Mathematical epidemiology is not an oxymoron

A brief description of the importance of communicable diseases in history and the development of mathematical modelling of disease transmission is given. This includes reasons for mathematical modelling, the history of mathematical modelling from the foundations laid in the late nineteenth century to the present, some of the accomplishments of mathematical modelling, and some challenges for the future. Our purpose is to demonstrate the importance of mathematical modelling for the understanding and management of infectious disease transmission.

[1]  J. Heesterbeek,et al.  The saturating contact rate in marriage- and epidemic models , 1993, Journal of mathematical biology.

[2]  F. Ball,et al.  Stochastic and deterministic models for SIS epidemics among a population partitioned into households. , 1999, Mathematical biosciences.

[3]  C. Castillo-Chavez.  Mathematical and Statistical Approaches to Aids Epidemiology , 1989 .

[4]  D. Schenzle.  An age-structured model of pre- and post-vaccination measles transmission. , 1984, IMA journal of mathematics applied in medicine and biology.

[5]  Kenneth L. Cooke,et al.  Stability analysis for a vector disease model , 1979 .

[6]  Zhilan Feng,et al.  Final and peak epidemic sizes for SEIR models with quarantine and isolation. , 2007, Mathematical biosciences and engineering : MBE.

[7]  M. Newman,et al.  Network theory and SARS: predicting outbreak diversity , 2004, Journal of Theoretical Biology.

[8]  D Greenhalgh,et al.  Subcritical endemic steady states in mathematical models for animal infections with incomplete immunity. , 2000, Mathematical biosciences.

[9]  Julien Arino,et al.  The Basic Reproduction Number in a Multi-city Compartmental Epidemic Model , 2003, POSTA.

[10]  N. Goel,et al.  Stochastic models in biology , 1975 .

[11]  Herbert W. Hethcote,et al.  Modeling the effects of varicella vaccination programs on the incidence of chickenpox and shingles , 1999, Bulletin of mathematical biology.

[12]  G. Webb,et al.  Diffusive epidemic models with spatial and age dependent heterogeneity , 1994 .

[13]  I. Nåsell.  Endemicity, persistence, and quasi-stationarity , 2002 .

[14]  Maia Martcheva,et al.  Vaccination strategies and backward bifurcation in an age-since-infection structured model. , 2002, Mathematical biosciences.

[15]  Odo Diekmann,et al.  The legacy of Kermack and McKendrick , 1995 .

[16]  L. Allen.  An introduction to stochastic processes with applications to biology , 2003 .

[17]  P. D. En'ko,et al.  On the course of epidemics of some infectious diseases. , 1989, International journal of epidemiology.

[18]  F. Brauer,et al.  Models for the spread of universally fatal diseases. , 1990, Journal of mathematical biology.

[19]  M. Newman.  Spread of epidemic disease on networks. , 2002, Physical review. E, Statistical, nonlinear, and soft matter physics.

[20]  O. Diekmann,et al.  Thresholds and travelling waves for the geographical spread of infection , 1978, Journal of mathematical biology.

[21]  M. E. Alexander,et al.  Modelling strategies for controlling SARS outbreaks , 2004, Proceedings of the Royal Society of London. Series B: Biological Sciences.

[22]  Julien Arino,et al.  Quarantine in a multi-species epidemic model with spatial dynamics. , 2007, Mathematical biosciences.

[23]  Alan S. Perelson,et al.  Mathematical Analysis of HIV-1 Dynamics in Vivo , 1999, SIAM Rev..

[24]  Herbert W. Hethcote,et al.  Dynamic models of infectious diseases as regulators of population sizes , 1992, Journal of mathematical biology.

[25]  W. O. Kermack,et al.  Contributions to the Mathematical Theory of Epidemics. II. The Problem of Endemicity , 1932 .

[26]  Odo Diekmann,et al.  Run for your life; a note on the asymptotic speed of propagation of an epidemic : (preprint) , 1979 .

[27]  S. Lal,et al.  Epidemiology and control of malaria , 1999, Indian journal of pediatrics.

[28]  O. Diekmann,et al.  Prelude to hopf bifurcation in an epidemic model: Analysis of a characteristic equation associated with a nonlinear Volterra integral equation , 1982, Journal of mathematical biology.

