  Transmission rates are key in understanding the spread of infectious
diseases. Using the framework of compartmental models, we introduce a simple
method that enables us to reconstruct time series of transmission rates
directly from incidence or disease-related mortality data. The reconstruction
exploits differential equations, which model the time evolution of infective
stages and strains. Being sensitive to initial values, the method produces
asymptotically correct solutions. The computations are fast, with time
complexity being quadratic. We apply the reconstruction to data of measles
(England and Wales, 1948-67), dengue (Thailand, 1982-99), and influenza (U.S.,
1910-27). The Measles example offers comparison with earlier work. Here we
re-investigate reporting corrections, include and exclude demographic
information. The dengue example deals with the failure of vector-control
measures in reducing dengue hemorrhagic fever (DHF) in Thailand. Two competing
mechanisms have been held responsible: strain interaction and demographic
transitions. Our reconstruction reveals that both explanations are possible,
showing that the increase in DHF cases is consistent with decreasing
transmission rates resulting from reduced vector counts. The flu example
focuses on the 1918/19 pandemic, examining the transmission rate evolution for
an invading strain. Our analysis indicates that the pandemic strain could have
circulated in the population for many months before the pandemic was initiated
by an event of highly increased transmission.
