# Nine challenges for deterministic epidemic models

Roberts, Mick, et al. “Nine challenges for deterministic epidemic models.” *Epidemics* 10 (2015): 49-53.

Deterministic models have been used in infectious disease epidemiology. These models are usually considered to be easier to use and handle than stochastic models. However, many infectious disease systems can be represented as individual-based stochastic processes. Also, analysis of the deterministic often yields information about the stochastic system solution. However, the authors outline nine challenges that arise when using deterministic models.

1. Understanding the endemic equilibrium

Endemic equilibriums occur when there is a balance between the transmission of infection and the renewal of susceptible individuals. The effective reproduction number (*Reff*) is equal to 1. Determining the underlying equation for the *Reff* would allow us to understand the effect of empirically observed mixing patterns. The overall challenge is to use the renewal equation to understand the relationship between *Reff* and R0.

2. Defining the stability of the endemic equilibrium

Typically, it is relatively straightforward to determine small amplitude linear perturbations from the equilibrium, however these characteristic equations are too complex to provide general stability results. For some deterministic models, finding and understanding the general paradigm for stability of endemic equilibrium.

3. Modeling multi-strain systems

Most models of strain dynamics have been adaptations of either the quasi-species model of mutation selection balance and the competitive exclusion principle. They include competition directly between strains in the host or indirect competition through a shared resource. We should extend these models to include a greater diversity of strains.

4. Modeling time-varying infectivity

Compartmental models are typically used in deterministic models, which involve constant transition rates between compartments. However, these models lack generality and removes the possibility of embedding more realistic infectivity profiles. Time-since-infection models are models which assume that individuals have a time-varying infectivity profile. Extending the time-since-infection framework to structured population of increasing complexity is the challenge.

5. Modeling superinfection

Superinfections occur when a host is infected with a heterologous strain of a pathogen they have yet to clear. Most deterministic microparasite models do not allow individuals to become infected a second time before recovering. We should work towards developing the general theory for superinfection modeling.

6. Constructing realistic spatially explicit models

Network models represent social contacts in space. In the definition of R0, spatially explicit models highlight the use of typical. An alternative to the spatial modeling paradigm is the metapopulation structures.

7. Exploring the interaction with non-communicable diseases

Cancers are typically considered to be non-communicable diseases. Thre may be an interaction between the pathogen or parasite. New mathematical models should be developed and used to evaluate the role that transmission agents have on developing the non-communicable diseases.

8. Defining the limitations of deterministic models

There are inherit limitations when using deterministic models. However, they often have been used without consider the full limitations of them. These may lead to misleading conclusions.

9. Developing robust approximations for stochastic models