[29]  J. Yorke,et al.  A Deterministic Model for Gonorrhea in a Nonhomogeneous Population , 1976 .

[30]  Gail S. K. Wolkowicz,et al.  Dynamical Systems and Their Applications in Biology , 2003 .

[31]  Herbert W. Hethcote,et al.  The Mathematics of Infectious Diseases , 2000, SIAM Rev..

[32]  John A. Jacquez,et al.  Structured mixing: heterogeneous mixing by the definition of activity groups , 1989 .

[33]  Ingemar Nåsell,et al.  The quasi-stationary distribution of the closed endemic sis model , 1996, Advances in Applied Probability.

[34]  C. Macken,et al.  Mitigation strategies for pandemic influenza in the United States. , 2006, Proceedings of the National Academy of Sciences of the United States of America.

[35]  P. Kaye.  Infectious diseases of humans: Dynamics and control , 1993 .

[36]  I. Nåsell.  The quasi-stationary distribution of the closed endemic sis model , 1996, Advances in Applied Probability.

[37]  D. Earn,et al.  A simple model for complex dynamical transitions in epidemics. , 2000, Science.

[38]  D. Cummings,et al.  Strategies for containing an emerging influenza pandemic in Southeast Asia , 2005, Nature.

[39]  C. Castillo-Chavez,et al.  Mathematical Approaches for Emerging and Reemerging Infectious Diseases: An Introduction , 2002 .

[40]  G. Webb.  Theory of Nonlinear Age-Dependent Population Dynamics , 1985 .

[41]  W. O. Kermack,et al.  Contributions to the Mathematical Theory of Epidemics. III. Further Studies of the Problem of Endemicity , 1933 .

[42]  Shigui Ruan,et al.  Dynamical behavior of an epidemic model with a nonlinear incidence rate , 2003 .

[43]  Denis Mollison,et al.  Spatial Contact Models for Ecological and Epidemic Spread , 1977 .

[44]  H. Weinberger.  Some Deterministic Models for the Spread of Genetic and Other Alterations , 1980 .

[45]  A. Benenson.  Control of communicable diseases in man : an official report of the American Public Health Association , 1995 .

[46]  M. Iannelli,et al.  Analytical and numerical results for the age-structured S-I-S epidemic model with mixed inter-intracohort transmission , 1992 .

[47]  S. A. Levin,et al.  On the role of long incubation periods in the dynamics of acquired immunodeficiency syndrome (AIDS) , 1989, Journal of mathematical biology.

[48]  R. May,et al.  Epidemiology. How viruses spread among computers and people. , 2001, Science.

[49]  C. Castillo-Chavez,et al.  A model for tuberculosis with exogenous reinfection. , 2000, Theoretical population biology.

[50]  K. Cooke,et al.  Vertically transmitted diseases , 1993 .

[51]  Y Cha,et al.  Existence and uniqueness of endemic states for the age-structured S-I-R epidemic model. , 1998, Mathematical biosciences.

[52]  Valerie Isham,et al.  Models for Infectious Human Diseases: Their Structure and Relation to Data , 1996 .

[53]  J. Chin,et al.  Control of Communicable Diseases Manual , 2002 .

[54]  Charles J. Mode,et al.  Stochastic Processes in Epidemiology: Hiv/Aids, Other Infectious Diseases and Computers , 2000 .

[55]  J. Velasco-Hernández,et al.  A simple vaccination model with multiple endemic states. , 2000, Mathematical biosciences.

[56]  David J. Bartholomew,et al.  Mathematical and Statistical Approaches , 2005 .

[57]  A. Nizam,et al.  Containing pandemic influenza with antiviral agents. , 2004, American journal of epidemiology.

[58]  J. Brownlee,et al.  Statistical Studies in Immunity: The Theory of an Epidemic , 1906 .

[59]  S. Busenberg,et al.  A general solution of the problem of mixing of subpopulations and its application to risk- and age-structured epidemic models for the spread of AIDS. , 1991, IMA journal of mathematics applied in medicine and biology.

[60]  C. Castillo-Chavez,et al.  Dispersal, disease and life-history evolution. , 2001, Mathematical biosciences.

[61]  I. Nåsell, Hybrid models of tropical infections , 1986 .

[62]  P van den Driessche,et al.  Backward bifurcation in epidemic control. , 1997, Mathematical biosciences.

