Modeling the Seasonal Epidemic - Part 2
In the last issue we discussed the various ways that seasonality can impact
infectious disease transmission; this month we’re 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
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 - we’d be happy to help!
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