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, (1992).   Google Scholar

[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.   Google Scholar

[3]

C. L. Barrett, S. G. Eubank and J. P. Smith, If smallpox strikes Portland,, Scientific American, 292 (2005), 54.  doi: 10.1038/scientificamerican0305-54.  Google Scholar

[4]

S. P. Blythe and C. Castillo-Chavez, Like-with-like preference and sexual mixing models,, Mathematical Biosciences, 96 (1989), 221.  doi: 10.1016/0025-5564(89)90060-6.  Google Scholar

[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.  doi: 10.1056/NEJMsa025075.  Google Scholar

[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.  doi: 10.1093/imammb/8.1.1.  Google Scholar

[7]

Centers for Disease Control and Prevention (CDC), Clinical evaluation tools for smallpox vaccine adverse reactions,, (2004). Available from: , (2004).   Google Scholar

[8]

J. Chin, "Control of Communicable Diseases Manual,", $17^th$ edition, (2002).   Google Scholar

[9]

N. Chitnis, J. M. Hyman, J. Restrepo and J. Li, DSDISP,, (2004). Available from: , (2004).   Google Scholar

[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.  doi: 10.1016/S0022-5193(03)00228-5.  Google Scholar

[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.  doi: 10.1016/j.mbs.2005.03.006.  Google Scholar

[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.   Google Scholar

[13]

M. Eichner, Analysis of historical data suggests long-lasting protective effects of smallpox vaccination,, American Journal of Epidemiology, 158 (2003), 717.  doi: 10.1093/aje/kwg225.  Google Scholar

[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.  doi: 10.1093/aje/kwg103.  Google Scholar

[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.  doi: 10.1038/nature02541.  Google Scholar

[16]

T. Fararo, Biased networks and the strength of weak ties,, Social Networks, 5 (1983), 1.  doi: 10.1016/0378-8733(83)90013-8.  Google Scholar

[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.  doi: 10.1073/pnas.1011250108.  Google Scholar

[18]

F. Fenner, D. A. Henderson, I. Arita, Z. Jezek and I. D. Ladnyi, Smallpox and its eradication,, World Health Organization Geneva, (1988).   Google Scholar

[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.  doi: 10.1038/nature02007.  Google Scholar

[20]

R. Gani and S. Leach, Transmission potential of smallpox in contemporary populations,, Nature, 414 (2001), 748.   Google Scholar

[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.  doi: 10.1016/j.mbs.2011.10.001.  Google Scholar

[22]

K. P. Hadeler and C. Castillo-Chavez, A core group model for disease transmission,, Mathematical Biosciences, 128 (1995), 41.  doi: 10.1016/0025-5564(94)00066-9.  Google Scholar

[23]

E. M. Halloran, I. M. Longini, Jr., A. Nizam and Y. Yang, Containing bioterrorist smallpox,, Science, 298 (2002), 1428.  doi: 10.1126/science.1074674.  Google Scholar

[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.  doi: 10.1016/0025-5564(87)90044-7.  Google Scholar

[25]

H. W. Hethcote and J. A. Yorke, "Gonorrhea Transmission Dynamics and Control,", Lecture Notes in Biomathematics, 56 (1984).   Google Scholar

[26]

J. M. Hyman and J. Li, Biased preference models for partnership formation,, in, (1995), 3137.  doi: 10.1515/9783110883237.3137.  Google Scholar

[27]

J. M. Hyman and J. Li, Disease transmission models with biased partnership selection,, Applied Numerical Mathematics, 24 (1997), 379.  doi: 10.1016/S0168-9274(97)00034-2.  Google Scholar

[28]

J. M. Hyman and J. Li, Behavior changes in SIS STD models with selective mixing,, SIAM Journal on Applied Mathematics, 57 (1997), 1082.  doi: 10.1137/S0036139995294123.  Google Scholar

[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.  doi: 10.1016/S0025-5564(98)10057-3.  Google Scholar

[30]

J. M. Hyman and E. A. Stanley, Using mathematical models to understand the AIDS epidemic,, Mathematical Biosciences, 90 (1988), 415.  doi: 10.1016/0025-5564(88)90078-8.  Google Scholar

