Two SIS epidemiologic models with delays.

The SIS epidemiologic models have a delay corresponding to the infectious period, and disease-related deaths, so that the population size is variable. The population dynamics structures are either logistic or recruitment with natural deaths. Here the thresholds and equilibria are determined, and stabilities are examined. In a similar SIS model with exponential population dynamics, the delay destabilized the endemic equilibrium and led to periodic solutions. In the model with logistic dynamics, periodic solutions in the infectious fraction can occur as the population approaches extinction for a small set of parameter values.

Simulations of pertussis epidemiology in the United States: effects of adult booster vaccinations.

An expanded pertussis (whooping cough) vaccination program which includes adult boosters every 10 yr is studied using computer simulations of two models. These age-structured pertussis transmission models include waning of both infection-acquired and vaccine-induced immunity, and vaccination of children corresponding to the vaccination coverage since 1940. Adult vaccinations cause a larger boost in the immunity level in the second model than in the first model. In the simulations the addition of adult pertussis booster vaccinations every 10 yr is beneficial in reducing adult incidence, but causes only modest reductions in the incidence in infants and young children. These simulations suggest that a careful cost effectiveness analysis is needed before implementation of an adult pertussis vaccination program.

Modeling the effects of varicella vaccination programs on the incidence of chickenpox and shingles.

Two possible dangers of an extensive varicella vaccination program are more varicella (chickenpox) cases in adults, when the complication rates are higher, and an increase in cases of zoster (shingles). Here an age-structured epidemiologic-demographic model with vaccination is developed for varicella and zoster. Parameters are estimated from epidemiological data. This mathematical and computer simulation model is used to evaluate the effects of varicella vaccination programs. Although the age distribution of varicella cases does shift in the simulations, this does not seem to be a danger because many of the adult cases occur after vaccine-induced immunity wanes, so they are mild varicella cases with fewer complications. In the simulations, zoster incidence increases in the first three decades after initiation of a vaccination program, because people who had varicella in childhood age without boosting, but then it decreases. Thus the simulations validate the second danger of more zoster cases.

An age-structured model for pertussis transmission.

The vaccination program for pertussis (whooping cough) in the United States consists of giving multiple doses of pertussis vaccine to young children. A demographic model with a steady-state age distribution is used as a basis for building an epidemiologic model for the transmission of pertussis. This age-structured model includes vaccination of infants and children for pertussis with waning of both infection-acquired and vaccine-induced immunity. Computer simulations of the mathematical model between 1940 and 2040 show the changes that took place during the implementation phase of the U.S. program and predict only minor future changes in the age distribution and incidence of pertussis if the vaccination program is maintained at the 1995 level. The sensitivities of these results to changes in demographic and epidemiologic parameters, vaccine efficacy, duration of protection, and levels of vaccination coverage are investigated.

Four SEI endemic models with periodicity and separatrices.

Periodic solutions have been found for some infectious disease models of the SI and SEI types. Here four SEI models with either disease-reduced or uniform reproduction are examined to determine the model features that do and do not lead to periodic solutions. The two SEI models with the simple mass action incidence beta XY can have periodic solutions for some parameter values, but the two SEI models with the standard mass action incidence lambda XY/N do not have periodic solutions. For some intermediate values of lambda in the SEI model with incidence lambda XY/N and uniform reproduction, the interior equilibrium is a saddle whose stable manifold separates the attractive regions for the disease-free equilibrium and the susceptible-free equilibrium.

An SIS epidemic model with variable population size and a delay.

The SIS epidemiological model has births, natural deaths, disease-related deaths and a delay corresponding to the infectious period. The thresholds for persistence, equilibria and stability are determined. The persistence of the disease combined with the disease-related deaths can cause the population size to decrease to zero, to remain finite, or to grow exponentially with a smaller growth rate constant. For some parameter values, the endemic infective-fraction equilibrium is asymptotically stable, but for other parameter values, it is unstable and a surrounding periodic solution appears by Hopf bifurcation.

Waning immunity and its effects on vaccination schedules.

