Last modified: 2014-03-31

#### Abstract

In the present paper we will generalize a simple SIR type model defined by Tuckwell and Williams [TW]: Math. Biosci. 208 (2007), pp.76—97. All the SIR type models share the following structure: the population is divided into some classes which represent the state of the contagious. Usually with S we denote the class of susceptible individuals, by I that of infectious and by R that of the removed, which become immune. If one is interested in modeling diseases where an initial incubation period is present, as well as a period when the individual is infectious, but still not aware to be ill, a simple SIR model is no more adequate. For example, the varicella disease is poorly described by a SIR model (see also Ebola and Influenza). To overcome this problem, one can introduce two additional classes, in order to define what we will call a SEIHR models, where E denotes the class of the exposed individuals, who therefore are in the incubation non infectious period, and H that of the individuals which are infectious and sick, so often are hospitalized. Note that in our model the class I will just contain infectious individuals still not aware to be sick. The class E is clearly motivated by the necessity to include an incubation period, while the introduction of the class H allows us to study the effects of the quarantine procedure, applicable just in the moment that a given individual becomes sick.

Following [TW], we will assume that all the individuals but at least one are initially in the class S. The time is discretized and the unit length is one day. We assume that every day any susceptible individual meet a given number of different individuals and if one of these is infectious the diseases is transmitted with a given probability. When an individual enter in the class E, will stay in this class for a given, fixed number of days, then he will pass to the following class I, stay there for a fixed, possible different, given number of days and so on. Once in the class R he will be no more susceptible and will be ``removed'' by the system. In this way we will have a simple model of a disease with an incubation period as well as a period when the individual is infectious, but still not aware to be ill. We will derive an explicit structure for this discrete time Markov chain and we will simulate some possible scenarios. Furthermore, we will derive a simple diffusion approximation of this model. The resulting stochastic differential equation will present multiple delays, due to the presence of the additional classes E and H. We will see, via numerical simulation, how the discrete time and the diffusion models are very close and we will apply both the models to the case of the varicella disease, with a good description of the multi-peak behavior of this disease.