[31]

J. M. Hyman and E. A. Stanley, The effect of social mixing patterns on the spread of AIDS,, in, 81 (1989), 190.  doi: 10.1007/978-3-642-46693-9_15.  Google Scholar

[32]

D. Hopkins, "The Greatest Killer: Smallpox in History,", University of Chicago Press, (1983).   Google Scholar

[33]

H. F. Hull, R. Danila and K. Ehresmann, Smallpox and bioterrorism: public-health responses,, Journal of Laboratory and Clinical Medicine, 142 (2003), 221.  doi: 10.1016/S0022-2143(03)00144-6.  Google Scholar

[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.  doi: 10.1016/0025-5564(88)90031-4.  Google Scholar

[35]

E. H. Kaplan, P. C. Cramton and A. D. Paltiel, Nonrandom mixing models of HIV transmission,, in, 83 (1989), 218.  doi: 10.1007/978-3-642-93454-4_10.  Google Scholar

[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.  doi: 10.1073/pnas.162282799.  Google Scholar

[37]

H. Knolle, A discrete branching process model for the spread of HIV via steady sexual partnerships,, Journal of Mathematical Biology, 48 (2004), 423.  doi: 10.1007/s00285-003-0241-7.  Google Scholar

[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, 83 (1989), 316.  doi: 10.1007/978-3-642-93454-4_16.  Google Scholar

[39]

M. Kretzschmar and M. Morris, Measures of concurrency in networks and the spread of infectious diseases,, Mathematical Biosciences, 133 (1996), 165.  doi: 10.1016/0025-5564(95)00093-3.  Google Scholar

[40]

M. Kretzschmar, D. Reinking, H. Brouwers, G. Zessen and J. Jager, Networks models: From paradigm to mathematical tool,, in, (1994), 561.   Google Scholar

[41]

M. Kretzschmar, S. van den Hof, J. Wallinga and J. van Wijngaarden, Ring vaccination and smallpox control,, EID, 10 (2004), 832.  doi: 10.3201/eid1005.030419.  Google Scholar

[42]

T. M. Mack, Smallpox in Europe 1950-1971,, The Journal of Infectious Diseases, 125 (1972), 161.  doi: 10.1093/infdis/125.2.161.  Google Scholar

[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).  doi: 10.1371/journal.ppat.1000745.  Google Scholar

[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.  doi: 10.3201/eid0706.010607.  Google Scholar

[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.  doi: 10.2188/jea.14.41.  Google Scholar

[46]

A. Nold, Heterogeneity in disease-transmission modeling,, Mathematical Biosciences, 52 (1980), 227.  doi: 10.1016/0025-5564(80)90069-3.  Google Scholar

[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.  doi: 10.3201/eid1004.030628.  Google Scholar

[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.  doi: 10.1001/jama.290.24.3215.  Google Scholar

[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.   Google Scholar

[50]

A. Rapoport and Y. Yuan, Some aspects of epidemics and social nets,, in, (1989), 327.   Google Scholar

[51]

S. Singh, Some aspects of the epidemiology of smallpox in Nepal,, World Health Organization, (1969).   Google Scholar

[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.  doi: 10.1016/j.mbs.2006.01.007.  Google Scholar

[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).   Google Scholar

[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.  doi: 10.1016/S0025-5564(02)00108-6.  Google Scholar

[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.   Google Scholar

[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).   Google Scholar

[57]

G. S. Zaric, Random vs. nonrandom mixing in network epidemic models,, Health Care Management Science, 5 (2002), 147.   Google Scholar

show all references

References:
[1]

R. M. Anderson and R. M. May, "Infectious Diseases of Humans: Dynamics and Control,", Oxford Science Publications, (1992).   Google Scholar

[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.   Google Scholar

[3]

C. L. Barrett, S. G. Eubank and J. P. Smith, If smallpox strikes Portland,, Scientific American, 292 (2005), 54.  doi: 10.1038/scientificamerican0305-54.  Google Scholar

[4]

