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## The mathematics of infectious diseases (2000)

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Venue: | SIAM Review |

Citations: | 464 - 4 self |

### Citations

384 |
Ordinary Differential Equations
- Hale
- 1969
(Show Context)
Citation Context ...s positively invariant, because no solution paths leave through any boundary. The right sides of (3.1) are smooth, so that initial value problems have unique solutions that exist on maximal intervals =-=[92]-=-. Since paths cannot leave D, solutions exist for all positive time. Thus the model is mathematically and epidemiologically well posed.s620 HERBERT W. HETHCOTE 3.2. Equilibria and Thresholds. The basi... |

343 |
On the definition and the computation of the basic reproduction ratio R0 in models for infectious diseases in heterogeneous populations
- Diekmann, Heesterbeek, et al.
- 1990
(Show Context)
Citation Context ... as the average number of secondary infections that occur when one infective is introduced into a completely susceptible host population [61]. Note that R0 is also called the basic reproduction ratio =-=[58]-=- or basic reproductive rate [12]. It is implicitly assumed that the infected outsider is in the host population for the entire infectious period and mixes with the host population in exactly the same ... |

181 |
Epidemic Modeling: An Introduction
- Daley, Gani
- 2005
(Show Context)
Citation Context ...whooping cough, diphtheria, smallpox, malaria, onchocerciasis, filariasis, rabies, gonorrhea, herpes, syphilis, and HIV/AIDS. The breadth of the subject is shown in the books on epidemiology modeling =-=[5, 9, 12, 18, 19, 20, 22, 33, 38, 39, 55, 56, 59, 80, 81, 90, 111, 113, 127, 137, 141, 151, 164, 167, 173, 181, 194, 196]-=-. Compartments with labels such as M, S, E, I, and R are often used for the epidemiological classes as shown in Figure 1. If a mother has been infected, then some IgG antibodies are transferred across... |

161 |
Population biology of infectious diseases
- Anderson, May
- 1979
(Show Context)
Citation Context ...44]. Models with a variable total population size are often more difficult to analyze mathematically because the population size is an additional variable which is governed by a differential equation =-=[7, 8, 29, 30, 35, 37, 83, 88, 153, 159, 171, 201]-=-. Some models of HIV/AIDS with varying population size have been considered [13, 39, 118, 132, 146]. Before looking at MSEIR models with age structures, we first consider an MSEIR model in a populatio... |

116 |
for Disease Control and Prevention: Summary of notifiable diseases
- Centers
(Show Context)
Citation Context ...S. The yearly incidences of CRS in the United States were between 22 and 67 in the 1970s, between 0 and 50 in the 1980s, 11 in 1990, 47 in 1991, 11 in 1992, and then between 4 and 8 from 1993 to 1999 =-=[43]-=-. Although there have been some increases in CRS cases associated with occasional rubella outbreaks, CRS has been at a relatively low level in the United States in recent years. In recent rubella outb... |

109 |
The Mathematical Theory of Infectious Diseases
- Bailey
- 1975
(Show Context)
Citation Context ...ped differential equation models for malaria as a host-vector disease in 1911 [173]. Other deterministic epidemiology models were then developed in papers by Ross, Ross and Hudson, Martini, and Lotka =-=[18, 60, 66]-=-. Starting in 1926 Kermack and McKendrick published papers on epidemic models and obtained the epidemic threshold result that the density of susceptibles must exceed a critical value in order for an e... |

108 |
for Disease Control and Prevention. Update: influenza activity—United States, September 28, 2008–April 4, 2009, and composition of the 2009–10 influenza vaccine
- Centers
(Show Context)
Citation Context ...whose mutations lead to new variants every few years [23]. For example, the A/Sydney/5/97(H3N2) variant entered the United States in 1998–1999 and was the dominant variant in the 1999–2000 flu season =-=[51]-=-. An infection or vaccination for one variant may give only partial immunity to another variant of the same subtype, so that flu vaccines must be reformulated almost every year. If an influenza virus ... |

99 |
for Diseases Control and Prevention. Progress toward elimination of rubella and congenital rubella syndrome: the Americas, 2003–2008. Morbidity and Mortality Weekly Report 2008; 57:1176–9
- Centers
(Show Context)
Citation Context ...is likely that polio will soon be eradicated worldwide. The WHO estimates that eradicating polio will save about $1.5 billion each year in immunization, treatment, and rehabilitation around the globe =-=[45]-=-. Measles is a serious disease of childhood that can lead to complications and death. For example, measles caused about 7,500 deaths in the United States in 1920 and still causes about 1 million death... |

83 |
for disease control and prevention. In Epidemiology and prevention of vaccine-preventable diseases. 11th edition. Edited by
- Centers
- 2009
(Show Context)
Citation Context ... disease of childhood that can lead to complications and death. For example, measles caused about 7,500 deaths in the United States in 1920 and still causes about 1 million deaths worldwide each year =-=[47, 48]-=-. Measles vaccinations are given to children between 6 and 18 months of age, but the optimal age of vaccination for measles seems to vary geographically [99]. Rubella (also called three-day measles or... |

81 |
The population dynamics of microparasites and their invertebrate hosts
- Anderson, May
- 1981
(Show Context)
Citation Context ...44]. Models with a variable total population size are often more difficult to analyze mathematically because the population size is an additional variable which is governed by a differential equation =-=[7, 8, 29, 30, 35, 37, 83, 88, 153, 159, 171, 201]-=-. Some models of HIV/AIDS with varying population size have been considered [13, 39, 118, 132, 146]. Before looking at MSEIR models with age structures, we first consider an MSEIR model in a populatio... |

