Time-Dependent Infectivity and Flexible Latent and Infectious Periods in Compartmental Models of Plant Disease

Compartmental models have become the dominant theoretical paradigm in mechanistic modeling of plant disease and offer well-known advantages in terms of analytic tractability, ease of simulation, and extensibility. However, underlying assumptions of constant rates of infection and of exponentially distributed latent and infectious periods are difficult to justify. Although alternative approaches, including van der Plank's seminal discrete time model and models based on the integro-differential formulation of Kermack and McKendrick's model, have been suggested for plant disease and relax these unrealistic assumptions, they are challenging to implement and to analyze. Here, we propose an extension to the susceptible, exposed, infected, and removed (SEIR) compartmental model, splitting the latent and infection compartments and thereby allowing time-varying infection rates and more realistic distributions of latent and infectious periods to be represented. Although the model is, in fact, more general, we specifically target plant disease by demonstrating how it can represent both the van der Plank model and the most commonly used variant of the Kermack and McKendrick (K & M) model (in which the infectivity response is delay Gamma distributed). We show how our reformulation retains the numeric and analytic tractability of SEIR models, and how it can be used to replicate earlier analyses of the van der Plank and K & M models. Our reformulation has the advantage of using elementary mathematical techniques, making implementation easier for the nonspecialist. We show a practical implication of these results for disease control. By taking advantage of the easy extensibility characteristic of compartmental models, we also investigate the effects of including additional biological realism. As an example, we show how the more realistic infection responses we consider interact with host demography and lead to divergent invasion thresholds when compared with the "standard" SEIR model. An ever-increasing number of analyses purportedly extract more biologically realistic invasion thresholds by adding additional biological detail to the SEIR model framework; we contend that our results demonstrate that extending a model that has such a simplistic representation of the infection dynamics may not, in fact, lead to more accurate results. Therefore, we suggest that modelers should carefully consider the underlying assumptions of the simplest compartmental models in their future work.