S. P. Blythe and C. Castillo-Chavez, Like-with-like preference and sexual mixing models,, Mathematical Biosciences, 96 (1989), 221.  doi: 10.1016/0025-5564(89)90060-6.  Google Scholar

[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.  doi: 10.1056/NEJMsa025075.  Google Scholar

[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.  doi: 10.1093/imammb/8.1.1.  Google Scholar

[7]

Centers for Disease Control and Prevention (CDC), Clinical evaluation tools for smallpox vaccine adverse reactions,, (2004). Available from: , (2004).   Google Scholar

[8]

J. Chin, "Control of Communicable Diseases Manual,", $17^th$ edition, (2002).   Google Scholar

[9]

N. Chitnis, J. M. Hyman, J. Restrepo and J. Li, DSDISP,, (2004). Available from: , (2004).   Google Scholar

[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.  doi: 10.1016/S0022-5193(03)00228-5.  Google Scholar

[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.  doi: 10.1016/j.mbs.2005.03.006.  Google Scholar

[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.   Google Scholar

[13]

M. Eichner, Analysis of historical data suggests long-lasting protective effects of smallpox vaccination,, American Journal of Epidemiology, 158 (2003), 717.  doi: 10.1093/aje/kwg225.  Google Scholar

[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.  doi: 10.1093/aje/kwg103.  Google Scholar

[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.  doi: 10.1038/nature02541.  Google Scholar

[16]

T. Fararo, Biased networks and the strength of weak ties,, Social Networks, 5 (1983), 1.  doi: 10.1016/0378-8733(83)90013-8.  Google Scholar

[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.  doi: 10.1073/pnas.1011250108.  Google Scholar

[18]

F. Fenner, D. A. Henderson, I. Arita, Z. Jezek and I. D. Ladnyi, Smallpox and its eradication,, World Health Organization Geneva, (1988).   Google Scholar

[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.  doi: 10.1038/nature02007.  Google Scholar

[20]

R. Gani and S. Leach, Transmission potential of smallpox in contemporary populations,, Nature, 414 (2001), 748.   Google Scholar

[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.  doi: 10.1016/j.mbs.2011.10.001.  Google Scholar

[22]

K. P. Hadeler and C. Castillo-Chavez, A core group model for disease transmission,, Mathematical Biosciences, 128 (1995), 41.  doi: 10.1016/0025-5564(94)00066-9.  Google Scholar

[23]

E. M. Halloran, I. M. Longini, Jr., A. Nizam and Y. Yang, Containing bioterrorist smallpox,, Science, 298 (2002), 1428.  doi: 10.1126/science.1074674.  Google Scholar

[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.  doi: 10.1016/0025-5564(87)90044-7.  Google Scholar

[25]

H. W. Hethcote and J. A. Yorke, "Gonorrhea Transmission Dynamics and Control,", Lecture Notes in Biomathematics, 56 (1984).   Google Scholar

[26]

J. M. Hyman and J. Li, Biased preference models for partnership formation,, in, (1995), 3137.  doi: 10.1515/9783110883237.3137.  Google Scholar

[27]

J. M. Hyman and J. Li, Disease transmission models with biased partnership selection,, Applied Numerical Mathematics, 24 (1997), 379.  doi: 10.1016/S0168-9274(97)00034-2.  Google Scholar

[28]

J. M. Hyman and J. Li, Behavior changes in SIS STD models with selective mixing,, SIAM Journal on Applied Mathematics, 57 (1997), 1082.  doi: 10.1137/S0036139995294123.  Google Scholar

[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.  doi: 10.1016/S0025-5564(98)10057-3.  Google Scholar

[30]

J. M. Hyman and E. A. Stanley, Using mathematical models to understand the AIDS epidemic,, Mathematical Biosciences, 90 (1988), 415.  doi: 10.1016/0025-5564(88)90078-8.  Google Scholar

[31]

J. M. Hyman and E. A. Stanley, The effect of social mixing patterns on the spread of AIDS,, in, 81 (1989), 190.  doi: 10.1007/978-3-642-46693-9_15.  Google Scholar

