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First, we study the effects of behavioral changes in a smallpox attack model. Response strategies to a smallpox bioterrorist attack have focused on interventions such as isolation, contact tracing, quarantine, ring vaccination, and mass vaccination. We formulate and analyze a mathematical model in which some individuals lower their daily contact activity rates once an epidemic has been identified in a community. We use computer simulations to analyze the effects of behavior change alone and in combination with other control measures. We demonstrate that the spread of the disease is highly sensitive to how rapidly people reduce their contact activity. Next, we study mixing patterns between age groups using social networks. The course of an epidemic through a population is determined by the interactions among individuals. To capture these elements of reality; we use the contact network simulations for the city of Portland, Oregon that were developed as part of the TRANSIMS/EpiSims project to study and identify mixing patterns. We analyze contact patterns between different age groups and identify those groups who are at higher risk of infection. We describe a new method for estimating transmission matrices that describe the mixing and the probability of transmission between the age groups. Our results show that the mixing structure within the population is divided into two main groups, adults and children. Finally, we study the effects of mixing patterns in the presence of population heterogeneity. We investigate the impact that different mixing assumptions have on the spread of a disease in an age-structured differential equation model. We use a realistic mixing population and investigate the impact that this mixing pattern has on epidemic outcomes when compared to a randomly mixing population. Furthermore, we investigate the impact on a smallpox epidemic of population heterogeneity such as differences in susceptibility and infectivity within the population for a smallpox epidemic outbreak. We find that different mixing assumptions lead to differences in disease prevalence and final epidemic size. We also demonstrate that a smallpox epidemic is greatly reduced by the level of residual immunity in the population.

High levels of notified pertussis in adolescents and adults, persisting severe disease (hospitalization and deaths) in infants despite high childhood immunization coverage, together with the availability of adult-formulated pertussis vaccines, have made alternate strategies for vaccine control of pertussis an important issue in Australia. An age-structured computer simulation model was used to compare the likely effects of adopting different vaccination strategies in Australia on pertussis transmission by age group over a 50 year time period. Epidemiological parameters and vaccination coverage in Australia were estimated from previous pertussis modeling studies and existing data. In the simulations, replacing the pertussis booster at 18 months with a booster dose for adolescents at an age between 12 and 17 years, assuming 80% coverage, led to decreases in pertussis cases of 30% in children of ages 0-23 months (who have the highest complication rates) and of 25% in adolescents, but an increase of 15% in cases in 2-4-year-old children. The simulations did not suggest any shift of pertussis cases into the adult child-bearing years. Varying parameter values in the simulations in a series of sensitivity analyses showed the model predictions to be robust over a plausible range. The results of these simulations suggest that the recent change in the Australian pertussis vaccination schedule, replacing the 18 month dose with a pertussis booster in 15-17-year-old adolescents, is very likely to reduce overall pertussis incidence in Australia without increasing the cost of the current vaccine program.

Influenza pandemics occur when a novel influenza strain, often of animal origin, becomes transmissible between humans. Domestic animal species such as poultry or swine in confined animal feeding operations (CAFOs) could serve as local amplifiers for such a new strain of influenza. A mathematical model is used to examine the transmission dynamics of a new influenza virus among three sequentially linked populations: the CAFO species, the CAFO workers (the bridging population), and the rest of the local human population. Using parameters based on swine data, simulations showed that when CAFO workers comprised 15-45% of the community, human influenza cases increased by 42-86%. Successful vaccination of at least 50% of CAFO workers cancelled the amplification. A human influenza epidemic due to a new virus could be locally amplified by the presence of confined animal feeding operations in the community. Thus vaccination of CAFO workers would be an effective use of a pandemic vaccine.

