An intuitive formulation for the reproductive number for the spread of diseases in heterogeneous populations.

The thresholds for mathematical epidemiology models specify the critical conditions for an epidemic to grow or die out. The reproductive number can provide significant insight into the transmission dynamics of a disease and can guide strategies to control its spread. We define the mean number of contacts, the mean duration of infection, and the mean transmission probability appropriately for certain epidemiological models, and construct a simplified formulation of the reproductive number as the product of these quantities. When the spread of the epidemic depends strongly upon the heterogeneity of the populations, the epidemiological models must account for this heterogeneity, and the expressions for the reproductive number become correspondingly more complex. We formulate several models with different heterogeneous structures and demonstrate how to define the mean quantities for an explicit expression for the reproductive number. In complex heterogeneous models, it seems necessary to define the reproductive number for each structured subgroup or cohort and then use the average of these reproductive numbers weighted by their heterogeneity to estimate the reproductive number for the total population.

[1]  X. Lin.  Qualitative analysis of an HIV transmission model. , 1991, Mathematical biosciences.

[2]  Herbert W. Hethcote,et al.  Optimal vaccination schedules in a deterministic epidemic model , 1973 .

[3]  Jia Li,et al.  The Diierential Infectivity and Staged Progression Models for the Transmission of Hiv , 1998 .

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

[5]  K Dietz,et al.  The basic reproduction ratio for sexually transmitted diseases: I. Theoretical considerations. , 1991, Mathematical biosciences.

[6]  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 .

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

[8]  Roy M. Anderson,et al.  Vaccination and herd immunity to infectious diseases , 1985, Nature.

[9]  K Dietz,et al.  Some bounds on estimates for reproductive ratios derived from the age-specific force of infection. , 1994, Mathematical biosciences.

[10]  R M May,et al.  A preliminary study of the transmission dynamics of the human immunodeficiency virus (HIV), the causative agent of AIDS. , 1986, IMA journal of mathematics applied in medicine and biology.

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

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

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

[14]  J. Heesterbeek,et al.  The basic reproduction ratio for sexually transmitted diseases. Part 2. Effects of variable HIV infectivity. , 1993, Mathematical biosciences.

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

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

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

[18]  Herbert W. Hethcote,et al.  Epidemiological models for heterogeneous populations: proportionate mixing, parameter estimation, and immunization programs , 1987 .

[19]  James M. Hyman,et al.  Disease transmission models with biased partnership selection , 1997 .

[20]  R. May,et al.  The transmission dynamics of human immunodeficiency virus (HIV). , 1988, Philosophical transactions of the Royal Society of London. Series B, Biological sciences.

[21]  J. Hyman,et al.  Threshold conditions for the spread of the HIV infection in age-structured populations of homosexual men. , 1994, Journal of theoretical biology.

[22]  H. Inaba.  Threshold and stability for an age-structured epidemic model , 1989 .

[23]  Roy M. Anderson,et al.  The Transmission Dynamics of Human Immunodeficiency Virus (HIV) , 1988 .

[24]  J. Botella de Maglia,et al.  [Prevention of malaria]. , 1999, Revista clinica espanola.

[25]  S. Blower,et al.  Uncertainty and sensitivity analysis of the basic reproductive rate. Tuberculosis as an example. , 1997, American journal of epidemiology.

[26]  R. M. May,et al.  Possible demographic consequences of AIDS in developing countries , 1988, Nature.

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

[28]  Roy M. Anderson,et al.  Transmission dynamics of HIV infection , 1987, Nature.

[29]  L. Dean,et al.  Effect of sexual behavior change on long-term human immunodeficiency virus prevalence among homosexual men. , 1994, American journal of epidemiology.

[30]  H. Inaba,et al.  Threshold and stability results for an age-structured epidemic model , 1990, Journal of mathematical biology.

[31]  K. Dietz,et al.  Models for Vector-Borne Parasitic Diseases , 1980 .

[32]  Klaus Dietz,et al.  The concept of RO in epidemic theory , 1996 .

[33]  G. G. Stokes.  "J." , 1890, The New Yale Book of Quotations.

[34]  S. Blower,et al.  Prophylactic vaccines, risk behavior change, and the probability of eradicating HIV in San Francisco. , 1994, Science.

[35]  G. Macdonald,et al.  The analysis of equilibrium in malaria. , 1952, Tropical diseases bulletin.

[36]  Lisa Sattenspiel,et al.  Modeling and analyzing HIV transmission: the effect of contact patterns , 1988 .

[37]  R. May,et al.  Infectious Diseases of Humans: Dynamics and Control , 1991, Annals of Internal Medicine.

[38]  K Dietz,et al.  The effect of household distribution on transmission and control of highly infectious diseases. , 1995, Mathematical biosciences.

[39]  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.

[40]  John A. Jacquez,et al.  Reproduction numbers and the stability of equilibria of SI models for heterogeneous populations , 1992 .

[41]  M Kremer,et al.  The effect of changing sexual activity on HIV prevalence. , 1998, Mathematical biosciences.