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Epidemic characteristics of two classic models and the dependence on the initial conditions
Modeling the role of healthcare access inequalities in epidemic outcomes
1.  Harvard T.H. Chan School of Public Health, Department of Biostatistics, Boston, MA, United States 
2.  SAL MCMSC, School of Human Evolution and Social Change, Arizona State University, Tempe, AZ, United States, United States 
3.  School of Mathematical and Statistical Sciences, Arizona State University, Tempe, AZ, United States 
References:
[1] 
E. Alirol, L. Getaz, B. Stoll, F. Chappuis and L. Loutan, Urbanisation and infectious diseases in a globalised world, The Lancet Infectious Diseases, 11 (2011), 131141. doi: 10.1016/S14733099(10)702231. 
[2] 
L. J. Allen, An Introduction to Stochastic Processes with Applications to Biology, $2^{nd}$ edition, Pearson Education, New Jersey, 2003. 
[3] 
J. P. Aparicio, A. F. Capurro and C. CastilloChávez, Markers of disease evolution: The case of tuberculosis, Journal of Theoretical Biology, 215 (2002), 227237. doi: 10.1006/jtbi.2001.2489. 
[4] 
P. H. Bamaiyi, The role of demographics and human activities in the spread of diseases, Current Trends in Technology and Sciences, 2 (2013), 253257. 
[5] 
S. P. Blythe and C. CastilloChavez, Likewithlike preference and sexual mixing models, Mathematical Biosciences, 96 (1989), 221238. 
[6] 
F. Brauer and C. CastilloChavez, Mathematical Models in Population Biology and Epidemiology, Springer, 2012. doi: 10.1007/9781461416869. 
[7] 
C. CastilloChavez, W. Huang and J. Li, Competitive exclusion in gonorrhea models and other sexually transmitted diseases, SIAM Journal on Applied Mathematics, 56 (1996), 494508. doi: 10.1137/S003613999325419X. 
[8] 
C. CastilloChavez, W. Huang and J. Li, The effects of females' susceptibility on the coexistence of multiple pathogen strains of sexually transmitted diseases, Journal of Mathematical Biology, 35 (1997), 503522. doi: 10.1007/s002850050063. 
[9] 
C. CastilloChavez, W. Huang and J. Li, Competitive exclusion and coexistence of multiple strains in an SIS STD model, SIAM Journal on Applied Mathematics, 59 (1999), 17901811. doi: 10.1137/S0036139997325862. 
[10] 
C. CastilloChavez and B. Song, Dynamical models of tuberculosis and their applications, Math. Biosci. Eng., 1 (2004), 361404. doi: 10.3934/mbe.2004.1.361. 
[11] 
A. Chen, E. Oster and H. Williams, Why is Infant Mortality Higher in the US than in Europe?, National Bureau of Economic Research, 2014. 
[12] 
K. C. Chow, X. Wang and C. CastilloChavez, A mathematical model of nosocomial infection and antibiotic resistance: Evaluating the efficacy of antimicrobial cycling programs and patient isolation on dual resistance, Mathematical and Theoretical Biology Institute archive, 2007. 
[13] 
C. Cohen, D. Horlacher and F. L. MacKellar, Is urbanization good for a nation's health,, 2010. Available from: , (). 
[14] 
C. Dye, Health and urban living, Science, American Association for the Advancement of Science, 319 (2008), 766769. doi: 10.1126/science.1150198. 
[15] 
D. N. Fisman, G. M. Leung and M. Lipsitch, Nuanced risk assessment for emerging infectious diseases, Lancet, 383 (2014). 
[16] 
K. Ford and A. Norris, Sexual networks of AfricanAmerican and Hispanic youth, Sexually Transmitted Diseases, 24 (1997), 327333. doi: 10.1097/0000743519970700000004. 
[17] 
K. Ford, W. Sohn and J. Lepkowski, American adolescents: Sexual mixing patterns, bridge partners, and concurrency, Sexually Transmitted Diseases, 29 (2002), 1319. doi: 10.1097/0000743520020100000003. 
