. Castillo-Chavez, A core group model for disease transmission, Mathematical Biosciences, 128 (1995), 41-55. doi: 10.1016/0025-5564(94)00066-9. [23] E. M. Halloran, I. M. Longini, Jr., A. Nizam and Y. Yang, Containing bioterrorist smallpox, Science, 298 (2002), 1428-1430. doi: 10.1126/science.1074674. [24] H. W. Hethcote and J. W. Ark, Epidemiological models for heterogeneous populations: Proportionate mixing, parameter estimation, and immunization programs, Mathematical Biosciences, 84 (1987), 85-118. doi: 10.1016/0025-5564(87)90044-7. [25] H. W. Hethcote and J. A. Yorke, "Gonorrhea Transmission Dynamics and Control," Lecture Notes in Biomathematics, 56, Springer-Verlag, New York, 1984. [26] J. M. Hyman and J. Li, Biased preference models for partnership formation, in "World Congress of Nonlinear Analysts '92: Proceedings of the First World Congress of Nonlinear Analysts" (ed. V. Lakshmikantham), Walter de Gruyter & Co., (1995), 3137-3148. doi: 10.1515/9783110883237.3137. [27] J. M. Hyman and J. Li, Disease transmission models with biased partnership selection, Applied Numerical Mathematics, 24 (1997), 379-392. doi: 10.1016/S0168-9274(97)00034-2. [28] J. M. Hyman and J. Li, Behavior changes in SIS STD models with selective mixing, SIAM Journal on Applied Mathematics, 57 (1997), 1082-1094. doi: 10.1137/S0036139995294123. [29] J. M. Hyman, J. Li and E. A. Stanley, The differential infectivity and staged progression models for the transmission of HIV, Mathematical Biosciences, 155 (1999), 77-109. doi: 10.1016/S0025-5564(98)10057-3. [30] J. M. Hyman and E. A. Stanley, Using mathematical models to understand the AIDS epidemic, Mathematical Biosciences, 90 (1988), 415-473. doi: 10.1016/0025-5564(88)90078-8. [31] J. M. Hyman and E. A. Stanley, The effect of social mixing patterns on the spread of AIDS, in "Mathematical Approaches to Problems in Resource Management and Epidemiology" (eds. C. Castillo-Chavez, S. A. Levin and C. A. Shoemaker), Lecture Notes in Biomathematics, 81, Springer, Berlin, (1989), 190-219. doi: 10.1007/978-3-642-46693-9_15. [32] D. Hopkins, "The Greatest Killer: Smallpox in History," University of Chicago Press, 1983. [33] H. F. Hull, R. Danila and K. Ehresmann, Smallpox and bioterrorism: public-health responses, Journal of Laboratory and Clinical Medicine, 142 (2003), 221-228. doi: 10.1016/S0022-2143(03)00144-6. [34] J. A. Jacquez, C. P. Simon, J. Koopman, L. Sattenspiel and T. Perry, Modeling and analyzing HIV transmission: The effect of contact patterns, Mathematical Biosciences, 92 (1988), 119-199. doi: 10.1016/0025-5564(88)90031-4. [35] E. H. Kaplan, P. C. Cramton and A. D. Paltiel, Nonrandom mixing models of HIV transmission, in "Mathematical and Statistical Approaches to AIDS Epidemiology" (ed. C. Castillo-Chavez), Lecture Notes in Biomath., 83, Springer, New York, (1989), 218-239. doi: 10.1007/978-3-642-93454-4_10. [36] E. H. Kaplan, D. L. Craft and L. M. Wein, Emergency response to a smallpox attack: The case for mass vaccination, Proceedings of the National Academy of Sciences, 99 (2002), 10935-10940. doi: 10.1073/pnas.162282799. [37] H. Knolle, A discrete branching process model for the spread of HIV via steady sexual partnerships, Journal of Mathematical Biology, 48 (2004), 423-443. doi: 10.1007/s00285-003-0241-7. [38] J. Koopman, J. A. Jacquez and T. Park, Selective contact within structured mixing with an application to HIV transmission risk from oral and anal sex, in "Mathematical and Statistical Approaches to AIDS Epidemiology" (ed. C. Castillo-Chavez), Lecture Notes in Biomathematics, 83, Springer, Berlin, (1989), 316-348. doi: 10.1007/978-3-642-93454-4_16. [39] M. Kretzschmar and M. Morris, Measures of concurrency in networks and