[63]  F. Ball,et al.  Epidemics with two levels of mixing , 1997 .

[64]  K. Dietz.  THE FIRST EPIDEMIC MODEL: A HISTORICAL NOTE ON P.D. EN'KO , 1988 .

[65]  E. B. Wilson,et al.  A Second Approximation to Soper's Epidemic Curve. , 1944, Proceedings of the National Academy of Sciences of the United States of America.

[66]  M. Bartlett,et al.  Stochastic Population Models in Ecology and Epidemiology. , 1961 .

[67]  Herbert W. Hethcote,et al.  Optimal ages of vaccination for measles , 1988 .

[68]  Carlos Castillo-Chavez,et al.  How May Infection-Age-Dependent Infectivity Affect the Dynamics of HIV/AIDS? , 1993, SIAM J. Appl. Math..

[69]  Carlos Castillo-Chavez,et al.  On the role of long incubation periods in the dynamics of Acquired Immunodeficiency Syndrome (AIDS). Part 2: multiple group models , 1990 .

[70]  R. Watson,et al.  On an epidemic in a stratified population , 1972, Journal of Applied Probability.

[71]  H. E. Soper.  The Interpretation of Periodicity in Disease Prevalence , 1929 .

[72]  A. Benenson.  CONTROL OF COMMUNICABLE DISEASES IN MAN , 1966 .

[73]  M E J Newman,et al.  Predicting epidemics on directed contact networks. , 2006, Journal of theoretical biology.

[74]  H. Hethcote.  Qualitative analyses of communicable disease models , 1976 .

[75]  B. Song,et al.  The Global Stability Analysis for an SIS Model with Age and Infection Age Structures , 2002 .

[76]  A. Perelson,et al.  Dynamics of HIV infection of CD4+ T cells. , 1993, Mathematical biosciences.

[77]  K. Cooke,et al.  The effect of integral conditions in certain equations modelling epidemics and population growth , 1980, Journal of mathematical biology.

[78]  H. Hethcote,et al.  Disease transmission models with density-dependent demographics , 1992, Journal of mathematical biology.

[79]  Horst R. Thieme,et al.  Asymptotic estimates of the solutions of nonlinear integral equations and asymptotic speeds for the spread of populations. , 1979 .

[80]  H. Hethcote.  PERIODICITY AND STABILITY IN EPIDEMIC MODELS: A SURVEY , 1981 .

[81]  J. Yorke,et al.  Gonorrhea Transmission Dynamics and Control , 1984 .

[82]  Julien Arino,et al.  Metapopulations epidemic models. A survey , 2006 .

[83]  H. Hethcote.  Three Basic Epidemiological Models , 1989 .

[84]  Horst R. Thieme,et al.  Stability Change of the Endemic Equilibrium in Age-Structured Models for the Spread of S—I—R Type Infectious Diseases , 1991 .

[85]  Fred Brauer,et al.  Backward bifurcations in simple vaccination models , 2004 .

[86]  P S Schenck,et al.  CONTROL OF COMMUNICABLE DISEASES. , 1914, American journal of public health.

[87]  P. Holgate,et al.  Branching Processes with Biological Applications , 1977 .

[88]  Klaus Dietz,et al.  Epidemics and Rumours: A Survey , 1967 .

[89]  D. Mollison.  Epidemic models : their structure and relation to data , 1996 .

[90]  R. May,et al.  Population biology of infectious diseases: Part II , 1979, Nature.

[91]  Denise E. Kirschner,et al.  Using Mathematics to Understand HIV Immune Dynamics , 1997 .

[92]  M. Keeling,et al.  Networks and epidemic models , 2005, Journal of The Royal Society Interface.

[93]  Daryl J. Daley,et al.  Epidemic Modelling: An Introduction , 1999 .

[94]  Johannes Müller,et al.  Optimal Vaccination Patterns in Age-Structured Populations , 1998, SIAM J. Appl. Math..

[95]  F. C. Hoppensteadt.  Mathematical theories of populations : demographics, genetics and epidemics , 1975 .

[96]  A. Dobson,et al.  Ecology of Infectious Diseases in Natural Populations , 1996 .

