Nine challenges for deterministic epidemic models

• We investigate the endemic equilibrium and its stability.
• We discuss multi-strain infections, time-varying infectivity and superinfection.
• We consider the need for advances in spatial epidemic models.
• We lack models that connect communicable and non-communicable diseases.
• We explore the connection with stochastic systems.

Deterministic models have a long history of being applied to the study of infectious disease epidemiology. We highlight and discuss nine challenges in this area. The first two concern the endemic equilibrium and its stability. We indicate the need for models that describe multi-strain infections, infections with time-varying infectivity, and those where superinfection is possible. We then consider the need for advances in spatial epidemic models, and draw attention to the lack of models that explore the relationship between communicable and non-communicable diseases. The final two challenges concern the uses and limitations of deterministic models as approximations to stochastic systems.