[18] 
S. Galea, Urbanization, urbanicity, and health, Journal of Urban Health, 79 (2002), S1S12. 
[19] 
S. Galea, N. Freudenberg and D. Vlahov, Cities and population health, Social Science & Medicine, 60 (2005), 10171033. doi: 10.1016/j.socscimed.2004.06.036. 
[20] 
M. G. M. Gomes, M. Lipsitch, A. R. Wargo, G. Kurath, C. Rebelo, G. F. Medley and A. Coutinho, A missing dimension in measures of vaccination impacts, PLoS Pathogens, 10 (2014). 
[21] 
T. Harpham and C. Molyneux, Urban health in developing countries: A review, Progress in Development Studies, 1 (2001), 113137. 
[22] 
H. W. Hethcote and J. A. Yorke, Gonorrhea Transmission Dynamics and Control, 56, Springer, Berlin, 1984. doi: 10.1007/9783662075449. 
[23] 
D. R. Holtgrave and R. A. Crosby, Social capital, poverty, and income inequality as predictors of gonorrhoea, syphilis, chlamydia and AIDS case rates in the United States, Sexually Transmitted Infections, 79 (2003), 6264. doi: 10.1136/sti.79.1.62. 
[24] 
M. J. Keeling and P. Rohani, Modeling Infectious Diseases in Humans and Animals, Princeton University Press, 2008. 
[25] 
M. J. Keeling and B. T. Grenfell, Effect of variability in infection period on the persistence and spatial spread of infectious diseases, Mathematical Biosciences, 147 (1998), 207226. doi: 10.1016/S00255564(97)001016. 
[26] 
D. A. Leon, Cities, urbanization and health, International Journal of Epidemiology, 37 (2008), 48. doi: 10.1093/ije/dym271. 
[27] 
J. Li, Z. Ma, S. P. Blythe and C. CastilloChavez, Coexistence of pathogens in sexuallytransmitted disease models, Journal of Mathematical Biology, 47 (2003), 547568. doi: 10.1007/s0028500302355. 
[28] 
M. Lipsitch, T. Cohen, B. Cooper, J. M. Robins, S. Ma, L. James, G. Gopalakrishna, S. K. Chew, C. C. Tan, M. H. Samore, et al., Transmission dynamics and control of severe acute respiratory syndrome, Science, 300 (2003), 19661970. doi: 10.1126/science.1086616. 
[29] 
A. L. Lloyd, Realistic distributions of infectious periods in epidemic models: Changing patterns of persistence and dynamics. Theoretical Population Biology, 60 (2001). 5971. 
[30] 
A. L. Lloyd, Destabilization of epidemic models with the inclusion of realistic distributions of infectious periods, Proceedings of the Royal Society of London. Series B: Biological Sciences, 268 (2011), 985993. 
[31] 
J. O. LloydSmith, S. J. Schreiber, P. E. Kopp and W. M. Getz, Superspreading and the effect of individual variation on disease emergence, Nature, 438 (2005), 355359. 
[32] 
K. Lönnroth, E. Jaramillo, B. G. Williams, C. Dye and M. Raviglione, Drivers of tuberculosis epidemics: the role of risk factors and social determinants, Social Science & Medicine, 68 (2009), 22402246. 
[33] 
M. Marmot, Social determinants of health inequalities, The Lancet, 365 (2005), 10991104. doi: 10.1016/S01406736(05)742343. 
[34] 
D. M. Morens and A. S. Fauci, Emerging infectious diseases: Threats to human health and global stability, PLoS Pathogens, 9 (2013), e1003467. doi: 10.1371/journal.ppat.1003467. 
[35] 
B. Morin, Variable susceptibility with an open population: A transport equation approach,, preprint, (). 
[36] 
B. R. Morin, C. CastilloChavez, S.F. Hsu Schmitz, A. Mubayi and X. Wang, Notes from the heterogeneous: A few observations on the implications and necessity of affinity, Journal of Biological Dynamics, 4 (2010), 456477. doi: 10.1080/17513758.2010.510212. 