[32]

D. Hopkins, "The Greatest Killer: Smallpox in History,", University of Chicago Press, (1983).   Google Scholar

[33]

H. F. Hull, R. Danila and K. Ehresmann, Smallpox and bioterrorism: public-health responses,, Journal of Laboratory and Clinical Medicine, 142 (2003), 221.  doi: 10.1016/S0022-2143(03)00144-6.  Google Scholar

[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.  doi: 10.1016/0025-5564(88)90031-4.  Google Scholar

[35]

E. H. Kaplan, P. C. Cramton and A. D. Paltiel, Nonrandom mixing models of HIV transmission,, in, 83 (1989), 218.  doi: 10.1007/978-3-642-93454-4_10.  Google Scholar

[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.  doi: 10.1073/pnas.162282799.  Google Scholar

[37]

H. Knolle, A discrete branching process model for the spread of HIV via steady sexual partnerships,, Journal of Mathematical Biology, 48 (2004), 423.  doi: 10.1007/s00285-003-0241-7.  Google Scholar

[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, 83 (1989), 316.  doi: 10.1007/978-3-642-93454-4_16.  Google Scholar

[39]

M. Kretzschmar and M. Morris, Measures of concurrency in networks and the spread of infectious diseases,, Mathematical Biosciences, 133 (1996), 165.  doi: 10.1016/0025-5564(95)00093-3.  Google Scholar

[40]

M. Kretzschmar, D. Reinking, H. Brouwers, G. Zessen and J. Jager, Networks models: From paradigm to mathematical tool,, in, (1994), 561.   Google Scholar

[41]

M. Kretzschmar, S. van den Hof, J. Wallinga and J. van Wijngaarden, Ring vaccination and smallpox control,, EID, 10 (2004), 832.  doi: 10.3201/eid1005.030419.  Google Scholar

[42]

T. M. Mack, Smallpox in Europe 1950-1971,, The Journal of Infectious Diseases, 125 (1972), 161.  doi: 10.1093/infdis/125.2.161.  Google Scholar

[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).  doi: 10.1371/journal.ppat.1000745.  Google Scholar

[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.  doi: 10.3201/eid0706.010607.  Google Scholar

[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.  doi: 10.2188/jea.14.41.  Google Scholar

[46]

A. Nold, Heterogeneity in disease-transmission modeling,, Mathematical Biosciences, 52 (1980), 227.  doi: 10.1016/0025-5564(80)90069-3.  Google Scholar

[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.  doi: 10.3201/eid1004.030628.  Google Scholar

[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.  doi: 10.1001/jama.290.24.3215.  Google Scholar

[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.   Google Scholar

[50]

A. Rapoport and Y. Yuan, Some aspects of epidemics and social nets,, in, (1989), 327.   Google Scholar

[51]

S. Singh, Some aspects of the epidemiology of smallpox in Nepal,, World Health Organization, (1969).   Google Scholar

[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.  doi: 10.1016/j.mbs.2006.01.007.  Google Scholar

[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).   Google Scholar

[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.  doi: 10.1016/S0025-5564(02)00108-6.  Google Scholar

[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.   Google Scholar

[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).   Google Scholar

[57]

G. S. Zaric, Random vs. nonrandom mixing in network epidemic models,, Health Care Management Science, 5 (2002), 147.   Google Scholar

[1]

Krzysztof Frączek, Leonid Polterovich. Growth and mixing. Journal of Modern Dynamics, 2008, 2 (2) : 315-338. doi: 10.3934/jmd.2008.2.315

[2]

Timothy C. Reluga, Jan Medlock, Alison Galvani. The discounted reproductive number for epidemiology. Mathematical Biosciences & Engineering, 2009, 6 (2) : 377-393. doi: 10.3934/mbe.2009.6.377

[3]

Robert E. Beardmore, Rafael Peña-Miller. Antibiotic cycling versus mixing: The difficulty of using mathematical models to definitively quantify their relative merits. Mathematical Biosciences & Engineering, 2010, 7 (4) : 923-933. doi: 10.3934/mbe.2010.7.923