63 |
The dynamics of cocirculating influenza strains conferring partial crossimmunity
- Andreasen, Lin, et al.
- 1997
(Show Context)
Citation Context ...he heterogeneous population [1, 103]. Many models with heterogeneity in the form of competing strains of infectious agents have been considered for diseases such as influenza, dengue, and myxomatosis =-=[17, 40, 41, 42, 63, 70, 73, 74, 76, 155, 160]-=-. HIV/AIDS is spread in a very heterogeneous population by heterosexual intercourse, homosexual intercourse, and sharing of needles by injecting drug users. Because of the great diversity and heteroge... |

58 |
A competitive exclusion principle for pathogen virulence
- Bremermann, Thieme
- 1989
(Show Context)
Citation Context ...44]. Models with a variable total population size are often more difficult to analyze mathematically because the population size is an additional variable which is governed by a differential equation =-=[7, 8, 29, 30, 35, 37, 83, 88, 153, 159, 171, 201]-=-. Some models of HIV/AIDS with varying population size have been considered [13, 39, 118, 132, 146]. Before looking at MSEIR models with age structures, we first consider an MSEIR model in a populatio... |

57 |
for Disease Control and Prevention. Ten great public health achievements: United States
- Centers
- 1900
(Show Context)
Citation Context ... cause significant early mortality in developing countries. Indeed, the longer life spans in developed countries seem to be primarily a result of the decline of mortality due to communicable diseases =-=[44]-=-. Models with a variable total population size are often more difficult to analyze mathematically because the population size is an additional variable which is governed by a differential equation [7,... |

57 | A simple model for complex dynamical transitions in epidemics
- Earn, Rohani, et al.
- 2000
(Show Context)
Citation Context ...c terms. Ferguson, Nokes, and Anderson [79] proposed finely age-stratified models with stochastic fluctuations that can shift the dynamics between biennial and triennial cycle attractors. Earn et al. =-=[71]-=- proposed a simple, time-forced SEIR model with slow variation in the average rate of recruitment of new susceptibles. In recent years HIV, which leads to AIDS, has emerged as an important new infecti... |

55 | Stochastic spatial models
- Durrett
- 1999
(Show Context)
Citation Context ... there is also a threshold condition, sosTHE MATHEMATICS OF INFECTIOUS DISEASES 645 that the disease dies out below the threshold and approaches an endemic stationary distribution above the threshold =-=[69]-=-. 10. Discussion. Mathematical epidemiology has now evolved into a separate area of population dynamics that is parallel to mathematical ecology. Epidemiology models are now used to combine complex da... |

53 |
Analysis of Infectious Disease Data
- Becker
- 1989
(Show Context)
Citation Context ...whooping cough, diphtheria, smallpox, malaria, onchocerciasis, filariasis, rabies, gonorrhea, herpes, syphilis, and HIV/AIDS. The breadth of the subject is shown in the books on epidemiology modeling =-=[5, 9, 12, 18, 19, 20, 22, 33, 38, 39, 55, 56, 59, 80, 81, 90, 111, 113, 127, 137, 141, 151, 164, 167, 173, 181, 194, 196]-=-. Compartments with labels such as M, S, E, I, and R are often used for the epidemiological classes as shown in Figure 1. If a mother has been infected, then some IgG antibodies are transferred across... |

52 |
Stochastic Population Models in Ecology and Epidemiology
- Bartlett
- 1960
(Show Context)
Citation Context ...whooping cough, diphtheria, smallpox, malaria, onchocerciasis, filariasis, rabies, gonorrhea, herpes, syphilis, and HIV/AIDS. The breadth of the subject is shown in the books on epidemiology modeling =-=[5, 9, 12, 18, 19, 20, 22, 33, 38, 39, 55, 56, 59, 80, 81, 90, 111, 113, 127, 137, 141, 151, 164, 167, 173, 181, 194, 196]-=-. Compartments with labels such as M, S, E, I, and R are often used for the epidemiological classes as shown in Figure 1. If a mother has been infected, then some IgG antibodies are transferred across... |

52 | Epidemiological models with age structure, proportionate mixing, and cross-immunity - Castillo-Chavez, Hethcote, et al. - 1989 |

52 |
for Disease Control and Prevention. Recommended community strategies and measurements to prevent obesity
- Centers
(Show Context)
Citation Context ...minate the few cases of vaccine-related paralytic polio each year, the United States now recommends the Salk injectable vaccine for the first four polio vaccinations, even though it is more expensive =-=[50]-=-. In the Americas, the last case of paralytic polio caused by the wild virus was in Peru in 1991. In 1988 the WHO set a goal of global polio eradication by the year 2000 [178]. Most countries are usin... |

49 |
Essai d’une nouvelle analyse de la mortalite causee par la petite verole
- Bernoulli
- 1982
(Show Context)
Citation Context ...and for the comparison of vaccination strategies with age-specific risk groups and age-dependent vaccination rates. Indeed, some of the early epidemiology models incorporated continuous age structure =-=[24, 136]-=-. Modern mathematical analysis of age-structured models appears to have started with Hoppensteadt [114], who formulated epidemiology models with both continuous chronological age and infection class a... |

47 | A preliminary study of the transmission dynamics of the human immunodeficiency virus (HIV), the causative agent of AIDS - Anderson, Medley, et al. - 1986 |

46 |
Transmission and control of arbovirus disease
- Dietz
- 1975
(Show Context)
Citation Context ...e basic reproduction number R0, which is defined as the average number of secondary infections produced when one infected individual is introduced into a host population where everyone is susceptible =-=[61]-=-. For many deterministic epidemiology models, an infection can get started in a fully susceptible population if and only if R0 > 1. Thus the basic reproduction number R0 is often considered as the thr... |

