causes acute pathologies in the host. When the pathogen first establishes an infection site, the immune

response in the host may be inactive for some initial period during which time the pathogen multiplies.

This initial delay period in the stimulation of the immune response can be a critical factor in host

mortality. For example, low levels of sporozites can lead to clinically significant malaria parasitization

while higher levels can cause death in a number of days. Studies with animals intravenously injected with

low doses of HIV revealed a production of neutralizing antibodies and high survival rates while those

inoculated with high doses rapidly developed clinical disease. Similar observations have been made for

measles where large inoculums, which may ocasionally result from airborne transmission, have been suggested

to increase the risk of vaccine failure.

For acute infections, the estimation of the time and extent of the pathogenic load can be of great

importance in developing effective intervention treatment programs for those individuals who have been

exposed to a pathogen. Mathematical models of the timing and extent of pathogen infection, and possible

reinfection, processes in an individual provide information on the early dynamics of acute infections, the

maximum expected pathogenic load, and the host immune response. Deterministic mathematical models of

infectious diseases are typically expressed as a series of coupled, nonlinear, first-order, initial value

ODEs [1]. Because these models are nonlinear, numerical solutions are typically obtained. Approximate

analytical solutions based on linearization have restricted utility because this approach eliminates all

of the interesting nonlinear effects incorporated in the model. Parameter perturbation is an alternative

method for generating approximate analytical solutions.

Asymptotic theory is used to develop an approximate solution to a nonlinear coupled ODE model of an

infectious disease in which a pathogen attacks a host whose immune system responds defensively. This

asymptotic model is then used to estimate the original model parameters based on initial-time measurements

of the pathogen or immune level. The model under consideration is that presented by Moshtashemi and Levins

[2]. There is a short time scale that characterizes the initial growth of the invading pathogen, referred

to as the initial layer, and a longer time scale that characterizes the immune system behavior after it has

reacted to the invasion, referred to as the remission layer. This process repeats with alternate periods of

short-time pathogen growth and long-time immune system dominance. Using singular perturbation theory, it is

possible to derive very accurate approximate analytical solutions in each of these regions starting from the

initial layer. The method of matched asymptotic expansions is used to describe the transient dynamics of the

interaction between the invading pathogen and the host immune system [3]. The advantage of using the

asymptotic model is the simplification that arises in the parameter estimation scheme when only the initial

time is considered. Uncertainty in the assay associated with pathogen loading or immune response measurements

is taken into consideration by the use of confidence interval estimation for the parameters of interest. The

result is a point estimate with uncertainty bounds of the maximum predicted pathogen loading and immune system

response based on information obtained early in the infection/immune response cycle that can be of great

utility in evaluating treatment options.

References

[1] Hethcote, The mathematics of infectious diseases, SIAM Rev., 42, 599, 2000.

[2] Mohtashemi and Levins, Transient dynamics and early diagnosis in infectious disease, J. Math. Biol.,

43, 446, 2001.

[3] Whitman and Ashrafiuon, Asymptotic theory of an infectiuos disease model, J. Math. Biol., 53, 287,

2006.