A host-parasite model is proposed that incorporates a nonlinear incidence rate. Under the influence of multiple infectious attacks, the model admits bistable regions such that the infection dies out if initial states lie in one region, and the population and parasites coexist if initial states lie in the other region. It is also found that parasites can drive the population to extinction for suitable parameters. It is verified that the model has a saddle-node bifurcation, Hopf bifurcations and a cusp of codimension 2 or higher codimension. Stable limit cycles and unstable limit cycles are examined as the infection-reduced reproduction rate varies. It is shown that the model goes through the change of stages of infection extinction, infection persistence, infection extinction, and the extinction of both parasites and the population as the contact coefficient increases.

Many infants whose mothers have rubella infections during their first trimester of pregnancy have birth defects called congenital rubella syndrome (CRS). China does not routinely vaccinate against rubella in the public sector, but may need to start as its 'one child per couple' policy changes the population age distribution and the dynamics of rubella epidemiology, so that the incidence of rubella in pregnant women increases. Computer simulations with demographic transitions and rubella transmission dynamics predict that, with no or limited rubella vaccination, CRS incidence in China in the 30 years after 2020 will be more than twice the level in 2005. Comparisons of rubella vaccination strategies using computer simulations show that routine vaccination of over 80% of 1-year-old children would be effective in reducing total CRS cases in 2005-2051 and eliminating rubella in China by 2051. Routine immunizations at higher levels and the addition of early mass vaccinations of 2-14-year-old children and women of childbearing ages would further reduce total CRS cases and speed up the elimination of rubella.

Models for the transmission of an infectious disease in one and two host populations with and without self-regulation are analyzed. Many unusual behaviors such as multiple positive equilibria and periodic solutions occur in previous models that use the mass-action (density-dependent) incidence. In contrast, the models formulated using the frequency-dependent (standard) incidence have the behavior of a classic endemic model, since below the threshold, the disease dies out, and above the threshold, the disease persists and the infectious fractions approach an endemic equilibrium. The results given here reinforce previous examples in which there are major differences in behavior between models using mass-action and frequency-dependent incidences.

Approximately one million adult pertussis cases occur annually in the US, and infants still die from pertussis. Computer simulations were used to predict the impact of vaccination of children, adults and/or adolescents, and household members of newborns (cocoon strategy). Childhood vaccination greatly reduced cases in children, but increased the incidence in adolescents and adults. Routine adolescent and adult vaccination had a large direct effect, whereas the cocoon strategy had a predominantly indirect effect on young infants. The number needed to vaccinate (NNV) to prevent a case of typical pertussis in the entire population was lowest for the adolescent strategy. The cocoon strategy had the lowest NNV to prevent a case of typical pertussis in young infants. The current vaccination schedule, local epidemiological data, age-specific cost of pertussis cases, and accessibility of the target population will determine which strategy has the highest likelihood of success in achieving the public health goal.

Thresholds, equilibria, and their stability are found for SIQS and SIQR epidemiology models with three forms of the incidence. For most of these models, the endemic equilibrium is asymptotically stable, but for the SIQR model with the quarantine-adjusted incidence, the endemic equilibrium is an unstable spiral for some parameter values and periodic solutions arise by Hopf bifurcation. The Hopf bifurcation surface and stable periodic solutions are found numerically.

Four modifications of a predator prey model to include an SIS or SIR parasitic infection are developed and analyzed. Thresholds are identified and global stability results are proved. When the disease persists in the prey population and the predators have a sufficient feeding efficiency to survive, the disease also persists in the predator population.

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.

Pertussis (whooping cough) incidence in the United States has oscillated with a period of about four years since data was first collected in 1922. An infection with pertussis confers immunity for several years, but then the immunity wanes, so that reinfection is possible. A pertussis reinfection is mild after partial loss of immunity, but the reinfection can be severe after complete loss of immunity. Three pertussis transmission models with waning of immunity are examined for periodic solutions. Equilibria and their stability are determined. Hopf bifurcation of periodic solutions around the endemic equilibrium can occur for some parameter values in two of the models. Periods of about four years are found for epidemiologically reasonable parameter values in two of these models.

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.

