Amath 504


AMATH 504
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Topics

March 31

Basic Epidemiology Models I

Introduction, types of infectious diseases, the SIS model, the SIR epidemic and SIR endemic models, expressions for Ro, parameter estimation
Readings:
Hethcote, H.W., The mathematics of infectious diseases, SIAM Review 42 (2000) 599-653.

2. April 2 -- Basic Epidemiology Models II

Comparisons of directly transmitted infectious diseases, vaccination programs for smallpox, polio, chickenpox (varicella), measles, and rubella
Readings: see topic 1
Hethcote, H.W., Measles and rubella in the United States, Am. J. Epidemiol. 117 (1983) 2-13.
Hethcote, H.W., Optimal ages of vaccination for measles, Math. Biosci. 89 (1988) 29-52.

3. April 7 -- Basic Epidemiology Models III

Purposes of epidemiology modeling, When is a disease eradicable? Dracunculiasis in Africa. Modeling influenza.
Readings: Purposes are discussed on pages 19-26 in H.W. Hethcote and J.W. Van Ark, Modeling HIV Transmission and AIDS in the United States, Lecture Notes in Biomathematics 95, Springer, Berlin, 1992.
Aylward B. et al. When is a disease eradicable? 100 years of lessons learned. Am. J. Publ. Health 90-10 (October 2000) 1515-1520. (this paper will be distributed)

4. April 9 -- A Predator-Prey Model with Infected Prey

How can an infectious disease affect a predator-prey process?
Readings:
H.W. Hethcote, W. Wang, L. Han, and Z. Ma, A Predator Prey Model with Infected Prey, Theoretical Population Biology 66 (2004) 259-268.

5. April 14 -- Gonorrhea Transmission Dynamics and Control I

Simple model, multi-group model, proportionate mixing, model with core group
Readings:
Hethcote, H. W. and Yorke, J. A., Gonorrhea Transmission Dynamics and Control, Lecture Notes in Biomathematics 56, Springer, Berlin, 1984, available as pdf file at instructor’s homepage

6. April 16 -- Gonorrhea Transmission Dynamics and Control II

Female-male model, seasonal oscillations, model for a heterogeneous population
Readings:
Hethcote, H. W. and Yorke, J. A., Gonorrhea Transmission Dynamics and Control, Lecture Notes in Biomathematics 56, Springer, Berlin, 1984.

7. April 21 -- Age structured epidemiology models and expressions for Ro

Demographic models with continuous age or age groups, MSEIR models, expressions for Ro, measles in Niger, pertussis in the USA
Readings:
Hethcote, H. W., The mathematics of infectious diseases, SIAM Review 42 (2000) 599-653.

8. April 23 -- Comparisons of Rubella Vaccination Strategies

The UK strategy vs. the USA strategy
Readings:
Hethcote, H. W., Measles and rubella in the United States, Am. J. Epidemiol. 117 (1983) 2-13.
Hethcote, H. W. Rubella, In Applied Mathematical Ecology, L. Gross, T.G. Hallam and S.A. Levin, eds., Springer-Verlag, Berlin, 1989, 212-234.
Hethcote, H. W., Rouderfer V. and Becker, N. Waning immunity and its effects on vaccination schedules, Math. Biosci., 124 (1994) 59-82.

9. April 28 -- Comparing Pertussis Vaccination Strategies

An epidemiologic-demographic model is formulated for pertussis transmission and vaccination in the USA. Computer simulations are used to predict the impact of vaccinations of children, adults, and/or adolescents, and household members (cocoon strategy). The numbers needed to vaccinate to prevent a typical case of pertussis in the entire population and in young infants were computed for five vaccination strategies. Modeling pertussis vaccination strategies in Australia.
Readings:
Van Rie, A. and Hethcote, H.W. Adolescent and adult pertussis vaccination: computer simulations of five new strategies, Vaccine 22 (2004) 3154-3165.
Hethcote, H. W., Horby, P., and McIntyre, P. Using computer simulations to compare pertussis vaccination strategies in Australia, Vaccine 22 (2004) 2181-2191.

10. April 30 -- Modeling the Effects of Varicella Vaccination

Computer simulations of an age-structured epidemiologic model have been used to study the effects of varicella vaccination strategies on cases of chickenpox (varicella) and shingles (herpes zoster) in the USA. Related papers on varicella and zoster in UK and Canada are reviewed.
Readings:
Schuette, M. C. and Hethcote, H. W. Modeling the effects of varicella vaccination programs on the incidence of chickenpox and shingles, Bull. Math. Biol. 61 (1999) 1031-1064.

