  An optimal filter for Poisson observations is developed as a variant of the
traditional Kalman filter. Poisson distributions are characteristic of
infectious diseases, which model the number of patients recorded as presenting
each day to a health care system. We develop both a linear and nonlinear
(extended) filter. The methods are applied to a case study of neonatal sepsis
and postinfectious hydrocephalus in Africa, using parameters estimated from
publicly available data. Our approach is applicable to a broad range of disease
dynamics, including both noncommunicable and the inherent nonlinearities of
communicable infectious diseases and epidemics such as from COVID-19.
