Modeling the Seasonal Epidemic - Part 2

In the last issue we discussed the various ways that seasonality can impact infectious disease transmission; this month were looking into the techniques to incorporate these periodic effects into mathematical models.

A function is said to be periodic if its behavior repeats over constant intervals. The most familiar periodic functions are trigonometric functions, particularly the sine wave. The continuous and gradual changes in the sine function mean that it can serve well to incorporate continuously varying conditions such as temperature or humidity.

In contrast, stepwise (or step) functions are discontinuous, defined as separate constant values over their distinct "pieces"; these can also be constructed to be periodic. The abrupt jump between the various pieces of the step function can be useful in defining events that have definite, abrupt start and end dates - school calendars, the July / August holiday in Europe, and so on.

Piecewise continuous functions can also be constructed of various sub-components so as to be periodic. For example, a function describing rainfall in Arizona (if such a thing could be reliably predicted) could be defined as zero for most of the year, then joined continuously to two bell-shaped curves coinciding with the late winter and summer monsoons.

Once the shape of the periodic function has been determined based upon the components of the system that show seasonality, it is essential to incorporate it into the appropriate portion of the model. When modeling a vector-borne disease and the periodic aspect is temperature or humidity, one application of the periodic function may be as a forcing term for vector reproduction or population size as these are directly impacted by environmental conditions, particularly when addressing container-breeding insects. When evaluating discontinuous conditions such as school sessions for direct-transmissible infections, the stepwise period function may be applied to the contact matrix (which would now no longer be held constant, but would vary throughout the model time-frame) or the beta transmission term. When modeling a fungal disease such as coccidioidomycosis (Valley Fever), the piecewise function describing monsoon seasonality could be applied to the disease individual attack rate - maintaining low or negligible levels throughout most of the year, then waxing and waning during at-risk periods.

Longer-term periodicity on the order of multiple years can be included in disease models explicitly through the incorporation of periodic functions applied to vector population growth, weather patterns, and so on. The annual forcing terms described above can then be combined with longer-term functions to generate compound periodic systems, like the hypothetical one shown below:

Longer-term fluctuations can also be brought in indirectly through the handling of issues such as waning immunity. For diseases for which naturally-acquired immunity fades after a number of years, it is possible to hold individuals in a "recovered / immune" compartment for a set period of time, following which they return to the susceptible component. This indirect periodicity allows the susceptible population to ebb and flow over multiple-year timeframes as outbreaks occur, immunity is conferred, and then susceptibility returns.

Next month we will cover the implications of seasonality and longer-term periodicity for the control and eradication of infectious disease. In the meantime, if your organization is wrestling with modeling seasonal epidemics and their control, contact us here at MathEcology a call - wed be happy to help!

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