11. May 5 -- Simulations of Rubella Vaccination Strategies in China

The demographic model for China reflects the effects of their policy of "one child per couple." Computer simulations of rubella transmission dynamics show the effects of the changing demographics and different vaccination policies on the incidence of rubella and Congenital Rubella Syndrome (CRS) in China.
Readings:
Gao, L. Q. and Hethcote, H. W. Simulations of rubella vaccination strategies in China, Mathematical Biosciences 202 (2006) 371-385.
Gao, L. Q. and Hethcote, H. W. A mathematical model and projection of various rubella vaccination strategies. Chinese Journal of Vaccines and Immunization 14-3 (2008) 193-197 (in Chinese with English abstract).

12. May 7 -- Biennial oscillations in Measles and Chaos

Annual oscillations and seasonality, biennial oscillations in measles and chaos
Readings:
Hethcote, H. W. and Levin, S. A. Periodicity in epidemiological models, In Applied Mathematical Ecology, L. Gross, T.G. Hallam and S.A. Levin, {eds.}, Springer, Berlin, 1989, 193-211.
Schaffer, W. M. Can nonlinear dynamics help us infer mechanisms in ecology and epidemiology? IMA J Math. Appl. Biol.Med 2 (1985) 221-252.
Schaffer, W. M. and Kot, M. Nearly one dimensional dynamics in an epidemic. J. Theor. Biol. 112 (1985) 403-427.

13. May 12 -- Adult Booster Vaccinations and Periodicity in Pertussis Incidence

With adult booster vaccinations in a pertussis model, simulations with seasonality produced damped oscillations, but yearly stochasticity yielded sustained oscillations.
Readings:
Hethcote, H. W. Simulations of pertussis epidemiology in the United States: Effects of adult booster doses, Math. Biosci. 158 (1999) 47-73.
Hethcote, H. W. Oscillations in an endemic model for pertussis, Canad. Appl. Math. Quart., 6 (1998) 61-88.
Hethcote, H. W. An age-structured model for pertussis transmission, Math. Biosci. 145 (1997) 89-136.

14. May 14 -- Epidemiology models with Variable Population Size

Horizontal incidences, waiting times, demographic structures, epidemiological-demographic interactions, infectious disease models
Readings:
Hethcote, H. W. A thousand and one epidemic models, In Frontiers in Mathematical Biology, S. Levin, ed., Lecture Notes in Biomathematics 100, Springer-Verlag, Berlin, 1994, 504-515.
Mena-Lorca, J. and Hethcote, H. W. Dynamic models of infectious diseases as regulators of population sizes, J. Math. Biology 30 (1992) 693-716.
Gao, L.Q. and Hethcote, H. W. Disease transmission models with density dependent demographics, J. Math. Biology 30 (1992) 717-731.
Zhou, J. and Hethcote, H. W. Population size dependent incidence in models for diseases without immunity, J. Math. Biol. 32 (1994) 809-834.
Related papers:
Anderson, R. M. and May, R. M. Population biology of infectious diseases I. Nature 280 (1979) 361-367.
May, R. M. and Anderson, R. M. Population biology of infectious diseases II. Nature 280 (1979) 455-461.

15. May 19 -- Periodicity in SEI Epidemiology Models for Fox Rabies

How does the choice of the standard incidence vs. the mass action incidence affect the behavior of SEI models for fox rabies?
Readings:
Anderson, R. M., Jackson, H. C., May, R. M. and Smith, A. D. Population dynamics of fox rabies in Europe. Nature 289 (1981) 765-777.
Gao, L.Q., Mena-Lorca, J. and Hethcote, H.W. Four SEI endemic models with periodicity and separatrices, Math. Biosci. 128 (1995) 157-184.
Gao, L.Q., Mena-Lorca, J. and Hethcote, H.W. Variations on a theme of SEI endemic models, In Differential Equations and Applications to Biology and to Industry, M. Martelli et. al., eds., World Scientific Publishing Co., Singapore, 1996, 191-207.