the spread of infectious diseases, Mathematical Biosciences, 133 (1996), 165-195. doi: 10.1016/0025-5564(95)00093-3. [40] M. Kretzschmar, D. Reinking, H. Brouwers, G. Zessen and J. Jager, Networks models: From paradigm to mathematical tool, in "Modeling the AIDS Epidemic: Planning, Policy, and Prediction" (eds. E.H. Kaplan and M.L. Brandeau), Raven Press, New York, (1994), 561-583. [41] M. Kretzschmar, S. van den Hof, J. Wallinga and J. van Wijngaarden, Ring vaccination and smallpox control, EID, 10 (2004), 832-841. doi: 10.3201/eid1005.030419. [42] T. M. Mack, Smallpox in Europe 1950-1971, The Journal of Infectious Diseases, 125 (1972), 161-169. doi: 10.1093/infdis/125.2.161. [43] B. Manicassamy, R. A. Medina, R. Hai, T. Tsibane, S. Stertz, E. Nistal-Villan, P. Palase, C. F. Basler and A. Garcia-Sastere, Protection of mice against lethal challenge with 2009 H1N1 influenza A virus by 1918-like and classical H1N1 based vaccines, PLoS Pathogens, 6 (2010), e1000745. doi: 10.1371/journal.ppat.1000745. [44] M. I. Meltzer, I. Damon, J. W. LeDuc and J. D. Millar, Modeling potential responses to smallpox as a bioterrorist weapon, Emerging Infectious Diseases, 7 (2001), 959-969. doi: 10.3201/eid0706.010607. [45] H. Nishiura and I. M. Tang, Modeling for a smallpox-vaccination policy against possible bioterrorism in Japan: The impact of long-lasting vaccinal immunity, Journal of Epidemiology, 14 (2004), 41-50. doi: 10.2188/jea.14.41. [46] A. Nold, Heterogeneity in disease-transmission modeling, Mathematical Biosciences, 52 (1980), 227-240. doi: 10.1016/0025-5564(80)90069-3. [47] J. T. F. Lau, H. Tsui, M. Lau and X. Yang, SARS transmission, risk factors, and prevention in Hong Kong, Emerging Infectious Diseases, 10 (2004), 587-592. doi: 10.3201/eid1004.030628. [48] X. Pang, Z. Zhu, F. Xu, J. Guo, X. Gong, D. Liu, Z. Liu, D. P. Chin and D. R. Feilin, Evaluation of control measures implemented in the severe acute respiratory syndrome outbreak in Beijing, 2003, JAMA, 290 (2003), 3215-3221. doi: 10.1001/jama.290.24.3215. [49] A. R. Rao, E. S. Jacob, S. Kamalakshi, S. Appaswamy and Bradbury, Epidemiological studies in smallpox. A study of intrafamilial transmission in a series of 254 infected families, Indian Journal of Medical Research, 56 (1968), 1826-1854. [50] A. Rapoport and Y. Yuan, Some aspects of epidemics and social nets, in "The Small World" (ed. M. Kochen), Ablex, Norwood, NJ, (1989), 327-348. [51] S. Singh, Some aspects of the epidemiology of smallpox in Nepal, World Health Organization, Geneva, (1969). [52] P. D. Stroud, S. J. Sydoriak, J. M. Riese, J. P. Smith, S. M. Mniszewski and P. R. Romero, Semi-empirical power-law scaling of new Infection rate to model epidemic dynamics with inhomogeneos mixing, Mathematical Biosciences, 203 (2006), 301-318. doi: 10.1016/j.mbs.2006.01.007. [53] P. Stroud, S. Del Valle, S. Sydoriak, J. Riese and S. Mniszewski, Spatial dynamics of pandemic influenza in a massive artificial society, Journal of Artificial Societies and Social Simulation, 10 (2007). [54] P. van den Driessche and J. Watmough, Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission, Mathematical Biosciences, 180 (2002), 29-48. doi: 10.1016/S0025-5564(02)00108-6. [55] J. X. Velasco-Hernandez and Y. H. Hsieh, Modeling the effect of treatment and behavioral change in HIV transmission dynamics, Journal of Mathematical Biology, 32 (1994), 233-249. [56] A. M. Wendelboe, A. Van Rie, S. Salmaso and J. A. Englund, Duration of immunity against pertussis after natural infection or vaccination, Pediatric Infectious Disease Journal, 24 (2005), S58-S61. [57] G. S. Zaric, Random vs. nonrandom mixing in network epidemic models, Health Care Management Science, 5 (2002), 147-155.

