**Author details**

### Raúl Fierro

24 Will-be-set-by-IN-TECH

apf rf

**Asymptotic and Estimated Power Function**

0 5 10 15

Delta

A wide class of discrete-time stochastic epidemic models was introduced and analyzed in this work, the convergence of the stochastic model to the deterministic one as the population increases was proved. Moreover, it was proved the convergence in distribution of a process, depending on the fluctuations between the stochastic and deterministic models, to the solution to a stochastic differential equation. The statistical analysis was focused in proving asymptotic normality of natural martingale estimators, for the parameters of the model, and these results were applied to hypothesis tests for the General Epidemic Model. As a consequence, a hypothesis test for the reproduction number was stated and numerical

The stochastic models considered here represent an alternative modeling to those using counting processes and having transitions occurring at random times. However, both are asymptotically consistent with their deterministic counterparts. The advantage of considering stochastic models at discrete times is that it is not necessary to observe the epidemic over a long period of time; however, in order to attain the asymptotical consistency mentioned, our model requires frequent observation when dealing with a large population. Another important conclusion is obtained from Subsection 4.2, where numerical simulations was carried out for the SIS epidemic model. Indeed, the simulation showed large fluctuations of the SIS stochastic epidemic model, for *n* = 10, regarding the deterministic one. However, by simulating the process for *n* = 1, 000 it could be appreciated the trajectories of the deterministic and stochastic models were quite closed. Moreover, Theorem 2 provides confidence bounds which give an insight of the fluctuations of the stochastic model regarding the deterministic one. The simulation in Subsection 4.2 for the SIS epidemic model shows the coherence between these bounds and the simulated process (Figure 3). As a conclusion, the deterministic and stochastic model are quite different for small populations, while for large populations both models perform similarly. However, it is worth noting, although the stochastic and deterministic trajectories are similar for large size populations, an attractor state for the deterministic model is not necessarily an attractor for the stochastic model. As shown

simulations validated the size populations under which the results are useful.

in Subsection 4.2, this is the case for the SIS epidemic model.

0.0 0.2 0.4 0.6 0.8 1.0

**Figure 6.** Grafics of *<sup>π</sup>*(Δ) and *<sup>π</sup><sup>n</sup>*(Δ) for a population size equals *<sup>n</sup>* <sup>=</sup> 300.

Probability of Rejection

**8. Conclusions**

*Instituto de Matemática, Pontificia Universidad Católica de Valparaíso, Centro de Investigación y Modelamiento de Fenómenos Aleatorios, Universidad de Valparaíso, Chile*