46 |
Qualitative analyses of communicable disease models
- Hethcote
- 1976
(Show Context)
Citation Context ... the number of new cases per unit time due to the S = Ns susceptibles. This form of the horizontal incidence is called the standard incidence, because it is formulated from the basic principles above =-=[96, 102]-=-. The simple mass action law ηIS = η(Ni)(Ns), with η as a mass action coefficient, has sometimes been used for the horizontal incidence. The parameter η has no direct epidemiological interpretation, b... |

45 |
The incidence of infectious disease under the influence of seasonal fluctuations
- Dietz
(Show Context)
Citation Context ...yses including the following seven modeling explanations, some of which involve age structure. Yorke and London [200] proposed SEIR models with seasonal forcing in delay differential equations. Dietz =-=[62]-=- proposed subharmonic resonance in a seasonally forced SEIR model using ordinary differential equations. Schenzle [177] used computer simulations to show that the measles outbreak patterns in England ... |

37 |
Mathematical Structures of Epidemics Systems
- Capasso
- 2008
(Show Context)
Citation Context |

37 |
Velasco-Hernández, “Competitive exclusion in a vector-host model for the dengue fever
- Feng, X
- 1997
(Show Context)
Citation Context ...he heterogeneous population [1, 103]. Many models with heterogeneity in the form of competing strains of infectious agents have been considered for diseases such as influenza, dengue, and myxomatosis =-=[17, 40, 41, 42, 63, 70, 73, 74, 76, 155, 160]-=-. HIV/AIDS is spread in a very heterogeneous population by heterosexual intercourse, homosexual intercourse, and sharing of needles by injecting drug users. Because of the great diversity and heteroge... |

34 |
Analysis of a dengue disease transmission model
- Esteva, Vargas
- 1998
(Show Context)
Citation Context ...he heterogeneous population [1, 103]. Many models with heterogeneity in the form of competing strains of infectious agents have been considered for diseases such as influenza, dengue, and myxomatosis =-=[17, 40, 41, 42, 63, 70, 73, 74, 76, 155, 160]-=-. HIV/AIDS is spread in a very heterogeneous population by heterosexual intercourse, homosexual intercourse, and sharing of needles by injecting drug users. Because of the great diversity and heteroge... |

33 |
A simulation model of the population dynamics and evolution of myxomatosis
- Dwyer, Levin, et al.
- 1990
(Show Context)
Citation Context |

32 |
Chaos and biological complexity in measles dynamics
- Bolker, Grenfell
- 1993
(Show Context)
Citation Context ...gland and Germany could be explained by the primary school yearly calenders and entry ages. Olson and Schaffer [169] proposed chaotic behavior in simple deterministic SEIR models. Bolker and Grenfell =-=[27]-=- proposed realistic age-structured models with seasonal forcing and stochastic terms. Ferguson, Nokes, and Anderson [79] proposed finely age-stratified models with stochastic fluctuations that can shi... |

29 |
A general solution of the problem of mixing sub-populations, and its application to risk-and age-structured epidemic models for the spread
- Busenberg, Castilla-Chavez
- 1991
(Show Context)
Citation Context ...els. It also contains papers on AIDS models with HIV class age, variable infectivity, distributions for the AIDS incubation period, heterogeneity, and structured mixing. Busenberg and Castillo-Chavez =-=[32]-=- found an R0 expression for an HIV model with variable infectivity and continuous chronological and HIV class age structure and proportionate mixing. Hyman, Li, and Stanley [120] generalized these res... |

27 |
Atlas of disease distribution: analytical approaches to epidemiological data
- Cliff, Haggett
- 1988
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27 |
Periodicity in epidemiological models
- Hethcote, Levin
- 1989
(Show Context)
Citation Context ...f Bailey’s book [18] is an important landmark), so that a tremendous variety of models have now been formulated, mathematically analyzed, and applied to infectious diseases. Reviews of the literature =-=[21, 39, 60, 65, 67, 102, 107, 109, 199]-=- show the rapid growth of epidemiology modeling. The recent models have involved aspects such as passive immunity, gradual loss of vaccine and disease-acquired immunity, stages of infection, vertical ... |

26 |
The Biomarhemarics of Malaria
- Bailey
- 1982
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25 |
The Coming Plague
- Garrett
- 1994
(Show Context)
Citation Context ...opathy (BSE, “mad cow disease”), Creutzfeldt-Jakob disease (CJD), kuru, and scrapie in sheep [168]. Recent popular books have given us exciting accounts of the emergence and detection of new diseases =-=[82, 168, 170, 183]-=-. It is clear that human or animal invasions ∗Received by the editors March 6, 2000; accepted for publication (in revised form) May 7, 2000; published electronically October 30, 2000. http://www.siam.... |

25 |
Three basic epidemiological models
- Hethcote
- 1986
(Show Context)
Citation Context ...can contribute to the design and analysis of epidemiological surveys, suggest crucial data that should be collected, identify trends, make general forecasts, and estimate the uncertainty in forecasts =-=[100, 111]-=-. Although a model for smallpox was formulated and solved by Daniel Bernoulli in 1760 in order to evaluate the effectiveness of variolation of healthy people with the smallpox virus [24], deterministi... |

23 |
den Driessche, Analysis of a disease transmission model in a population with varying size
- Busenberg, van
- 1990
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Citation Context |

23 |
A model for dengue disease with variable human population
- Esteva, Vargas
- 1999
(Show Context)
Citation Context |

22 |
Schenzle D. Proportionate mixing models for age-dependent infection transmission
- Dietz
- 1985
(Show Context)
Citation Context ...en by (5.11) R0 = � ∞ 0 � ã ˜ b(ã)ρe −D(ã)−qã−γã 0 b(y)e γy−k � y 0 b(α)dα dydã � ã 0 b(y)e γy dydã. This expression is similar to previous R0 expressions for SIR models with constant population size =-=[40, 68]-=-. The expression (5.11) for R0 can also be used for SIRS and SIS models, but the equations for the positive k when R0 > 1 would be different. Proofs of stability and persistence for the models in this... |