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##### References:
 [1] R. M. Anderson and R. M. May, "Infectious Diseases of Humans: Dynamics and Control," Oxford Science Publications, Oxford University Press, USA, 1992. [2] I. Arita, Duration of immunity after smallpox vaccination: A study on vaccination policy against smallpox bioterrorism in Japan, Japan Journal of Infectious Diseases, 55 (2002), 112-116. [3] C. L. Barrett, S. G. Eubank and J. P. Smith, If smallpox strikes Portland, Scientific American, 292 (2005), 54-61. doi: 10.1038/scientificamerican0305-54. [4] S. P. Blythe and C. Castillo-Chavez, Like-with-like preference and sexual mixing models, Mathematical Biosciences, 96 (1989), 221-238. doi: 10.1016/0025-5564(89)90060-6. [5] S. A. Bozzette, R. Boer, V. Bhatnagar, J. L. Brower, E. B. Keeler, S. C. Morton and M. A. Stoto, A model for a smallpox vaccination policy, The New England Journal of Medicine, 348 (2003), 416-425. doi: 10.1056/NEJMsa025075. [6] S. Busenberg and C. Castillo-Chavez, A general solution of the problem of mixing of sub-populations and its application to risk- and age-structured epidemic models for the spread of AIDS, IMA J. Math. Appl. Med. Biol., 8 (1991), 1-29. doi: 10.1093/imammb/8.1.1. [7] Centers for Disease Control and Prevention (CDC), Clinical evaluation tools for smallpox vaccine adverse reactions, (2004). Available from: http://www.bt.cdc.gov/agent/smallpox/vaccination/clineval/ [Online; accessed 19-July-2011]. [8] J. Chin, "Control of Communicable Diseases Manual," $17^th$ edition, American Public Health Association, Washington, D. C., 2002. [9] N. Chitnis, J. M. Hyman, J. Restrepo and J. Li, DSDISP, (2004). Available from: http://math.lanl.gov/~mac/dsdisp/ [Online; accessed 19-January-2012]. [10] G. Chowell, P. W. Fenimore, M. A. Castillo-Garsow and C. Castillo-Chavez, SARS outbreaks in Ontario, Hong Kong and Singapore: The role of diagnosis and isolation as a control mechanism, Emerging Infectious Diseases, 10 (2004), 1-8. doi: 10.1016/S0022-5193(03)00228-5. [11] S. Del Valle, H. Hethcote, J. M. Hyman and C. Castillo-Chavez, Effects of behavioral changes in a smallpox attack model, Mathematical Biosciences, 195 (2005), 228-251. doi: 10.1016/j.mbs.2005.03.006. [12] S. Y. Del Valle, J. M. Hyman, S. G. Eubank and H. W. Hethcote, Mixing patterns between age groups using social networks, Social Networks, 29 (2007), 539-554. [13] M. Eichner, Analysis of historical data suggests long-lasting protective effects of smallpox vaccination, American Journal of Epidemiology, 158 (2003), 717-723. doi: 10.1093/aje/kwg225. [14] M. Eichner and K. Dietz, Transmission potential of smallpox: Estimates based on detailed data from an outbreak, American Journal of Epidemiology, 158 (2003), 110-117. doi: 10.1093/aje/kwg103. [15] S. Eubank, H. Guclu, V. S. Anil Kumar, M. V. Marathe, A. Srinivasan, Z. Toroczkai and N. Wang, Modeling disease outbreaks in realistic urban social networks, Nature, 429 (2004), 180-184. doi: 10.1038/nature02541. [16] T. Fararo, Biased networks and the strength of weak ties, Social Networks, 5 (1983), 1-11. doi: 10.1016/0378-8733(83)90013-8. [17] E. P. Fenichel, C. Castillo-Chavez, M. G. Ceddia, G. Chowell, P. A. Gonzalez Parra, G. J. Hickling, G. Holloway, R. Horan, B. Morin, C. Perrings, M. Springborn, L. Velazquez and C. Villalobos, Adaptive human behavior in epidemiological models, PNAS, 108 (2011), 6306-6311. doi: 10.1073/pnas.1011250108. [18] F. Fenner, D. A. Henderson, I. Arita, Z. Jezek and I. D. Ladnyi, Smallpox and its eradication, World Health Organization Geneva, Switzerland, (1988). [19] N. M. Ferguson, M. J. Keeling, W. J. Edmunds, R. Gani, B. T. Grenfell, R. M. Anderson and S. Leach, Planning for smallpox outbreaks, Nature, 425 (2003), 681-685. doi: 10.1038/nature02007. [20] R. Gani and S. Leach, Transmission potential of smallpox in contemporary populations, Nature, 