[97]  Xinghuo Pang,et al.  Evaluation of control measures implemented in the severe acute respiratory syndrome outbreak in Beijing, 2003. , 2003, JAMA.

[98]  K. Dietz,et al.  The Incidence of Infectious Diseases under the Influence of Seasonal Fluctuations , 1976 .

[99]  R. May,et al.  Population Biology of Infectious Diseases , 1982, Dahlem Workshop Reports.

[100]  Andrew P. Dobson,et al.  Ecology of Infectious Diseases in Natural Populations: Frontmatter , 1995 .

[101]  J. Robins,et al.  Transmission Dynamics and Control of Severe Acute Respiratory Syndrome , 2003, Science.

[102]  P. Waltman.  Deterministic Threshold Models in the Theory of Epidemics , 1974, Lecture Notes in Biomathematics.

[103]  Carlos Castillo-Chavez,et al.  Stability and bifurcation for a multiple-group model for the dynamics of HIV/AIDS transmission , 1992 .

[104]  J. Arino,et al.  A multi-city epidemic model , 2003 .

[105]  A L Lloyd,et al.  Realistic distributions of infectious periods in epidemic models: changing patterns of persistence and dynamics. , 2001, Theoretical population biology.

[106]  O. Diekmann,et al.  On the definition and the computation of the basic reproduction ratio R0 in models for infectious diseases in heterogeneous populations , 1990, Journal of mathematical biology.

[107]  James Watmough,et al.  A simple SIS epidemic model with a backward bifurcation , 2000, Journal of mathematical biology.

[108]  Frank Ball,et al.  Epidemics Among a Population of Households , 2002 .

[109]  H. Hethcote,et al.  An age-structured model for pertussis transmission. , 1997, Mathematical biosciences.

[110]  E. Bickerstaff.  CONTROL OF COMMUNICABLE DISEASES IN MAN , 1971 .

[111]  Mark E. J. Newman,et al.  The Structure and Function of Complex Networks , 2003, SIAM Rev..

[112]  Stephen P. Ellner,et al.  Detecting nonlinearity and chaos in epidemic data , 1993 .

[113]  D Greenhalgh,et al.  Vaccination campaigns for common childhood diseases. , 1990, Mathematical biosciences.

[114]  S. Busenberg,et al.  Analysis of a disease transmission model in a population with varying size , 1990, Journal of mathematical biology.

[115]  G. Chowell,et al.  SARS outbreaks in Ontario, Hong Kong and Singapore: the role of diagnosis and isolation as a control mechanism , 2003, Journal of Theoretical Biology.

[116]  Odo Diekmann,et al.  The velocity of spatial population expansion , 1990 .

[117]  Pejman Rohani,et al.  Appropriate Models for the Management of Infectious Diseases , 2005, PLoS medicine.

[118]  A. J. Hall.  Infectious diseases of humans: R. M. Anderson & R. M. May. Oxford etc.: Oxford University Press, 1991. viii + 757 pp. Price £50. ISBN 0-19-854599-1 , 1992 .

[119]  Horst R. Thieme,et al.  Endemic Models with Arbitrarily Distributed Periods of Infection I: Fundamental Properties of the Model , 2000, SIAM J. Appl. Math..

[120]  Horst R. Thieme,et al.  Demographic change and persistence of HIV/AIDS in heterogeneous population , 1991 .

[121]  K. Hadeler,et al.  A core group model for disease transmission. , 1995, Mathematical biosciences.

[122]  A. Nizam,et al.  Containing Pandemic Influenza at the Source , 2005, Science.

[123]  Roy M. Anderson,et al.  Population dynamics of fox rabies in Europe , 1981, Nature.

[124]  I. Nåsell.  Hybrid Models of Tropical Infections , 1985, Lecture Notes in Biomathematics.

[125]  W. O. Kermack,et al.  A contribution to the mathematical theory of epidemics , 1927 .

[126]  K. Dietz,et al.  A structured epidemic model incorporating geographic mobility among regions. , 1995, Mathematical biosciences.

[127]  Frank Hoppenstaedt.  Mathematical Theories of Populations: Demographics, Genetics and Epidemics , 1975 .

[128]  S. Cornell,et al.  Dynamics of the 2001 UK Foot and Mouth Epidemic: Stochastic Dispersal in a Heterogeneous Landscape , 2001, Science.