22 |
A Thousand and One Epidemic Models
- Hethcote
- 1994
(Show Context)
Citation Context ...f Bailey’s book [18] is an important landmark), so that a tremendous variety of models have now been formulated, mathematically analyzed, and applied to infectious diseases. Reviews of the literature =-=[21, 39, 60, 65, 67, 102, 107, 109, 199]-=- show the rapid growth of epidemiology modeling. The recent models have involved aspects such as passive immunity, gradual loss of vaccine and disease-acquired immunity, stages of infection, vertical ... |

20 |
Ordinary differential equations, Walter de Gruyter
- Amann
- 1990
(Show Context)
Citation Context ...the only positively invariant subset of the set with ˙V = 0. If there is a finite maximum age (so that all forward paths have compact closure), then either Corollary 2.3 in [162] or Corollary 18.5 in =-=[4]-=- (Liapunov–Lasalle theorems for semiflows) implies that all paths in the feasible region approach the disease-free steady state. If R0 > 1, then we have ˙ V > 0 for points sufficiently close to the di... |

20 |
Competitive exclusion in gonorrhea models and other sexually-transmitted diseases
- Castilla-Chavez, Huang, et al.
- 1993
(Show Context)
Citation Context |

19 |
The role of mathematical models in the study of HIV transmission and the epidemiology of AIDS
- Anderson
- 1988
(Show Context)
Citation Context ..., homosexual intercourse, and sharing of needles by injecting drug users. Because of the great diversity and heterogeneity among those at risk of HIV/AIDS, modeling this disease is a challenging task =-=[6, 12, 39, 104, 115, 118, 119, 126, 130]-=-. For HIV/AIDS models with a continuous distribution of sexual activity levels and with various preference mixing functions, the proportionate mixing has been shown to be the only separable solution, ... |

19 |
for Disease Control and Prevention. Global measles mortality, 2000–2008. MMWR Morb Mortal Wkly Rep 2009
- Centers
(Show Context)
Citation Context ... disease of childhood that can lead to complications and death. For example, measles caused about 7,500 deaths in the United States in 1920 and still causes about 1 million deaths worldwide each year =-=[47, 48]-=-. Measles vaccinations are given to children between 6 and 18 months of age, but the optimal age of vaccination for measles seems to vary geographically [99]. Rubella (also called three-day measles or... |

19 |
An age-structured model for pertussis transmission
- Hethcote
- 1997
(Show Context)
Citation Context ...with Age Groups. This demographic model with age groups has been developed from the initial boundary value problem in the previous section for use in age-structured epidemiologic models for pertussis =-=[105]-=-. It consists of a system of n ordinary differential equations for the sizes of the n age groups defined by the age intervals [ai−1,ai], where 0 = a0 <a1 <a2 < ··· <an−1 <an = ∞. A maximum age is not ... |

19 |
den Driessche, Some epidemiological models with delays, Differential Equations and Applications to Industry
- van
- 1994
(Show Context)
Citation Context ...rns is infected vertically [33]. Models with population size–dependent contact functions have also been considered [29, 171, 190, 191, 201]. Various forms of nonlinear incidences have been considered =-=[112, 147, 148, 149]-=-. See [107] for a survey of mechanisms including nonlinear incidences that can lead to periodicity in epidemiological models. A common assumption is that the movements out of the M, E, and I compartme... |

17 |
Disease transmission models with density-dependent demographics
- Gao, Hethecote
- 1992
(Show Context)
Citation Context ... in a herd [57], because disease transmission primarily occurs locally from nearby animals. For more information about the differences in models using these two forms of the horizontal incidence, see =-=[83, 84, 85, 96, 110, 159]-=-. Vertical incidence, which is the infection rate of newborns by their mothers, is sometimes included in epidemiology models by assuming that a fixed fraction of the newborns is infected vertically [3... |

17 |
den Driessche, Periodicity and stability in epidemic models: A survey
- Hethcote, Stech, et al.
- 1981
(Show Context)
Citation Context ...f Bailey’s book [18] is an important landmark), so that a tremendous variety of models have now been formulated, mathematically analyzed, and applied to infectious diseases. Reviews of the literature =-=[21, 39, 60, 65, 67, 102, 107, 109, 199]-=- show the rapid growth of epidemiology modeling. The recent models have involved aspects such as passive immunity, gradual loss of vaccine and disease-acquired immunity, stages of infection, vertical ... |

16 |
Epidemiologic interference of virus populations
- Dietz
- 1979
(Show Context)
Citation Context |

15 |
for Disease Control and Prevention. Measles outbreaks. http://www.cdc.gov/ measles/outbreaks.html
- Centers
- 2009
(Show Context)
Citation Context ... Most of the unvaccinated cases were people belonging to a religious denomination that routinely does not accept vaccination. The 2,961 measles cases included 3 measles-related deaths. Reprinted from =-=[52]-=-.sTHE MATHEMATICS OF INFECTIOUS DISEASES 607 contacts of a typical infective during the infectious period. Here the replacement number at time zero is σso, which is the product of the contact number σ... |

14 |
Modelling heterogeneous mixing in infectious diseases dynamics," Models for infectious human diseases Their structure and relation to data
- Hethcote
- 1996
(Show Context)
Citation Context ...9]. One example of separable mixing is proportionate mixing, in which the contacts of a person of age a are distributed over those of other ages in proportion to the activity levels of the other ages =-=[103, 174]-=-. If l(a) is the average number of people contacted by a person of age a per unit time, u(a) is the steady state age distribution for the population, and D = � ∞ 0 l(a)u(a)da is the total number of co... |