414 (2001), 748-751. [21] J. Glasser, Z. Feng, A. Moylan, R. Germundsson, S. Del Valle and C. Castillo-Chavez, Mixing in age-structured population models of infectious diseases, Mathematical Biosciences, 235 (2012), 1-7. doi: 10.1016/j.mbs.2011.10.001. [22] K. P. Hadeler and C. Castillo-Chavez, A core group model for disease transmission, Mathematical Biosciences, 128 (1995), 41-55. doi: 10.1016/0025-5564(94)00066-9. [23] E. M. Halloran, I. M. Longini, Jr., A. Nizam and Y. Yang, Containing bioterrorist smallpox, Science, 298 (2002), 1428-1430. doi: 10.1126/science.1074674. [24] H. W. Hethcote and J. W. Ark, Epidemiological models for heterogeneous populations: Proportionate mixing, parameter estimation, and immunization programs, Mathematical Biosciences, 84 (1987), 85-118. doi: 10.1016/0025-5564(87)90044-7. [25] H. W. Hethcote and J. A. Yorke, "Gonorrhea Transmission Dynamics and Control," Lecture Notes in Biomathematics, 56, Springer-Verlag, New York, 1984. [26] J. M. Hyman and J. Li, Biased preference models for partnership formation, in "World Congress of Nonlinear Analysts '92: Proceedings of the First World Congress of Nonlinear Analysts" (ed. V. Lakshmikantham), Walter de Gruyter & Co., (1995), 3137-3148. doi: 10.1515/9783110883237.3137. [27] J. M. Hyman and J. Li, Disease transmission models with biased partnership selection, Applied Numerical Mathematics, 24 (1997), 379-392. doi: 10.1016/S0168-9274(97)00034-2. [28] J. M. Hyman and J. Li, Behavior changes in SIS STD models with selective mixing, SIAM Journal on Applied Mathematics, 57 (1997), 1082-1094. doi: 10.1137/S0036139995294123. [29] J. M. Hyman, J. Li and E. A. Stanley, The differential infectivity and staged progression models for the transmission of HIV, Mathematical Biosciences, 155 (1999), 77-109. doi: 10.1016/S0025-5564(98)10057-3. [30] J. M. Hyman and E. A. Stanley, Using mathematical models to understand the AIDS epidemic, Mathematical Biosciences, 90 (1988), 415-473. doi: 10.1016/0025-5564(88)90078-8. [31] J. M. Hyman and E. A. Stanley, The effect of social mixing patterns on the spread of AIDS, in "Mathematical Approaches to Problems in Resource Management and Epidemiology" (eds. C. Castillo-Chavez, S. A. Levin and C. A. Shoemaker), Lecture Notes in Biomathematics, 81, Springer, Berlin, (1989), 190-219. doi: 10.1007/978-3-642-46693-9_15. [32] D. Hopkins, "The Greatest Killer: Smallpox in History," University of Chicago Press, 1983. [33] H. F. Hull, R. Danila and K. Ehresmann, Smallpox and bioterrorism: public-health responses, Journal of Laboratory and Clinical Medicine, 142 (2003), 221-228. doi: 10.1016/S0022-2143(03)00144-6. [34] J. A. Jacquez, C. P. Simon, J. Koopman, L. Sattenspiel and T. Perry, Modeling and analyzing HIV transmission: The effect of contact patterns, Mathematical Biosciences, 92 (1988), 119-199. doi: 10.1016/0025-5564(88)90031-4. [35] E. H. Kaplan, P. C. Cramton and A. D. Paltiel, Nonrandom mixing models of HIV transmission, in "Mathematical and Statistical Approaches to AIDS Epidemiology" (ed. C. Castillo-Chavez), Lecture Notes in Biomath., 83, Springer, New York, (1989), 218-239. doi: 10.1007/978-3-642-93454-4_10. [36] E. H. Kaplan, D. L. Craft and L. M. Wein, Emergency response to a smallpox attack: The case for mass vaccination, Proceedings of the National Academy of Sciences, 99 (2002), 10935-10940. doi: 10.1073/pnas.162282799. [37] H. Knolle, A discrete branching process model for the spread of HIV via steady sexual partnerships, Journal of Mathematical Biology, 48 (2004), 423-443. doi: 10.1007/s00285-003-0241-7. [38] J. Koopman, J. A. Jacquez and T. Park, Selective contact within structured mixing with an application to HIV transmission risk from oral and anal sex, in "Mathematical and Statistical Approaches to AIDS Epidemiology" (ed. C. Castillo-Chavez), Lecture Notes in Biomathematics, 83, Springer, Berlin, (1989), 