[37] 
O. PattersonLomba, E. Goldstein, A. GómezLiévano, C. CastilloChavez and S. Towers, Per capita incidence of sexually transmitted infections increases systematically with urban population size: A crosssectional study, Sexually Transmitted Infections,91 (2015), 610614. doi: 10.1136/sextrans2014051932. 
[38] 
K. Pickett and R. Wilkinson, The Spirit Level: Why More Equal Societies Almost Always do Better, London: Allen Lane, 2009. 
[39] 
E. Saez and G. Zucman, Wealth inequality in the united states since 1913: Evidence from capitalized income tax data, The Quarterly Journal of Economics, 131 (2016), 519578. doi: 10.1093/qje/qjw004. 
[40] 
C. Stephens, Healthy cities or unhealthy islands? The health and social implications of urban inequality, Environment and Urbanization, 8 (1996), 930. doi: 10.1177/095624789600800211. 
[41] 
S. H. Strogatz, Nonlinear Dynamics and Chaos: With Applications to Physics, Biology, Chemistry, and Engineering, AddisonWesley Pub., 1994. 
[42] 
P. van den Driessche and J. Watmough, Reproduction numbers and subthreshold endemic equilibria for compartmental models of disease transmission, Math. Biosci., 180 (2002), 2948. doi: 10.1016/S00255564(02)001086. 
[43] 
, United Nations, World Urbanization Prospects: The 2011 Revision,, 2012. Available from: , (). 
[44] 
J. Wallinga and M. Lipsitch, How generation intervals shape the relationship between growth rates and reproductive numbers, Proceedings of the Royal Society B: Biological Sciences, 274 (2007), 599604. 
[45] 
R. G. Wilkinson, Socioeconomic determinants of health. Health inequalities: Relative or absolute material standards?, BMJ: British Medical Journal, 314 (1997), 591595. doi: 10.1136/bmj.314.7080.591. 
[46] 
R. G. Wilkinson and K. E. Pickett, Income inequality and population health: A review and explanation of the evidence, Social Science & Medicine, 62 (2006), 17681784. doi: 10.1016/j.socscimed.2005.08.036. 
[47] 
P. Zhang and P. M. Atkinson, Modelling the effect of urbanization on the transmission of an infectious disease, Mathematical Biosciences, 211 (2008), 166185. doi: 10.1016/j.mbs.2007.10.007. 
show all references
References:
[1] 
E. Alirol, L. Getaz, B. Stoll, F. Chappuis and L. Loutan, Urbanisation and infectious diseases in a globalised world, The Lancet Infectious Diseases, 11 (2011), 131141. doi: 10.1016/S14733099(10)702231. 
[2] 
L. J. Allen, An Introduction to Stochastic Processes with Applications to Biology, $2^{nd}$ edition, Pearson Education, New Jersey, 2003. 
[3] 
J. P. Aparicio, A. F. Capurro and C. CastilloChávez, Markers of disease evolution: The case of tuberculosis, Journal of Theoretical Biology, 215 (2002), 227237. doi: 10.1006/jtbi.2001.2489. 
[4] 
P. H. Bamaiyi, The role of demographics and human activities in the spread of diseases, Current Trends in Technology and Sciences, 2 (2013), 253257. 
[5] 
S. P. Blythe and C. CastilloChavez, Likewithlike preference and sexual mixing models, Mathematical Biosciences, 96 (1989), 221238. 
[6] 
F. Brauer and C. CastilloChavez, Mathematical Models in Population Biology and Epidemiology, Springer, 2012. doi: 10.1007/9781461416869. 
[7] 
C. CastilloChavez, W. Huang and J. Li, Competitive exclusion in gonorrhea models and other sexually transmitted diseases, SIAM Journal on Applied Mathematics, 56 (1996), 494508. doi: 10.1137/S003613999325419X. 
[8] 
C. CastilloChavez, W. Huang and J. Li, The effects of females' susceptibility on the coexistence of multiple pathogen strains of sexually transmitted diseases, Journal of Mathematical Biology, 35 (1997), 503522. doi: 10.1007/s002850050063. 