13 |
Models for the spread of universally fatal diseases
- Brauer
- 1990
(Show Context)
Citation Context ...etimes included in epidemiology models by assuming that a fixed fraction of the newborns is infected vertically [33]. Models with population size–dependent contact functions have also been considered =-=[29, 171, 190, 191, 201]-=-. Various forms of nonlinear incidences have been considered [112, 147, 148, 149]. See [107] for a survey of mechanisms including nonlinear incidences that can lead to periodicity in epidemiological m... |

13 |
Endemic thresholds and stability in a class of age-structured epidemics
- Busenberg, Cooke, et al.
- 1988
(Show Context)
Citation Context ...analyzed for many epidemiology models with age structure; more references are cited in the following papers. These SIS and SIR models with continuous age structure have included vertical transmission =-=[33, 34, 72]-=-, age-dependent disease transmission [14, 61, 91, 189], infection class age [186, 197], cross immunity [40], intercohort transmission [35, 36, 53, 124, 125], short infectious period [15, 16], and opti... |

13 |
Epidemics and rumours: a survey
- Dietz
- 1967
(Show Context)
Citation Context ...ped differential equation models for malaria as a host-vector disease in 1911 [173]. Other deterministic epidemiology models were then developed in papers by Ross, Ross and Hudson, Martini, and Lotka =-=[18, 60, 66]-=-. Starting in 1926 Kermack and McKendrick published papers on epidemic models and obtained the epidemic threshold result that the density of susceptibles must exceed a critical value in order for an e... |

12 |
Vaccination against rubella and measles: quantitative investigations of different policies
- Anderson, May
- 1983
(Show Context)
Citation Context ... and optimal vaccination patterns [86, 87, 94, 135, 165, 175]. Age-structured models have been used in the epidemiology modeling of many diseases [12]. Dietz [61, 64], Hethcote [98], Anderson and May =-=[10, 11]-=-, and Rouderfer, Becker, and Hethcote [174] used continuous age-structured models for the evaluation of measles and rubella vaccination strategies. Tudor [192] found threshold conditionssTHE MATHEMATI... |

12 |
A model of the spread of HIV infection and the demographic impact of AIDS. Stat Med
- Bongaarts
- 1989
(Show Context)
Citation Context ...DS, has emerged as an important new infectious disease. Many age-structured models have been developed for HIV/AIDS. May and Anderson [154] found R0 for some simple HIV transmission models. Bongaarts =-=[28]-=- and May, Anderson, and McLean [156] used models with age structure to examine the demographic effects of AIDS in African countries. The book [39] by Castillo-Chavez contains a review of HIV/AIDS mode... |

11 |
Vertically transmitted diseases, Biomathematics Volume 23
- Busenberg, Cooke
- 1993
(Show Context)
Citation Context |

10 |
The Use of Epidemic Models
- Becker
- 1978
(Show Context)
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10 |
Mathematical Models for Infectious Disease Statistics
- Dietz, Schenzle
- 1985
(Show Context)
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10 |
Threshold and stability results for an epidemic models with an age-structured meeting rate
- Greenhalgh
- 1988
(Show Context)
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9 |
Existence and uniqueness of endemic states for the age-structured S-I-R epidemic model
- Cha, Iannelli, et al.
- 1998
(Show Context)
Citation Context ...ix with their own age group [103]. For more general mixing, the endemic steady state might not be unique, but some conditions that guarantee existence, uniqueness, and local stability have been given =-=[53, 125]-=-. Because the basic reproduction number for the MSEIR model does not depend on δ or on whether recovered people have no, temporary, or permanent immunity, the expression (5.9) for R0 also works for th... |

9 |
G.P(1996).Mass vaccination to control chickenpox: The influence of zoster(population dynamics/infectious diseases/mathematical modeling/varicella/immunization).Medicinal sciences.93
- Furguson, Anderson
(Show Context)
Citation Context ...NFECTIOUS DISEASES 643 for a measles model with age groups. Hethcote [99] considered optimal ages of vaccination for measles on three continents. Halloran et al. [93], Ferguson, Anderson, and Garnett =-=[78]-=-, and Schuette and Hethcote [179] used age-structured models to study the effects of varicella (chickenpox) vaccination programs. Grenfell and Anderson [89] and Hethcote [105, 106] have used age-struc... |

9 |
Dynamical Complexity in Age-Structured Models of the Transmission of the Measles Virus: Epidemiological Implications at High Levels of Vaccine Uptake
- Ferguson, Nokes, et al.
- 1996
(Show Context)
Citation Context ...posed chaotic behavior in simple deterministic SEIR models. Bolker and Grenfell [27] proposed realistic age-structured models with seasonal forcing and stochastic terms. Ferguson, Nokes, and Anderson =-=[79]-=- proposed finely age-stratified models with stochastic fluctuations that can shift the dynamics between biennial and triennial cycle attractors. Earn et al. [71] proposed a simple, time-forced SEIR mo... |

8 |
for Disease Control and Prevention. Update: Raccoon Rabies Epizootic
- Centers
- 1999
(Show Context)
Citation Context ...me estimated speeds of propagation are 30–60 kilometers per year for fox rabies in Europe starting in 1939 [166], 18–24 miles per year for raccoon rabies in the Eastern United States starting in 1977 =-=[49]-=-, about 140 miles per year for the plague in Europe in 1347–1350 [166], and worldwide in one year for influenza in the 20th century [176]. Epidemiology models with spatial structures have been used to... |