A relatively comprehensive age-specific transmission model is used to determine the effect of various factors on the optimal vaccination ages in one-dose and two-dose vaccination schedules. Motivated by the situation for measles, the model allows the duration of immunity of newborns to depend on the level of immunity of the mother at the time of the birth and allows for waning immunity as well as boosting of immunity by exposure to the disease. It is found that a significant amount of waning of disease-acquired immunity is plausible when boosting occurs but this is not an important factor in determining optimal vaccination schedules. On the other hand, plausible rates of loss of vaccine-induced immunity can have a substantial effect on the optimal vaccination schedule, particularly when there is no boosting of immunity. For two-dose schedules the optimal vaccination ages depend significantly on the level of vaccination coverage achieved. In the presence of plausible rates of loss of vaccine-induced immunity for measles, it is found that the vaccination coverage required to eradicate the disease is substantially higher than previously suggested.

Population size dependent incidence in models for diseases without immunity.

Epidemiological models of SIS type are analyzed to determine the thresholds, equilibria, and stability. The incidence term in these models has a contact rate which depends on the total population size. The demographic structures considered are recruitment-death, generalized logistic, decay and growth. The persistence of the disease combined with disease-related deaths and reduced reproduction of infectives can greatly affect the population dynamics. For example, it can cause the population size to decrease to zero or to a new size below its carrying capacity or it can decrease the exponential growth rate constant of the population.

An epidemiological model for HIV/AIDS with proportional recruitment.

A model for HIV transmission is formulated for a homosexual population of varying size, with recruitment into the susceptible class proportional to the active population size and with stages of progression to AIDS. Analysis of this model includes identifying the threshold that determines whether the disease dies out or proportions remain endemic and establishing criteria that determine whether the population size decays asymptotically exponentially to zero or grows asymptotically exponentially to infinity. In an analogous heterosexual model, the conservation of heterosexual contacts is shown to imply that this two-sex model reduces to the one-sex model.

Dynamic models of infectious diseases as regulators of population sizes.

Five SIRS epidemiological models for populations of varying size are considered. The incidences of infection are given by mass action terms involving the number of infectives and either the number of susceptibles or the fraction of the population which is susceptible. When the population dynamics are immigration and deaths, thresholds are found which determine whether the disease dies out or approaches an endemic equilibrium. When the population dynamics are unbalanced births and deaths proportional to the population size, thresholds are found which determine whether the disease dies out or remains endemic and whether the population declines to zero, remains finite or grows exponentially. In these models the persistence of the disease and disease-related deaths can reduce the asymptotic population size or change the asymptotic behavior from exponential growth to exponential decay or approach to an equilibrium population size.

Disease transmission models with density-dependent demographics.

The models considered for the spread of an infectious disease in a population are of SIRS or SIS type with a standard incidence expression. The varying population size is described by a modification of the logistic differential equation which includes a term for disease-related deaths. The models have density-dependent restricted growth due to a decreasing birth rate and an increasing death rate as the population size increases towards its carrying capacity. Thresholds, equilibria and stability are determined for the systems of ordinary differential equations for each model. The persistence of the infectious disease and disease-related deaths can lead to a new equilibrium population size below the carrying capacity and can even cause the population to become extinct.

A simulation model of AIDS in San Francisco: I. Model formulation and parameter estimation.

A model is formulated for the spread of the human immunodeficiency virus (HIV) and the subsequent development of acquired immunodeficiency syndrome (AIDS) in the population of homosexual men in San Francisco. The dynamic simulation model includes sexually very active and active subpopulations, migration, and a staged progression of HIV-infected persons to AIDS and death. Numerous data sources are used to estimate parameter values in the model. In a companion paper, simulations using the model and parameter estimates are found that are consistent with HIV and AIDS incidence data.

A simulation model of AIDS in San Francisco: II. Simulations, therapy, and sensitivity analysis.