[9] 
C. CastilloChavez, W. Huang and J. Li, Competitive exclusion and coexistence of multiple strains in an SIS STD model, SIAM Journal on Applied Mathematics, 59 (1999), 17901811. doi: 10.1137/S0036139997325862. 
[10] 
C. CastilloChavez and B. Song, Dynamical models of tuberculosis and their applications, Math. Biosci. Eng., 1 (2004), 361404. doi: 10.3934/mbe.2004.1.361. 
[11] 
A. Chen, E. Oster and H. Williams, Why is Infant Mortality Higher in the US than in Europe?, National Bureau of Economic Research, 2014. 
[12] 
K. C. Chow, X. Wang and C. CastilloChavez, A mathematical model of nosocomial infection and antibiotic resistance: Evaluating the efficacy of antimicrobial cycling programs and patient isolation on dual resistance, Mathematical and Theoretical Biology Institute archive, 2007. 
[13] 
C. Cohen, D. Horlacher and F. L. MacKellar, Is urbanization good for a nation's health,, 2010. Available from: , (). 
[14] 
C. Dye, Health and urban living, Science, American Association for the Advancement of Science, 319 (2008), 766769. doi: 10.1126/science.1150198. 
[15] 
D. N. Fisman, G. M. Leung and M. Lipsitch, Nuanced risk assessment for emerging infectious diseases, Lancet, 383 (2014). 
[16] 
K. Ford and A. Norris, Sexual networks of AfricanAmerican and Hispanic youth, Sexually Transmitted Diseases, 24 (1997), 327333. doi: 10.1097/0000743519970700000004. 
[17] 
K. Ford, W. Sohn and J. Lepkowski, American adolescents: Sexual mixing patterns, bridge partners, and concurrency, Sexually Transmitted Diseases, 29 (2002), 1319. doi: 10.1097/0000743520020100000003. 
[18] 
S. Galea, Urbanization, urbanicity, and health, Journal of Urban Health, 79 (2002), S1S12. 
[19] 
S. Galea, N. Freudenberg and D. Vlahov, Cities and population health, Social Science & Medicine, 60 (2005), 10171033. doi: 10.1016/j.socscimed.2004.06.036. 
[20] 
M. G. M. Gomes, M. Lipsitch, A. R. Wargo, G. Kurath, C. Rebelo, G. F. Medley and A. Coutinho, A missing dimension in measures of vaccination impacts, PLoS Pathogens, 10 (2014). 
[21] 
T. Harpham and C. Molyneux, Urban health in developing countries: A review, Progress in Development Studies, 1 (2001), 113137. 
[22] 
H. W. Hethcote and J. A. Yorke, Gonorrhea Transmission Dynamics and Control, 56, Springer, Berlin, 1984. doi: 10.1007/9783662075449. 
[23] 
D. R. Holtgrave and R. A. Crosby, Social capital, poverty, and income inequality as predictors of gonorrhoea, syphilis, chlamydia and AIDS case rates in the United States, Sexually Transmitted Infections, 79 (2003), 6264. doi: 10.1136/sti.79.1.62. 
[24] 
M. J. Keeling and P. Rohani, Modeling Infectious Diseases in Humans and Animals, Princeton University Press, 2008. 
[25] 
M. J. Keeling and B. T. Grenfell, Effect of variability in infection period on the persistence and spatial spread of infectious diseases, Mathematical Biosciences, 147 (1998), 207226. doi: 10.1016/S00255564(97)001016. 
[26] 
D. A. Leon, Cities, urbanization and health, International Journal of Epidemiology, 37 (2008), 48. doi: 10.1093/ije/dym271. 
[27] 
J. Li, Z. Ma, S. P. Blythe and C. CastilloChavez, Coexistence of pathogens in sexuallytransmitted disease models, Journal of Mathematical Biology, 47 (2003), 547568. doi: 10.1007/s0028500302355. 
[28] 
M. Lipsitch, T. Cohen, B. Cooper, J. M. Robins, S. Ma, L. James, G. Gopalakrishna, S. K. Chew, C. C. Tan, M. H. Samore, et al., Transmission dynamics and control of severe acute respiratory syndrome, Science, 300 (2003), 19661970. doi: 10.1126/science.1086616. 