8 |
Analytical threshold and stability results on age-structured epidemic models with vaccination
- Greenhalgh
- 1988
(Show Context)
Citation Context ...ransmission [14, 61, 91, 189], infection class age [186, 197], cross immunity [40], intercohort transmission [35, 36, 53, 124, 125], short infectious period [15, 16], and optimal vaccination patterns =-=[86, 87, 94, 135, 165, 175]-=-. Age-structured models have been used in the epidemiology modeling of many diseases [12]. Dietz [61, 64], Hethcote [98], Anderson and May [10, 11], and Rouderfer, Becker, and Hethcote [174] used cont... |

8 |
Vaccination campaigns for common childhood diseases
- Greenhalgh
- 1990
(Show Context)
Citation Context ...ransmission [14, 61, 91, 189], infection class age [186, 197], cross immunity [40], intercohort transmission [35, 36, 53, 124, 125], short infectious period [15, 16], and optimal vaccination patterns =-=[86, 87, 94, 135, 165, 175]-=-. Age-structured models have been used in the epidemiology modeling of many diseases [12]. Dietz [61, 64], Hethcote [98], Anderson and May [10, 11], and Rouderfer, Becker, and Hethcote [174] used cont... |

7 |
The Effects of Averaging on the Basic Reproductive Ratio
- Adler
- 1992
(Show Context)
Citation Context ...t has been shown that estimates of R0, under the false assumption that a heterogeneously mixing population is homogeneously mixing, are not greater than the actual R0 for the heterogeneous population =-=[1, 103]-=-. Many models with heterogeneity in the form of competing strains of infectious agents have been considered for diseases such as influenza, dengue, and myxomatosis [17, 40, 41, 42, 63, 70, 73, 74, 76,... |

7 |
Comparison of deterministic and stochastic
- Allen, Burgin
(Show Context)
Citation Context ...the books and survey papers listed in the introduction. We refer the reader to other sources for information on stochastic epidemiology models [18, 20, 56, 59, 66, 81, 128, 167], discrete time models =-=[2, 3]-=-, models involving macroparasites [12, 59, 90], genetic heterogeneity [12, 90], plant disease models [137, 194], and wildlife disease models [90]. Age-structured epidemiology models with either contin... |

7 |
Disease regulation of age-structured host populations, Theor
- Andreasen
- 1989
(Show Context)
Citation Context ...ructure; more references are cited in the following papers. These SIS and SIR models with continuous age structure have included vertical transmission [33, 34, 72], age-dependent disease transmission =-=[14, 61, 91, 189]-=-, infection class age [186, 197], cross immunity [40], intercohort transmission [35, 36, 53, 124, 125], short infectious period [15, 16], and optimal vaccination patterns [86, 87, 94, 135, 165, 175]. ... |

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The effect of age-dependent host mortality on the dynamics of an endemic disease
- Andreasen
- 1993
(Show Context)
Citation Context ...ssion [33, 34, 72], age-dependent disease transmission [14, 61, 91, 189], infection class age [186, 197], cross immunity [40], intercohort transmission [35, 36, 53, 124, 125], short infectious period =-=[15, 16]-=-, and optimal vaccination patterns [86, 87, 94, 135, 165, 175]. Age-structured models have been used in the epidemiology modeling of many diseases [12]. Dietz [61, 64], Hethcote [98], Anderson and May... |

7 |
Global behavior of an age-structured epidemic model
- Busenberg, Ianelli, et al.
- 1991
(Show Context)
Citation Context ...ous age structure have included vertical transmission [33, 34, 72], age-dependent disease transmission [14, 61, 91, 189], infection class age [186, 197], cross immunity [40], intercohort transmission =-=[35, 36, 53, 124, 125]-=-, short infectious period [15, 16], and optimal vaccination patterns [86, 87, 94, 135, 165, 175]. Age-structured models have been used in the epidemiology modeling of many diseases [12]. Dietz [61, 64... |

7 |
Pertussis in England and Wales: an investigation of transmission dynamics and control by man vaccination
- Grenfell, Anderson
- 1989
(Show Context)
Citation Context ...t al. [93], Ferguson, Anderson, and Garnett [78], and Schuette and Hethcote [179] used age-structured models to study the effects of varicella (chickenpox) vaccination programs. Grenfell and Anderson =-=[89]-=- and Hethcote [105, 106] have used age-structured models in evaluating pertussis (whooping cough) vaccination programs. Irregular and biennial oscillations of measles incidences have led to various ma... |

7 |
Theoretical epidemiologic and morbidity effects of routine varicella immunization of preschool children in the united states
- Halloran, Cochi, et al.
- 1994
(Show Context)
Citation Context ...eshold conditionssTHE MATHEMATICS OF INFECTIOUS DISEASES 643 for a measles model with age groups. Hethcote [99] considered optimal ages of vaccination for measles on three continents. Halloran et al. =-=[93]-=-, Ferguson, Anderson, and Garnett [78], and Schuette and Hethcote [179] used age-structured models to study the effects of varicella (chickenpox) vaccination programs. Grenfell and Anderson [89] and H... |

7 |
An immunization model for a heterogeneous population, Theoret
- Hethcote
- 1978
(Show Context)
Citation Context ...ã at time t. The division by the total population size � ∞ 0 0 U(a, t)da makes the contact rate λ(a, t) independent of the population size, so the contact number is independent of the population size =-=[57, 97, 102, 159]-=-. One example of separable mixing is proportionate mixing, in which the contacts of a person of age a are distributed over those of other ages in proportion to the activity levels of the other ages [1... |

7 |
Measles and rubella in the United States
- Hethcote
- 1983
(Show Context)
Citation Context ... Because the goal of a rubella vaccination program is to prevent rubella infections in pregnant women, special vaccination strategies such as vaccination of 12 to 14-year-old girls are sometimes used =-=[98, 101]-=-. The estimates above based on R0 values suggest that herd immunity would be achieved if the immune fraction in the population were greatersTHE MATHEMATICS OF INFECTIOUS DISEASES 615 Fig. 9 After a me... |

