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A mathematical model for predicting the geographic spread of new infectious agents
1179
Mathematical and Computer Modelling Reports Mar/d Biosci. Vol. 90. pp. 26S.786, 1988
LATERAL
INHIBITION
MODELS GEORGE
OF DEVELOPMENTAL
PROCESSES
F. OSTER
Department8 of Biophysics, Entomology, and Zoology, University of California, Berkeley, CA 94720, U.S.A. Abstract-Most mathematical models for embryological pattern formation depend on the phenomenon of local autocatalysis with lateral inhibition (LALI). While the underlying physical and chemical mechanisms hypothesized by the models may be quite different, they all predict very similar kinds of spatial patterns. Therefore, since the underlying mechanism cannot in general be deduced from the pattern itself, other criteria must be applied in evaluating the usefulness of pattern formation models. The author points out how LALI is implemented in neural, chemical, and mechanical models of development, and suggests some general properties of LALI models that may impose limitations on organ shapes in ontogeny and phylogeny.
Moth/
Biosci. Vol. 90, pp. 287-303,
1988
A STOCHASTIC MODEL FOR CHEMOSENSORY CELL MOVEMENT: APPLICATION TO NEUTROPHIL AND MACROPHAGE PERSISTENCE AND ORIENTATION R. T. TRANQUILLO, E. S. FISHER,B. E. FARRELLand D. A. LAUFFENBURGER Department of Chemical Engineering, University of Pennsylvania, Philadelphia, PA 19104, U.S.A. Abstract-Two central features of leukocyte chemosensory movement behavior demand fundamental theoretical understanding. In uniform concentrations of chemoattractant, these cells exhibit a persistent random walk, with a characteristic “persistence time” between significant changes in direction. In chemoattractant concentration gradients, they demonstrate a biased random walk, with an “orientation bias” characterizing the fraction of cells moving up the gradient. A coherent picture of cell-movement responses to chemoattractant requires that both the persistence time and the orientation bias be explained within a unifying framework. In this paper we offer the possibility that “noise” in the cellular signal perception/response mechanism can simultaneously account for these two key phenomena. In particular, we report on a stochastic mathematical model for cell locomotion based on kinetic fluctuations in chemoattractant receptor binding. This model proves to be capable of simulating cell paths similar to those observed experimentally for two cell types examined to date: neutrophils and alveolar macrophages, under conditions of uniform chemoattractant concentrations as well as chemoattractant concentration gradients. Further, this model can quantitatively predict both cell persistence time and dependence of orientation bias on gradient size. The model also successfully predicts that an increase in persistence time is associated with a decrease in orientation for typical system parameter values, as is observed for alveolar macrophages in comparison to neutrophils. Thus, the concept of signal “noise” can quantitatively unify the major characteristics of leukocyte random motility and chemotaxis. The same level of noise large enough to account for the observed frequency of turning in uniform environments is simultaneously small enough to allow for the observed degree of directional bias in gradients. This suggest that chemosensory cell movement behavior may be based on a “usefully” imperfect integrated signal response system, which allows both random and directed searches under appropriate conditions.
Math/ Biosci.
Vol. 90, pp. 367-383,
1988
A MATHEMATICAL MODEL FOR PREDICTING THE GEOGRAPHIC SPREAD OF NEW INFECTIOUS AGENTS IRA M. LONGINI JR Department of Statistics and Biometry, Emory University, Atlanta, GA 30322, U.S.A.
Abstract-A mathematical model for the temporal and geographic spread of an epidemic in a network of populations is presented. The model is formulated on a continuous state space in discrete time for an
Mathematical and Computer Modelling Reports
1180
infectious disease that confers immunity following infection. The model allows for a general distribution of both the latent and infectious periods. An epidemic threshold theorem is given along with methods for finding the final attack rate when a single closed population is modeled. The model is first applied to analyzing the spread of influenza in single, closed populations in England and Wales and Greater London for the years 1958-1973. Then the model is used to predict the spread of Hong Kong influenza in 1968-1969 among 52 of the world’s major cities. The prediction for the whole network of cities is based on air-transport data and on the estimated parameters from the ascending limb of the reported epidemic curve in Hong Kong, the first city to experience a major influenza epidemic in 1968. Finally, extensions and future uses of a model for temporal-geographic spread of infectious agents is discussed.
Mmhl
Biosci. Vol. 90, pp. 385-396, 1988
MATHEMATICAL
MODELING
OF IMMUNITY
TO MALARIA
JOANL. ARON
Department
of Population Dynamics, School of Hygiene and Public Health, The Johns Hopkins University, 615 North Wolfe Street, Baltimore, MD 21205, U.S.A.