316-348. doi: 10.1007/978-3-642-93454-4_16. [39] M. Kretzschmar and M. Morris, Measures of concurrency in networks and the spread of infectious diseases, Mathematical Biosciences, 133 (1996), 165-195. doi: 10.1016/0025-5564(95)00093-3. [40] M. Kretzschmar, D. Reinking, H. Brouwers, G. Zessen and J. Jager, Networks models: From paradigm to mathematical tool, in "Modeling the AIDS Epidemic: Planning, Policy, and Prediction" (eds. E.H. Kaplan and M.L. Brandeau), Raven Press, New York, (1994), 561-583. [41] M. Kretzschmar, S. van den Hof, J. Wallinga and J. van Wijngaarden, Ring vaccination and smallpox control, EID, 10 (2004), 832-841. doi: 10.3201/eid1005.030419. [42] T. M. Mack, Smallpox in Europe 1950-1971, The Journal of Infectious Diseases, 125 (1972), 161-169. doi: 10.1093/infdis/125.2.161. [43] B. Manicassamy, R. A. Medina, R. Hai, T. Tsibane, S. Stertz, E. Nistal-Villan, P. Palase, C. F. Basler and A. Garcia-Sastere, Protection of mice against lethal challenge with 2009 H1N1 influenza A virus by 1918-like and classical H1N1 based vaccines, PLoS Pathogens, 6 (2010), e1000745. doi: 10.1371/journal.ppat.1000745. [44] M. I. Meltzer, I. Damon, J. W. LeDuc and J. D. Millar, Modeling potential responses to smallpox as a bioterrorist weapon, Emerging Infectious Diseases, 7 (2001), 959-969. doi: 10.3201/eid0706.010607. [45] H. Nishiura and I. M. Tang, Modeling for a smallpox-vaccination policy against possible bioterrorism in Japan: The impact of long-lasting vaccinal immunity, Journal of Epidemiology, 14 (2004), 41-50. doi: 10.2188/jea.14.41. [46] A. Nold, Heterogeneity in disease-transmission modeling, Mathematical Biosciences, 52 (1980), 227-240. doi: 10.1016/0025-5564(80)90069-3. [47] J. T. F. Lau, H. Tsui, M. Lau and X. Yang, SARS transmission, risk factors, and prevention in Hong Kong, Emerging Infectious Diseases, 10 (2004), 587-592. doi: 10.3201/eid1004.030628. [48] X. Pang, Z. Zhu, F. Xu, J. Guo, X. Gong, D. Liu, Z. Liu, D. P. Chin and D. R. Feilin, Evaluation of control measures implemented in the severe acute respiratory syndrome outbreak in Beijing, 2003, JAMA, 290 (2003), 3215-3221. doi: 10.1001/jama.290.24.3215. [49] A. R. Rao, E. S. Jacob, S. Kamalakshi, S. Appaswamy and Bradbury, Epidemiological studies in smallpox. A study of intrafamilial transmission in a series of 254 infected families, Indian Journal of Medical Research, 56 (1968), 1826-1854. [50] A. Rapoport and Y. Yuan, Some aspects of epidemics and social nets, in "The Small World" (ed. M. Kochen), Ablex, Norwood, NJ, (1989), 327-348. [51] S. Singh, Some aspects of the epidemiology of smallpox in Nepal, World Health Organization, Geneva, (1969). [52] P. D. Stroud, S. J. Sydoriak, J. M. Riese, J. P. Smith, S. M. Mniszewski and P. R. Romero, Semi-empirical power-law scaling of new Infection rate to model epidemic dynamics with inhomogeneos mixing, Mathematical Biosciences, 203 (2006), 301-318. doi: 10.1016/j.mbs.2006.01.007. [53] P. Stroud, S. Del Valle, S. Sydoriak, J. Riese and S. Mniszewski, Spatial dynamics of pandemic influenza in a massive artificial society, Journal of Artificial Societies and Social Simulation, 10 (2007). [54] P. van den Driessche and J. Watmough, Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission, Mathematical Biosciences, 180 (2002), 29-48. doi: 10.1016/S0025-5564(02)00108-6. [55] J. X. Velasco-Hernandez and Y. H. Hsieh, Modeling the effect of treatment and behavioral change in HIV transmission dynamics, Journal of Mathematical Biology, 32 (1994), 233-249. [56] A. M. Wendelboe, A. Van Rie, S. Salmaso and J. A. Englund, Duration of immunity against pertussis after natural infection or vaccination, Pediatric Infectious Disease Journal, 24 (2005), S58-S61. [57] G. S. Zaric, Random vs. nonrandom mixing in network epidemic models, Health Care Management Science, 5 (2002), 147-155.
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