7 |
Simulations of pertussis epidemiology in the United States: Effects of adult booster vaccinations
- Hethcote
- 1999
(Show Context)
Citation Context ...ng the SIR model formula (6.8) for R0 in the pertussis computer simulation programs with the baseline parameter sets, the values of the basic reproduction number R0 are 5.4 for the pertussis model in =-=[105, 106]-=- and 3.7 for the second pertussis model in [106]. In the first model each pertussis booster moves the individual back up one vaccinated or removed class, but for those in the second model who have had... |

6 |
Some discrete-time SI
- Allen
- 1994
(Show Context)
Citation Context ...the books and survey papers listed in the introduction. We refer the reader to other sources for information on stochastic epidemiology models [18, 20, 56, 59, 66, 81, 128, 167], discrete time models =-=[2, 3]-=-, models involving macroparasites [12, 59, 90], genetic heterogeneity [12, 90], plant disease models [137, 194], and wildlife disease models [90]. Age-structured epidemiology models with either contin... |

6 |
Instability in SIR-model with age-dependent susceptibility
- Andreasen
- 1995
(Show Context)
Citation Context ...tate would usually be stable, this may not be true in unusual cases. For example, the endemic steady state can be unstable in the agestructured SIR model when b(a) is decreasing and ˜b(ã) is constant =-=[16]-=- and when ˜b(ã) is concentrated at a certain age while b(a) is constant [189]. Some types of mixing cannot be written in the separable form b(a) ˜b(ã). For example, in preferred mixing, certain age gr... |

6 |
How does transmission de-pend on population size
- Jong, Diekmann, et al.
- 1995
(Show Context)
Citation Context ... have a similar number of daily contacts). The standard incidence is also a better formulation than the simple mass action law for animal populations such as mice in a mouse-room or animals in a herd =-=[57]-=-, because disease transmission primarily occurs locally from nearby animals. For more information about the differences in models using these two forms of the horizontal incidence, see [83, 84, 85, 96... |

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The evaluation of rubella vaccination strategies
- DIETZ
- 1981
(Show Context)
Citation Context ...4, 125], short infectious period [15, 16], and optimal vaccination patterns [86, 87, 94, 135, 165, 175]. Age-structured models have been used in the epidemiology modeling of many diseases [12]. Dietz =-=[61, 64]-=-, Hethcote [98], Anderson and May [10, 11], and Rouderfer, Becker, and Hethcote [174] used continuous age-structured models for the evaluation of measles and rubella vaccination strategies. Tudor [192... |

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Frauenthal, Mathematical Modeling in Epidemiology
- C
- 1980
(Show Context)
Citation Context |

6 |
Four SEI endemic models with periodicity and separatrices
- Gao, Mena-Lorca, et al.
- 1995
(Show Context)
Citation Context ... in a herd [57], because disease transmission primarily occurs locally from nearby animals. For more information about the differences in models using these two forms of the horizontal incidence, see =-=[83, 84, 85, 96, 110, 159]-=-. Vertical incidence, which is the infection rate of newborns by their mothers, is sometimes included in epidemiology models by assuming that a fixed fraction of the newborns is infected vertically [3... |

5 |
The first epidemic model: historical note on P
- Dietz
- 1988
(Show Context)
Citation Context ...ped differential equation models for malaria as a host-vector disease in 1911 [173]. Other deterministic epidemiology models were then developed in papers by Ross, Ross and Hudson, Martini, and Lotka =-=[18, 60, 66]-=-. Starting in 1926 Kermack and McKendrick published papers on epidemic models and obtained the epidemic threshold result that the density of susceptibles must exceed a critical value in order for an e... |

5 |
Struchiner C. Epidemiologic effects of vaccine with complex direct effects in an agestructured population
- EM, Watelet
(Show Context)
Citation Context ...ransmission [14, 61, 91, 189], infection class age [186, 197], cross immunity [40], intercohort transmission [35, 36, 53, 124, 125], short infectious period [15, 16], and optimal vaccination patterns =-=[86, 87, 94, 135, 165, 175]-=-. Age-structured models have been used in the epidemiology modeling of many diseases [12]. Dietz [61, 64], Hethcote [98], Anderson and May [10, 11], and Rouderfer, Becker, and Hethcote [174] used cont... |

5 |
Epidemic disease in
- Hamer
- 1906
(Show Context)
Citation Context ...rministic epidemiology modeling seems to have started in the 20th century. In 1906 Hamer formulated and analyzed a discrete time model in his attempt to understand the recurrence of measles epidemics =-=[95]-=-. His model may have been the first to assume that the incidence (number of new cases per unit time) depends on the product of the densities of the susceptibles and infectives. Ross was interested in ... |

5 | Optimal age of vaccination for measles - Hethcote - 1988 |

5 | Ark, Modeling HIV Transmission and - Hethcote, Van - 1992 |

4 |
Incorporating spatial components into models of epidemic spread, in: D. Mollison (Ed.), Epidemic Models: Their Structure and Relation to Data
- Cliff
- 1996
(Show Context)
Citation Context ... year for influenza in the 20th century [176]. Epidemiology models with spatial structures have been used to describe spatial heterogeneity [12, 96, 110] and the spatial spread of infectious diseases =-=[38, 54, 59, 90, 166, 193]-=-. There seem to be two types of spatial epidemiology models [163, 193]. Diffusion epidemiology models are formulated from nonspatial models by adding diffusion terms corresponding to the random moveme... |