[4]

Lidong Wang, Xiang Wang, Fengchun Lei, Heng Liu. Mixing invariant extremal distributional chaos. Discrete & Continuous Dynamical Systems - A, 2016, 36 (11) : 6533-6538. doi: 10.3934/dcds.2016082

[5]

A. Crannell. A chaotic, non-mixing subshift. Conference Publications, 1998, 1998 (Special) : 195-202. doi: 10.3934/proc.1998.1998.195

[6]

Zhi Lin, Katarína Boďová, Charles R. Doering. Models & measures of mixing & effective diffusion. Discrete & Continuous Dynamical Systems - A, 2010, 28 (1) : 259-274. doi: 10.3934/dcds.2010.28.259

[7]

Werner Creixell, Juan Carlos Losada, Tomás Arredondo, Patricio Olivares, Rosa María Benito. Serendipity in social networks. Networks & Heterogeneous Media, 2012, 7 (3) : 363-371. doi: 10.3934/nhm.2012.7.363

[8]

Yuki Kumagai. Social networks and global transactions. Journal of Dynamics & Games, 2019, 6 (3) : 211-219. doi: 10.3934/jdg.2019015

[9]

Rui Kuang, Xiangdong Ye. The return times set and mixing for measure preserving transformations. Discrete & Continuous Dynamical Systems - A, 2007, 18 (4) : 817-827. doi: 10.3934/dcds.2007.18.817

[10]

Nir Avni. Spectral and mixing properties of actions of amenable groups. Electronic Research Announcements, 2005, 11: 57-63.

[11]

Richard Miles, Thomas Ward. A directional uniformity of periodic point distribution and mixing. Discrete & Continuous Dynamical Systems - A, 2011, 30 (4) : 1181-1189. doi: 10.3934/dcds.2011.30.1181

[12]

Hadda Hmili. Non topologically weakly mixing interval exchanges. Discrete & Continuous Dynamical Systems - A, 2010, 27 (3) : 1079-1091. doi: 10.3934/dcds.2010.27.1079

[13]

Piotr Oprocha, Paweł Potorski. Topological mixing, knot points and bounds of topological entropy. Discrete & Continuous Dynamical Systems - B, 2015, 20 (10) : 3547-3564. doi: 10.3934/dcdsb.2015.20.3547

[14]

Jean René Chazottes, F. Durand. Local rates of Poincaré recurrence for rotations and weak mixing. Discrete & Continuous Dynamical Systems - A, 2005, 12 (1) : 175-183. doi: 10.3934/dcds.2005.12.175

[15]

Matúš Dirbák. Minimal skew products with hypertransitive or mixing properties. Discrete & Continuous Dynamical Systems - A, 2012, 32 (5) : 1657-1674. doi: 10.3934/dcds.2012.32.1657

[16]

Dmitri Scheglov. Absence of mixing for smooth flows on genus two surfaces. Journal of Modern Dynamics, 2009, 3 (1) : 13-34. doi: 10.3934/jmd.2009.3.13

[17]

Jian Li. Localization of mixing property via Furstenberg families. Discrete & Continuous Dynamical Systems - A, 2015, 35 (2) : 725-740. doi: 10.3934/dcds.2015.35.725

[18]

Oliver Knill. Singular continuous spectrum and quantitative rates of weak mixing. Discrete & Continuous Dynamical Systems - A, 1998, 4 (1) : 33-42. doi: 10.3934/dcds.1998.4.33

[19]

Makoto Mori. Higher order mixing property of piecewise linear transformations. Discrete & Continuous Dynamical Systems - A, 2000, 6 (4) : 915-934. doi: 10.3934/dcds.2000.6.915

[20]

Benoît Saussol. Recurrence rate in rapidly mixing dynamical systems. Discrete & Continuous Dynamical Systems - A, 2006, 15 (1) : 259-267. doi: 10.3934/dcds.2006.15.259

2018 Impact Factor: 1.313

Metrics

  • PDF downloads (19)
  • HTML views (0)
  • Cited by (0)

Other articles
by authors

[Back to Top]