Abstract-A comparison of two epidemiological models of immunity to malaria shows that different characterizations of immunity boosted by exposure to infection generate qualitatively different results. Attempts to control disease by reducing transmission or increasing the recovery rate can produce an increase in prevalence in the compartmental model with discrete epidemiological states. However, the parasite density always decreases in response to disease control in the model with continuous epidemiological variables. Each model accounts for some epidemiological patterns. The increase in prevalence seen in the compartmental model is in accord with observed effects of variation in transmission. Parasite suppression in areas of antimalarial drug use is consistent with the effect of an increased recovery rate in the density model. Future work should combine the two approaches, perhaps by using the compartmental model over the low to moderate range of infection rates and switching to the density model at high infection rates. In any case, the validation of models needs to take account of the usage of antimalarial drugs as well as the intensity of transmission.
MoM
Biosci. Vol. 90, pp. 415473,
1988
USING MATHEMATICAL MODELS TO UNDERSTAND AIDS EPIDEMIC
THE
JAMESM. HYMANand E. ANN STANLEY
Center for Nonlinear Studies, Theoretical Division, MS-B284, Los Alamos National Laboratory, Los Alamos, NM 87545, U.S.A. Abstract-The most urgent public-health problem today is to devise effective strategies to minimize the destruction caused by the AIDS epidemic. This complex problem will involve medical advances and new public-health and education initiatives. Mathematical models based on the underlying transmission mechanisms of the AIDS virus can help the medical/scientific community understand and anticipate its spread in different populations and evaluate the potential effectiveness of different approaches for bringing the epidemic under control. Before we can use models to predict the future, we must carefully test them against the past spread of the infection and for sensitivity to parameter changes. The long and extremely variable incubation period and the low probability of transmitting the AIDS virus in a single contact imply that population structure and variations in infectivity both play an important role in its spread. The population structure is caused by differences between people in numbers of sexual partners and the use of intravenous drugs and because of the way in which people mix among age, ethnic, and social groups. We use a simplified approach to investigate the effects of variation in incubation periods and infectivity specific to the AIDS virus, and we compare a model of random partner choices with a model in which partners both come from similar behavior groups.
Mathematical and Computer Modelling Reports Mar/d Biosci. Vol. 90. pp. 26S.786, 1988
LATERAL
INHIBITION
MODELS GEORGE
OF DEVELOPMENTAL
PROCESSES
F. OSTER
Department8 of Biophysics, Entomology, and Zoology, University of California, Berkeley, CA 94720, U.S.A. Abstract-Most mathematical models for embryological pattern formation depend on the phenomenon of local autocatalysis with lateral inhibition (LALI). While the underlying physical and chemical mechanisms hypothesized by the models may be quite different, they all predict very similar kinds of spatial patterns. Therefore, since the underlying mechanism cannot in general be deduced from the pattern itself, other criteria must be applied in evaluating the usefulness of pattern formation models. The author points out how LALI is implemented in neural, chemical, and mechanical models of development, and suggests some general properties of LALI models that may impose limitations on organ shapes in ontogeny and phylogeny.
Moth/
Biosci. Vol. 90, pp. 287-303,
1988
A STOCHASTIC MODEL FOR CHEMOSENSORY CELL MOVEMENT: APPLICATION TO NEUTROPHIL AND MACROPHAGE PERSISTENCE AND ORIENTATION R. T. TRANQUILLO, E. S. FISHER,B. E. FARRELLand D. A. LAUFFENBURGER Department of Chemical Engineering, University of Pennsylvania, Philadelphia, PA 19104, U.S.A. Abstract-Two central features of leukocyte chemosensory movement behavior demand fundamental theoretical understanding. In uniform concentrations of chemoattractant, these cells exhibit a persistent random walk, with a characteristic “persistence time” between significant changes in direction. In chemoattractant concentration gradients, they demonstrate a biased random walk, with an “orientation bias” characterizing the fraction of cells moving up the gradient. A coherent picture of cell-movement responses to chemoattractant requires that both the persistence time and the orientation bias be explained within a unifying framework. In this paper we offer the possibility that “noise” in the cellular signal perception/response mechanism can simultaneously account for these two key phenomena. In particular, we report on a stochastic mathematical model for cell locomotion based on kinetic fluctuations in chemoattractant receptor binding. This model proves to be capable of simulating cell paths similar to those observed experimentally for two cell types examined to date: neutrophils and alveolar macrophages, under conditions of uniform chemoattractant concentrations as well as chemoattractant concentration gradients. Further, this model can quantitatively predict both cell persistence time and dependence of orientation bias on gradient size. The model also successfully predicts that an increase in persistence time is associated with a decrease in orientation for typical system parameter values, as is observed for alveolar macrophages in comparison to neutrophils. Thus, the concept of signal “noise” can quantitatively unify the major characteristics of leukocyte random motility and chemotaxis. The same level of noise large enough to account for the observed frequency of turning in uniform environments is simultaneously small enough to allow for the observed degree of directional bias in gradients. This suggest that chemosensory cell movement behavior may be based on a “usefully” imperfect integrated signal response system, which allows both random and directed searches under appropriate conditions.