The HIV and AIDS incidences each year for homosexual men in San Francisco are estimated from data. A computer simulation model for HIV transmission dynamics and progression to AIDS is used to reconstruct the HIV epidemic. Using some a priori parameter estimates, simulations are found that give good fits to the incidence data. In the stimulations the populations is divided into risk groups whose sexual activities are found to be strongly connected. There is saturation in the high-risk group, but changes in sexual behavior are more important in obtaining adequate fits. The simulation modeling yields useful parameter estimates, but the remaining uncertainty in parameter values implies that the simulation forecasts are also uncertain. Changes in HIV incidence lead to changes in AIDS incidence about 6-10 years later. Simulation models with and without zidovudine treatment both fit the incidence data; thus the effects of therapy on AIDS incidence are unclear. The fits of the simulation model are most sensitive to the yearly migration rate, the number of stages in the progression to AIDS, and the average number of new sexual partners per month; thus better estimates of these parameters would be desirable.

Some epidemiological models with nonlinear incidence.

Epidemiological models with nonlinear incidence rates can have very different dynamic behaviors than those with the usual bilinear incidence rate. The first model considered here includes vital dynamics and a disease process where susceptibles become exposed, then infectious, then removed with temporary immunity and then susceptible again. When the equilibria and stability are investigated, it is found that multiple equilibria exist for some parameter values and periodic solutions can arise by Hopf bifurcation from the larger endemic equilibrium. Many results analogous to those in the first model are obtained for the second model which has a delay in the removed class but no exposed class.

Statistical analysis of the stages of HIV infection using a Markov model.

We use a staged Markov model to estimate the distribution and mean length of the incubation period for acquired immunodeficiency syndrome (AIDS) from a cohort of 603 human immunodeficiency virus (HIV) infected individuals who have been followed through various stages of infection. The model partitions the infected period into four progressive stages: (1) infected but antibody-negative; (2) antibody-positive but asymptomatic; (3) pre-AIDS symptoms and/or abnormal haematologic indicator; and (4) clinical AIDS. We also model a fifth stage: death due to AIDS. The estimated mean (median) waiting times in each stage of infection are stage 1, 2.2 (1.5) months; stage 2, 52.6 (36.5) months; stage 3, 62.9 (43.6) months; and stage 4, 23.6(16.3) months. We estimate the mean AIDS incubation period (from infection to development of clinical AIDS) as 9.8 years with a 95 per cent confidence interval of [8.4, 11.2] years. The paper also considers the estimated density function of the AIDS incubation period and the estimated survival functions for individuals in each stage of infection. This work represents one of the most complete statistical descriptions to date of the natural history of HIV infection.

Epidemiological models with age structure, proportionate mixing, and cross-immunity.

Infection by one strain of influenza type A provides some protection (cross-immunity) against infection by a related strain. It is important to determine how this influences the observed co-circulation of comparatively minor variants of the H1N1 and H3N2 subtypes. To this end, we formulate discrete and continuous time models with two viral strains, cross-immunity, age structure, and infectious disease dynamics. Simulation and analysis of models with cross-immunity indicate that sustained oscillations cannot be maintained by age-specific infection activity level rates when the mortality rate is constant; but are possible if mortalities are age-specific, even if activity levels are independent of age. Sustained oscillations do not seem possible for a single-strain model, even in the presence of age-specific mortalities; and thus it is suggested that the interplay between cross-immunity and age-specific mortalities may underlie observed oscillations.

An epidemiological model with a delay and a nonlinear incidence rate.

An epidemiological model with both a time delay in the removed class and a nonlinear incidence rate is analysed to determine the equilibria and their stability. This model is for diseases where individuals are first susceptible, then infected, then removed with temporary immunity and then susceptible again when they lose their immunity. There are multiple equilibria for some parameter values, and, for certain of these, periodic solutions arise by Hopf bifurcation from the large nontrivial equilibrium state.

Dynamical behavior of epidemiological models with nonlinear incidence rates.

Epidemiological models with nonlinear incidence rates lambda IpSq show a much wider range of dynamical behaviors than do those with bilinear incidence rates lambda IS. These behaviors are determined mainly by p and lambda, and secondarily by q. For such models, there may exist multiple attractive basins in phase space; thus whether or not the disease will eventually die out may depend not only upon the parameters, but also upon the initial conditions. In some cases, periodic solutions may appear by Hopf bifurcation at critical parameter values.