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 βXY can have periodic solutions for some parameter values, but the two SEI models with the standard mass action incidence λXY N do not have periodic solutions. For some intermediate values of λ in the SEI model with incidence λXY 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.

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.

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.

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.

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.

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.

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.

The impact of individual and community behavioral changes in response to an outbreak of a disease with high mortality is often not appreciated. Response strategies to a smallpox bioterrorist attack have focused on interventions such as isolation of infectives, contact tracing, quarantine of contacts, ring vaccination, and mass vaccination. We formulate and analyze a mathematical model in which some individuals lower their daily contact activity rates once an epidemic has been identified in a community. Transmission parameters are estimated from data and an expression is derived for the effective reproduction number. We use computer simulations to analyze the effects of behavior change alone and in combination with other control measures. We demonstrate that the spread of the disease is highly sensitive to how rapidly people reduce their contact activity rates and to the precautions that the population takes to reduce the transmission of the disease. Even gradual and mild behavioral changes can have a dramatic impact in slowing an epidemic. When behavioral changes are combined with other interventions, the epidemic is shortened and the number of smallpox cases is reduced. We conclude that for simulations of a smallpox outbreak to be useful, they must consider the impact of behavioral changes. This is especially true if the model predictions are being used to guide public health policy.

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.

Mathematical models have become important tools in analyzing the spread and control of infectious diseases. Although chronic diseases such as cancer and heart disease receive more attention in developed countries, infectious diseases are the most important causes of suffering and mortality in developing countries. Recently, the human immunodeficiency virus (HIV), which is the etiological agent for acquired immunodeficiency syndrome (AIDS), has become an important sexually-transmitted disease throughout the world. Tuberculosis is again becoming a problem because drug-resistant strains have evolved. Understanding the transmission characteristics of infectious diseases in communities, regions and countries can lead to better approaches to decreasing the transmission of these diseases. Mathematical models are useful in building and testing theories, and in comparing, planning, implementing and evaluating various detection, prevention, therapy and control programs. See Hethcote and Van Ark [30] for a discussion of the purposes and limitations of epidemiological modeling.

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.

For many reasons it is important to consider differences and similarities in the racial/ ethnic (r/e) patterns of AIDS cases in the United States (U.S.). Although most AIDS cases in the U.S. have occurred in white homosexual men, the per capita risk of AIDS is generally higher in Blacks and Hispanics. Understanding AIDS case rates and growth rates in r/e subgroups of a risk group can lead to more focused educational or prevention programs. Moreover, cross-checking r/e distributions between related risk groups and between regions provides a simple measure of the confidence which can be placed in the data. Time trends in r/e AIDS cases are analyzed in Chapter 10.

Since homosexual men who are also intravenous drug users interact sexually with homosexual men and share needles with intravenous drug users, they are a potential transfer linkage between the HIV epidemics in homosexual men and intravenous drug users. One theory is that these HIV epidemics are not crucially linked by homosexual intravenous drug users so that neither epidemic is feeding or sustaining the other. A simulation model with HIV transmission and progression to AIDS is formulated for three risk groups. Evidence based on fitting this model to AIDS incidence data in New York City supports the theory that neither HIV epidemic sustains the other. Thus the HIV epidemics in homosexual men and intravenous drug users are only weakly linked and can be modeled as separate epidemics.

Transmissions of HIV have occurred after exposure to people in the pre—antibody period (Peterman et al., 1985). The latent period (i.e., from infection to infectiousness) for HIV infection is short enough to be ignored in a model. The infectious period is unknown, but the virus seems to persist in the host indefinitely since it can be isolated from the blood for many years after the infection (Curran et al., 1988). No evidence to the contrary exists; therefore HIV infectivity is assumed to continue for life. Since individuals are either susceptible to HIV infection or infectious, HIV is an SI (susceptible-infected) disease (Hethcote, 1978, 1989c). In contrast, gonorrhea is an SIS disease since people can recover and return to the susceptible class.

