# Age-structured Epidemiological Models with Multiple Infections and Its Application to Pertussis

#### Abstract

This dissertation presents several age-structured epidemiological models with multiple infections, including formulation of the models and mathematical analysis of their dynamical behavior, as well as applications to pertussis transmission and evaluation of the effectiveness of widely used pertussis vaccines. Pertussis was a common disease of infants and children in the pre-vaccine era. After the development and administration of whole-cell and acellular vaccines, there was a dramatic drop of reported pertussis incidence in the United States. However, an increasing trend is observed since 1970s despite continued use of vaccination. The population with the highest incidence has also shifted from infants to adolescents. Various mathematical models have been used to investigate the possible causes and to study the impact of different possible vaccination strategies. Besides the conventional DTaP vaccination series, one Tdap booster dose for people aged 11 through 18 years old who have completed the childhood doses is recommended in this country. Aside from the direct benefit of reducing the incidence among adolescents, there is also interest in evaluating any possible indirect effect of Tdap on the most vulnerable infant group. This dissertation is mainly focused on modeling the effect of Tdap dose on this age group. The thesis begins with a brief review of some widely-used mathematical models for pertussis. Chapter 2 of this dissertation deals with an age-structured deterministic model with two infections. It accounts for passively acquired maternal antibodies that decay and infection-induced immunity that wanes. The age-dependent force of infection is based on proportionate or preferential mixing. The model is then extended to incorporate two more re-infections and a sequence of vaccination compartments in Chapter 3. The partial differential equation system is then analyzed mathematically to determine the local and global stability of the disease-free steady state when mixing function is separable (i.e., proportionate). This gives an analytical expression for the threshold quantity R0. This expression shows that the basic reproduction number depends only on the first infection in a model with multiple infections. In Chapter 4, the partial differential equations system is reduced to an ordinary differential equation system by partitioning the continuous age spectrum into nine disjoint age groups and assuming constant parameter values within each age interval. The general mixing function in its continuous form is discretized into a general mixing matrix without imposing any specific assumption on the mixing pattern. This enables the use of empirical mixing matrix in simulations that will likely produce more accurate and reliable modeling results. In this work, age-dependent transmission probability estimated in earlier study and realistic mixing data are used for determining the parameter values. Simulations show that the combination of DTaP and Tdap doses fails to bring R0 below 1, which means, for the given set of parameters, pertussis will remain endemic in spite of sustained high coverage vaccination. It is also shown that both DTaP and Tdap vaccines have remarkable effect on reducing the incidences of the age groups being directly vaccinated, while the Tdap adolescent booster dose can provide some limited indirect protection for infants as well (<5%). If a 1% death rate is assumed, this corresponds to prevention of 469 yearly infant deaths.

#### Degree

Ph.D.

Feng, Purdue University.

#### Subject Area

Applied Mathematics|Epidemiology|Public health

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