[129]  N. Ling.  The Mathematical Theory of Infectious Diseases and its applications , 1978 .

[130]  W. Mcneill.  Plagues and Peoples , 1977, The Review of Politics.

[131]  Christl A. Donnelly,et al.  The Foot-and-Mouth Epidemic in Great Britain: Pattern of Spread and Impact of Interventions , 2001, Science.

[132]  Haiyun Zhao,et al.  Epidemiological Models with Non-Exponentially Distributed Disease Stages and Applications to Disease Control , 2007, Bulletin of mathematical biology.

[133]  J. Yorke,et al.  Recurrent outbreaks of measles, chickenpox and mumps. I. Seasonal variation in contact rates. , 1973, American journal of epidemiology.

[134]  R. Ross,et al.  Prevention of malaria. , 2012, BMJ.

[135]  H. Hethcote.  A Thousand and One Epidemic Models , 1994 .

[136]  H. McCallum,et al.  How should pathogen transmission be modelled? , 2001, Trends in ecology & evolution.

[137]  H. Hethcote,et al.  Modeling HIV Transmission and AIDS in the United States , 1992 .

[138]  K. P. Hadeler,et al.  Pair formation models with maturation period , 1993 .

[139]  M. Bartlett,et al.  Stochastic Population Models in Ecology and Epidemiology. , 1962 .

[140]  Geoffrey L. Smith.  Viruses, Plagues, and History , 1998, Nature Medicine.

[141]  R. May,et al.  How Viruses Spread Among Computers and People , 2001, Science.

[142]  A. Perelson,et al.  A Model for the Immune System Response to HIV: AZT Treatment Studies , 1993 .

[143]  L. Wahl,et al.  Perspectives on the basic reproductive ratio , 2005, Journal of The Royal Society Interface.

[144]  I. Nåsell,et al.  On the quasi-stationary distribution of the stochastic logistic epidemic. , 1999, Mathematical biosciences.

[145]  Alexander Grey,et al.  The Mathematical Theory of Infectious Diseases and Its Applications , 1977 .

[146]  Norman T. J. Bailey,et al.  The Mathematical Theory of Infectious Diseases , 1975 .

[147]  Z. Grossman,et al.  Oscillatory phenomena in a model of infectious diseases. , 1980, Theoretical population biology.

[148]  D. Cummings,et al.  Strategies for mitigating an influenza pandemic , 2006, Nature.

[149]  V. Capasso.  Mathematical Structures of Epidemic Systems , 1993, Lecture Notes in Biomathematics.

[150]  L. Meyers.  Contact network epidemiology: Bond percolation applied to infectious disease prediction and control , 2006 .

[151]  Horst R. Thieme,et al.  Endemic Models with Arbitrarily Distributed Periods of Infection II: Fast Disease Dynamics and Permanent Recovery , 2000, SIAM J. Appl. Math..

[152]  L. Rass,et al.  The asymptotic speed of propagation of the deterministic non-reducible n-type epidemic , 1986, Journal of mathematical biology.

[153]  S. Levin,et al.  Periodicity in Epidemiological Models , 1989 .

[154]  Carlos Castillo-Chavez,et al.  Backwards bifurcations and catastrophe in simple models of fatal diseases , 1998, Journal of mathematical biology.

[155]  Thierry Champion,et al.  The ∞-Wasserstein Distance: Local Solutions and Existence of Optimal Transport Maps , 2008, SIAM J. Math. Anal..

[156]  F. Fenner.  Smallpox and its eradication , 1988 .

[157]  K. Hadeler,et al.  Models for Infectious Human Diseases: Optimal vaccination patterns in age-structured populations II: optimal strategies , 1996 .

[158]  J. E. WilkinsJr..  The differential difference equation for epidemics , 1945 .

[159]  G. Webb,et al.  A diffusive age-structured SEIRS epidemic model , 1996 .

[160]  Carlos Castillo-Chavez,et al.  Mathematical approaches for emerging and reemerging infectious diseases , 2002 .

[161]  Herbert W. Hethcote,et al.  NONLINEAR OSCILLATIONS IN EPIDEMIC MODELS , 1981 .

[162]  E B Wilson,et al.  The Epidemic Curve. , 1942, Proceedings of the National Academy of Sciences of the United States of America.