The largest number of AIDS cases in the United States have occurred in homosexual men and intravenous drug users (IVDUs). The transmission mechanisms for HIV in these groups are needle-sharing between IVDUs and homosexual intercourse. Since homosexual men who are also IVDUs have both possible transmission mechanisms and interact with both homosexual men and IVDUs, they could be an important transmission link between these two groups. It is conceivable that a primary HIV epidemic in one of these groups could sustain a secondary HIV epidemic in the other group through the linkage provided by the homosexual IVDUs. The goal of this chapter is to evaluate the relative importance of HIV transmissions by homosexual IVDUs and to decide if they form an essential linkage between the HIV epidemics in homosexual men and IVDUs.

Chapters 2 and 3 have focused on modeling the transmission dynamics of HIV and the progression to AIDS for homosexual men. That model is now expanded in Section 7.1 to include five major risk groups: homosexual (and bisexual) men, homosexual (and bisexual) men who are intravenous drug users, intravenous drug users (IVDUs), heterosexual partners of IVDUs and neonates of female IVDUs and heterosexual partners. In this model the homosexual men have homosexual partnerships within their own group and also with homosexual—IVDUs. The IVDUs have needle-sharing partnerships with other IVDUs and also with homosexual—IVDUs. Obviously, the heterosexual partners of IVDUs have sexual partnerships with people who are IVDUs. Perinatal transmission occurs from the female IVDUs and the female heterosexual partners. Figure 7.1 summarizes the risk groups and possible routes of transmission of HIV infection. The fitting procedure for the five-group model is outlined in Section 7.2. This model with the first three risk groups is used for New York City in Chapter 8. The full model is used for regional comparisons in Chapter 10.

San Francisco (SF) has been chosen as the site for modeling HIV transmission in homosexual men because very good data are available there. The large open community of homosexual men in SF has been the source of numerous sexual behavior surveys. Additionally, data are available in SF for AIDS incidence each year and for HIV incidence as early as 1978. In this application of the simulation model, men who have sex with men (homosexual and bisexual men) are called homosexual men. Since most (85%) of the reported AIDS cases in SF have occurred in homosexual men, the modeling here does not include transmission between homosexual/bisexual men and others such as their heterosexual partners or homosexual intravenous drug users.

The goal of this chapter is to use the computer simulation model developed in Chapter 7 to analyze HIV transmission and AIDS in risk groups in the four major regions of the United States shown in Figure 10.1. Each region is divided into three to five subregions based on similarities and differences in AIDS incidences in parts of the region. Whenever there are no significant differences in the time trends in AIDS incidences between racial/ethnic (r/e) groups or subregions, AIDS cases in the groups or subregions are aggregated. Thus the first step in each region is to aggregate into as few subpopulations as possible. Aggregation simplifies the simulation process since it leads to fewer populations to fit. Of course, the HIV and AIDS incidences in the groups or subregions which were aggregated could be recovered at any time by using the fraction of cases which occurred in these groups or subregions.

Since homosexual men in San Francisco (SF) are probably the most studied AIDS population in the world, it is useful to demonstrate that one can simulate HIV and AIDS in this population in a manner consistent with all available information. The analysis in Chapter 5 and this chapter is important not only for a better understanding of HIV and AIDS in homosexual men in SF but also as a foundation for further work. It provides a basis for simulation modeling in later chapters of other geographic locations and other risk groups.

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.

For systems of linear differential and difference equations, it is possible to find explicit solutions (see introductory differential and difference equations textbooks). The difference equations in Figure 3.2 are nonlinear because the incidences involve products of susceptible fractions and the numbers of infectives. For systems of nonlinear differential and difference equations, it is not generally possible to find explicit solutions, but it is possible to find numerical solutions as a function of time. The numerical solutions of the system of nonlinear difference equations in Figure 3.2 are called the simulations of the HIV transmission dynamics model in Figure 3.1.