16. May 21 -- Effects of Behavioral Change in Smallpox and Influenza Models

How do changes in behavior affect the spread of diseases?
Readings:
Del Valle, S., Hethcote, H. W., Hyman, J. M., and Castillo-Chavez, C. Effects of Behavioral Changes in a Smallpox Attack Model, Mathematical Biosciences 195 (2005) 228-251.
Del Valle, S., Mniszewski, S. M., Stroud, P. D., Riese, J. M., and Sydoriak, S. Can Temporal Behavioral Changes Generate Waves during a Pandemic?, Submitted for Publication.
Stroud, P., Del Valle, S., Sydoriak, S., Riese J., and Mniszewski, S. Spatial Dynamics of Pandemic Influenza in a Massive Artificial Society, Journal of Artificial Societies and Social Simulation (2007) .
Stroud, P. D., Sydoriak, S. J., Riese, J. M., Mniszewski, S. M., Romero, P. R., Smith, J. P., and Del Valle, S. Household Sizes Predict Pandemic Influenza Hot-Spots, Submitted for Publication.

17. May 26 -- Periodicity in SIRS models with Delays

How does the behavior of models formulated with delays differ from that of models formulated with exponential waiting times?
Readings:
Hethcote, H. W., Stech, H. W., and van den Driessche, P. Nonlinear oscillations in epidemic models, SIAM J. Appl. Math. 40 (1981) 1-9.
Hethcote, H. W., Stech, H. W., and van den Dreissche, P. Stability analysis for models of diseases without immunity, J. Math. Biology 13 (1981) 185-198.
Hethcote, H. W. and Tudor, D. W. Integral equation models for endemic infectious diseases, J. Math. Biology 9 (1980) 37-48.
Hethcote, H. W. and Levin, S. A. Periodicity in epidemiological models, In Applied Mathematical Ecology, L. Gross, T.G. Hallam and S.A. Levin, eds., Springer-Verlag, Berlin Heidelberg NewYork, 1989, 193-211.
Hethcote, H. W., Stech, H. W., and van den Dreissche, P. Periodicity and stability in epidemic models: A survey, In Differential Equations and Applications in Ecology, Epidemics and Population Problems, Claremont Conference Proceedings, S. Busenberg and K. Cooke, eds., Academic Press, New York, 1981, 65-82.
Hethcote, H. W. and van den Driessche, P. Two SIS epidemiologic models with delays, J. Math. Biol. 40 (2000) 3-26.
Hethcote, H. W. and van den Driessche, P. An SIS epidemic model with variable population size and a delay, J. Math. Biol. 34 (1995) 177-194.

18. May 28 -- Periodicity in Epidemiology Models with Nonlinear Incidence

How do models with nonlinear incidences give rise to periodic solutions?
Readings:
Liu, W. M., Hethcote, H. W., and Levin, S. A. Dynamical behavior of epidemiological models with nonlinear incidence rates, J. Math. Biology 25 (1987) 359-380.
Hethcote, H. W., Lewis, M. A., and van den Driessche, P. An epidemiological model with a delay and a nonlinear incidence rate, J. Math. Biology 27 (1989) 49-64.
Hethcote, H. W. and van den Driessche, P. Some epidemiological models with nonlinear incidence, J. Math. Biology 29 (1991) 271-287.

19. & 20. June 2 and June 4 -- Topics chosen from the following:

A. Effects of Isolation in 3 Endemic Models

SIR models with isolation used as an intervention.
Readings:
Hethcote, H. W., Ma, Z., and Liao, S. Effects of Quarantine in Six Endemic Models for Infectious Diseases, Mathematical Biosciences 180 (2002) 141-160.

B. Species Coexistence and Periodicity in Host-Host-Pathogen Models

How does the form of the incidence term affect the behavior?
Readings:
Hethcote, H. W., Wang, W., and Li, Y. Species coexistence and periodicity in host-host-pathogen models, J Math Biology 51 (2005) 629-660.

C. Competing Species with an Infectious Disease

Grey squirrels, red squirrels, and parapoxvirus in the United Kingdom
Readings:
Saenz, R. A. and Hethcote, H. W. Competing species models with an infectious disease, Mathematical Biosciences and Engineering 3 (2006) 219-235.

D. Periodic Traveling Waves in SIRS Endemic Models

Can infection wave fronts travel geographically around a continent?
Readings:
Li, T., Li, Y., and Hethcote, H. W. Periodic traveling waves in SIRS endemic models, Mathematical and Computer Modelling 49 (2009) 393-401.