4 |
Density dependence in parasite t.ransmission dynamics, Parasi.t ology Today [XI Freail
- Dietz
- 1983
(Show Context)
Citation Context |

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Ark, Epidemiological models with heterogeneous populations: Proportionate mixing, parameter estimation and immunization programs
- Hethcote, Van
- 1987
(Show Context)
Citation Context ... in a herd [57], because disease transmission primarily occurs locally from nearby animals. For more information about the differences in models using these two forms of the horizontal incidence, see =-=[83, 84, 85, 96, 110, 159]-=-. Vertical incidence, which is the infection rate of newborns by their mothers, is sometimes included in epidemiology models by assuming that a fixed fraction of the newborns is infected vertically [3... |

3 |
Like with like preference and sexual mixing models
- Blythe, Castillo-Chavez
- 1989
(Show Context)
Citation Context ...ence mixing functions, the proportionate mixing has been shown to be the only separable solution, and expressions for the basic reproduction number R0 in the proportionate mixing case have been found =-=[26, 32]-=-. Expressions for R0 have also been found for HIV/AIDS models using groups of people based on their sexual behavior, e.g., homosexual men, bisexual men, heterosexual women, and heterosexual men, with ... |

3 |
Analysis of nonlinear integro-differential equations arising in agedependent epidemic models
- El-Doma
(Show Context)
Citation Context ...analyzed for many epidemiology models with age structure; more references are cited in the following papers. These SIS and SIR models with continuous age structure have included vertical transmission =-=[33, 34, 72]-=-, age-dependent disease transmission [14, 61, 91, 189], infection class age [186, 197], cross immunity [40], intercohort transmission [35, 36, 53, 124, 125], short infectious period [15, 16], and opti... |

3 |
Variations on a theme of SEI endemic models, Differential Equations and Applications to Biology persistence
- Gao, Mena-Lorca, et al.
- 1996
(Show Context)
Citation Context |

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Modeling AIDS prevention programs in a population of homosexual in modeling the AIDS epidemic
- Hethcote
- 1994
(Show Context)
Citation Context ..., homosexual intercourse, and sharing of needles by injecting drug users. Because of the great diversity and heterogeneity among those at risk of HIV/AIDS, modeling this disease is a challenging task =-=[6, 12, 39, 104, 115, 118, 119, 126, 130]-=-. For HIV/AIDS models with a continuous distribution of sexual activity levels and with various preference mixing functions, the proportionate mixing has been shown to be the only separable solution, ... |

3 |
Hopf bifurcation in models for pertussis epidemiology
- Hethcote, Li
- 1999
(Show Context)
Citation Context ... model and σ =1.8 for the second pertussis model. This phenomenon that σ<R0 at the endemic equilibrium also holds for three relatively simple pertussis models based on ordinary differential equations =-=[108]-=-. For the pertussis model with four removed groups in [108], the three infective classes with decreasing infectivity are I, Im, and Iw, where the infective classes Im and Iw are nonempty as soon as pe... |

2 |
eds., Age related changes in the rate of disease transmission: Implication for the design of vaccination programs
- Anderson, May
- 1985
(Show Context)
Citation Context ... and optimal vaccination patterns [86, 87, 94, 135, 165, 175]. Age-structured models have been used in the epidemiology modeling of many diseases [12]. Dietz [61, 64], Hethcote [98], Anderson and May =-=[10, 11]-=-, and Rouderfer, Becker, and Hethcote [174] used continuous age-structured models for the evaluation of measles and rubella vaccination strategies. Tudor [192] found threshold conditionssTHE MATHEMATI... |

2 |
for Disease Control and Prevention. Rubella outbreak— Westchester County
- Centers
- 1997
(Show Context)
Citation Context ...ella outbreaks in the United States, most cases occurred among unvaccinated persons aged at least 20 years and among persons who were foreign born, primarily Hispanics (63% of reported cases in 1997) =-=[46]-=-. Although it does not solve the problem of unvaccinated immigrants, the rubella vaccination program for children has reduced the incidence of rubella and CRS in the United States to very low levels. ... |

2 |
On a nonlinear integral equation modelling an epidemic in an age-structured population
- Gripenberg
- 1983
(Show Context)
Citation Context ...ructure; more references are cited in the following papers. These SIS and SIR models with continuous age structure have included vertical transmission [33, 34, 72], age-dependent disease transmission =-=[14, 61, 91, 189]-=-, infection class age [186, 197], cross immunity [40], intercohort transmission [35, 36, 53, 124, 125], short infectious period [15, 16], and optimal vaccination patterns [86, 87, 94, 135, 165, 175]. ... |

1 |
et al., Evidence for the persistence of infectious disease agents in isolated human populations
- Black, Hierholzer, et al.
- 1974
(Show Context)
Citation Context ...chickenpox, but they can transmit the VZV. Indeed, it was found that some isolated Amazon tribes had no antibodies to diseases such as measles, mumps, and rubella, but they did have antibodies to VZV =-=[25]-=-. Thus it appears that the persistence of VZV in these small isolated populations has occurred because VZV can be dormant in people for many years and then be spread in the population by a case of shi... |

1 |
Competitive exclusion and multiple strains in an
- Castillo-Chavez, Huang, et al.
- 1999
(Show Context)
Citation Context |

1 |
Viral Infections of Humans, 2nd ed
- Evans
- 1982
(Show Context)
Citation Context ...tible fractions so and s∞ at the beginning and end of epidemics, this formula can be used to estimate contact numbers for specific diseases [100]. Using blood samples from freshmen at Yale University =-=[75]-=-, the fractions susceptible to rubella at the beginning and end of the freshman year were found to be 0.25 and 0.090, so the epidemic formula above gives σ ≈ 6.4. The fractions so =0.49 and s∞ =0.425 ... |