[29] 
A. L. Lloyd, Realistic distributions of infectious periods in epidemic models: Changing patterns of persistence and dynamics. Theoretical Population Biology, 60 (2001). 5971. 
[30] 
A. L. Lloyd, Destabilization of epidemic models with the inclusion of realistic distributions of infectious periods, Proceedings of the Royal Society of London. Series B: Biological Sciences, 268 (2011), 985993. 
[31] 
J. O. LloydSmith, S. J. Schreiber, P. E. Kopp and W. M. Getz, Superspreading and the effect of individual variation on disease emergence, Nature, 438 (2005), 355359. 
[32] 
K. Lönnroth, E. Jaramillo, B. G. Williams, C. Dye and M. Raviglione, Drivers of tuberculosis epidemics: the role of risk factors and social determinants, Social Science & Medicine, 68 (2009), 22402246. 
[33] 
M. Marmot, Social determinants of health inequalities, The Lancet, 365 (2005), 10991104. doi: 10.1016/S01406736(05)742343. 
[34] 
D. M. Morens and A. S. Fauci, Emerging infectious diseases: Threats to human health and global stability, PLoS Pathogens, 9 (2013), e1003467. doi: 10.1371/journal.ppat.1003467. 
[35] 
B. Morin, Variable susceptibility with an open population: A transport equation approach,, preprint, (). 
[36] 
B. R. Morin, C. CastilloChavez, S.F. Hsu Schmitz, A. Mubayi and X. Wang, Notes from the heterogeneous: A few observations on the implications and necessity of affinity, Journal of Biological Dynamics, 4 (2010), 456477. doi: 10.1080/17513758.2010.510212. 
[37] 
O. PattersonLomba, E. Goldstein, A. GómezLiévano, C. CastilloChavez and S. Towers, Per capita incidence of sexually transmitted infections increases systematically with urban population size: A crosssectional study, Sexually Transmitted Infections,91 (2015), 610614. doi: 10.1136/sextrans2014051932. 
[38] 
K. Pickett and R. Wilkinson, The Spirit Level: Why More Equal Societies Almost Always do Better, London: Allen Lane, 2009. 
[39] 
E. Saez and G. Zucman, Wealth inequality in the united states since 1913: Evidence from capitalized income tax data, The Quarterly Journal of Economics, 131 (2016), 519578. doi: 10.1093/qje/qjw004. 
[40] 
C. Stephens, Healthy cities or unhealthy islands? The health and social implications of urban inequality, Environment and Urbanization, 8 (1996), 930. doi: 10.1177/095624789600800211. 
[41] 
S. H. Strogatz, Nonlinear Dynamics and Chaos: With Applications to Physics, Biology, Chemistry, and Engineering, AddisonWesley Pub., 1994. 
[42] 
P. van den Driessche and J. Watmough, Reproduction numbers and subthreshold endemic equilibria for compartmental models of disease transmission, Math. Biosci., 180 (2002), 2948. doi: 10.1016/S00255564(02)001086. 
[43] 
, United Nations, World Urbanization Prospects: The 2011 Revision,, 2012. Available from: , (). 
[44] 
J. Wallinga and M. Lipsitch, How generation intervals shape the relationship between growth rates and reproductive numbers, Proceedings of the Royal Society B: Biological Sciences, 274 (2007), 599604. 
[45] 
R. G. Wilkinson, Socioeconomic determinants of health. Health inequalities: Relative or absolute material standards?, BMJ: British Medical Journal, 314 (1997), 591595. doi: 10.1136/bmj.314.7080.591. 
[46] 
R. G. Wilkinson and K. E. Pickett, Income inequality and population health: A review and explanation of the evidence, Social Science & Medicine, 62 (2006), 17681784. doi: 10.1016/j.socscimed.2005.08.036. 
[47] 
P. Zhang and P. M. Atkinson, Modelling the effect of urbanization on the transmission of an infectious disease, Mathematical Biosciences, 211 (2008), 166185. doi: 10.1016/j.mbs.2007.10.007. 
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