A Review of Compartmental Mathematical Models Used in Diabetology

This chapter is devoted to a review of compartmental mathematical models used for diabetes. Usually, compartmental models are used for communicable diseases for which dynamics is governed by the evolution of the disease based on the number of susceptible people that may become infected after a contact with infectious people. However, disease transmission is not a mandatory condition for the evolution from one compartment to another. In the case of diabetes, the main problem is not with diabetes but rather with complications of diabetes. Consequently, it is interesting to consider the evolution of diabetic people from the stage of diabetes without complication (compartment D) to the stage of diabetes with complications (compartment C) in order to see how to avoid or at least to delay as far as possible occurrence of complications. Stressing that dynamic should not be regarded in the usual way of transmitted diseases but rather as a probabilistic evolution from one stage to another, many authors have considered the evolution from pre-diabetes or diabetes without complications to diabetes with complications. Different mathematical tools were and are used, including discrete mathematics, ordinary differential equations, partial differential equations, optimal control, numerical approximations and stochastic approaches.