Math/ Biosci.
Vol. 90, pp. 367-383,
1988
A MATHEMATICAL MODEL FOR PREDICTING THE GEOGRAPHIC SPREAD OF NEW INFECTIOUS AGENTS IRA M. LONGINI JR Department of Statistics and Biometry, Emory University, Atlanta, GA 30322, U.S.A.
Abstract-A mathematical model for the temporal and geographic spread of an epidemic in a network of populations is presented. The model is formulated on a continuous state space in discrete time for an
Mathematical and Computer Modelling Reports
1180
infectious disease that confers immunity following infection. The model allows for a general distribution of both the latent and infectious periods. An epidemic threshold theorem is given along with methods for finding the final attack rate when a single closed population is modeled. The model is first applied to analyzing the spread of influenza in single, closed populations in England and Wales and Greater London for the years 1958-1973. Then the model is used to predict the spread of Hong Kong influenza in 1968-1969 among 52 of the world’s major cities. The prediction for the whole network of cities is based on air-transport data and on the estimated parameters from the ascending limb of the reported epidemic curve in Hong Kong, the first city to experience a major influenza epidemic in 1968. Finally, extensions and future uses of a model for temporal-geographic spread of infectious agents is discussed.
Mmhl
Biosci. Vol. 90, pp. 385-396, 1988
MATHEMATICAL
MODELING
OF IMMUNITY
TO MALARIA
JOANL. ARON
Department
of Population Dynamics, School of Hygiene and Public Health, The Johns Hopkins University, 615 North Wolfe Street, Baltimore, MD 21205, U.S.A.
Abstract-A comparison of two epidemiological models of immunity to malaria shows that different characterizations of immunity boosted by exposure to infection generate qualitatively different results. Attempts to control disease by reducing transmission or increasing the recovery rate can produce an increase in prevalence in the compartmental model with discrete epidemiological states. However, the parasite density always decreases in response to disease control in the model with continuous epidemiological variables. Each model accounts for some epidemiological patterns. The increase in prevalence seen in the compartmental model is in accord with observed effects of variation in transmission. Parasite suppression in areas of antimalarial drug use is consistent with the effect of an increased recovery rate in the density model. Future work should combine the two approaches, perhaps by using the compartmental model over the low to moderate range of infection rates and switching to the density model at high infection rates. In any case, the validation of models needs to take account of the usage of antimalarial drugs as well as the intensity of transmission.
MoM
Biosci. Vol. 90, pp. 415473,
1988
USING MATHEMATICAL MODELS TO UNDERSTAND AIDS EPIDEMIC
THE
JAMESM. HYMANand E. ANN STANLEY
Center for Nonlinear Studies, Theoretical Division, MS-B284, Los Alamos National Laboratory, Los Alamos, NM 87545, U.S.A. Abstract-The most urgent public-health problem today is to devise effective strategies to minimize the destruction caused by the AIDS epidemic. This complex problem will involve medical advances and new public-health and education initiatives. Mathematical models based on the underlying transmission mechanisms of the AIDS virus can help the medical/scientific community understand and anticipate its spread in different populations and evaluate the potential effectiveness of different approaches for bringing the epidemic under control. Before we can use models to predict the future, we must carefully test them against the past spread of the infection and for sensitivity to parameter changes. The long and extremely variable incubation period and the low probability of transmitting the AIDS virus in a single contact imply that population structure and variations in infectivity both play an important role in its spread. The population structure is caused by differences between people in numbers of sexual partners and the use of intravenous drugs and because of the way in which people mix among age, ethnic, and social groups. We use a simplified approach to investigate the effects of variation in incubation periods and infectivity specific to the AIDS virus, and we compare a model of random partner choices with a model in which partners both come from similar behavior groups.