Infectious diseases continue to be a significant public health problem, especially in developing nations where limited resources and other factors prevent sustained administration of effective public health programs. The need to develop reliable mechanistic models for the spread of infectious disease is two-fold. From a public health perspective, it is clear that reliable models will aid in program decision making and control of infectious disease spread. From a scientific perspective, the identification of an appropriate mechanistic model can help improve our understanding of the important factors affecting the spread of infectious disease.

While the spread of infectious disease between individuals is an inherently stochastic process, it is remarkable to observe that distinct patterns, both temporal and spatial, emerge in aggregate with sufficiently large populations. For example, measles incidence appears to be correlated with school term holidays, while also displaying annual and biennial patterns in time[1]. Furthermore, spatial patterns of synchrony and traveling waves have been observed in data for dengue[2] and influenza[3]. The emergence of these patterns implies that there are fundamental temporal and spatial influences that need to be further understood. In previous work, we have demonstrated a nonlinear programming approach for estimating seasonal drivers using models for a single community, and showed that this seasonality reflects school term holiday schedules.

Here, we extend this approach to the estimation of spatially-coupled models and seek to understand both temporal and spatial drivers influencing the spread of infectious disease. We present a nonlinear programming formulation for reliable and efficient parameter estimation of these models, including the spread within communities and the spread between communities. These estimation problems include very large-scale models and a large parameter space. Therefore, we require a scalable optimization approach and a sufficiently large data set. We present an interior-point internal decomposition strategy[4] that allows efficient solution of these large estimation problems in parallel, exploiting both spatial and temporal structure in the model. Estimation is performed over two extensive sets of data. The first, from England and Wales, includes weekly reported cases of scarlet fever, whooping cough, measles, and others diseases for hundreds of cities. The second set of data, from Thailand, includes monthly case counts and annual age distributions of various infectious diseases (including measles, chickenpox, influenza, and dengue) from 1972 to present. In addition to advancing our understanding of the fundamental drivers through estimation from case data, these improved models open the door for optimal planning of control programs using rigorous dynamic models in a mathematical programming framework.

[1] B.F. Finkenstadt and B.T. Grenfell. Time series modelling of childhood diseases: a dynamical systems approach. Journal of the Royal Statistical Society, Series C, 49:187–205, 2000.

[2] D.A.T. Cummings, R.A. Irizarry, N.E. Huang, T.P. Endy, A. Nisalak, K. Ungchusak, and D.S. Burke. Travelling waves in the occurrence of dengue haemorrhagic fever in thailand. Nature, 427(6972):344–347, 2004.

[3] C. Viboud, O. N. Bjornstad, D. L. Smith, L. Simonsen, M. A. Miller, and B.T. Grenfell. Synchrony, Waves, and Spatial Hierarchies in the Spread of Influenza . Science, 12:447 – 451, 5772.

[4] Laird, C. D., Biegler, L. T., “Large-Scale Nonlinear Programming for Multi-scenario Optimization”, accepted for publication in proceedings of the International Conference on High Performance Scientific Computing, Hanoi, Vietnam, 2006.