## 6 Conclusions and Future Work  
In this work, we proposed the SEIR-H epidemiological model with time-varying parameters in the form of step functions. This model is based on the classical SEIR model maintaining its simplicity in the spread dynamics of the disease. However, the SEIR-H model contains the dynamics of the occupations of normal beds and ICU in hospitals in order to guide decisions based on available hospital resources. This model was designed to have as few parameters as possible to be estimated, while still representing the dynamics observed in Paraguay.  
We summarise the contributions of this work as follows:  
* Integration of health system dynamics into the SEIR model, without losing the simplicity of the SEIR model.
* The methodology of parameter estimation for the time-varying parameters using a moving time-window, which allows us to determine the variability of the parameters according to the spread dynamics and social behaviour.
* The inclusion of the reported cases that were travellers from abroad in the inflow of reported cases, which is important at the beginning of an epidemic.
* Model assessment and statistical analysis of the error as a function of the projection horizon.  
* Discussion of spread dynamics in Paraguay using the estimated parameters and trajectories obtained.  
The proposed model has two main limitations. The first limitation is the assumption that all cases end up being reported ignoring the under-reporting of cases, which can be quite significant [36]. The second limitation is to ignore the limits of hospital capacity, both in terms of available care units for medium and severe cases. This issue is especially important for projections of deaths when the health system is overburdened [37]. Future work can address these aspects by adding a model for under-reported cases and by limiting the capacity of the health system.  
Although, the SEIR-H model is fitted to the case of Paraguay, this model can be useful for any country or region to have a macroscopic overview of the dynamics of an epidemic like COVID-19, especially in contexts of limited availability of data detailing the heterogeneity of the population.  
Conceptualization, H.H.S. and C.S.A.; methodology, H.H.S., C.S.A., S.G. and P.P.-E.; software, H.H.S., P.P.-E. and C.G.; validation, H.H.S., C.S.A., S.G., P.P.-E., L.S.-C. and G.S.; formal analysis, H.H.S., C.S.A., S.G., E.D.L.S. and P.P.-E.; investigation, H.H.S., C.S.A., S.G. and P.P.-E.; resources, G.S. and L.S.-C.; data curation, H.H.S., C.S.A., P.P.-E., L.S.-C., S.I., M.E.P. and G.S.; writing--original draft preparation, H.H.S., C.S.A., S.G. and P.P.-E.; writing--review and editing, H.H.S., C.S.A., S.G., L.S.-C., M.G.-T., G.S., E.D.L.S. and J.L.V.N.; visualization, H.H.S., C.S.A., C.G. and P.P.-E.; supervision, H.H.S. and J.L.V.N.; project administration, H.H.S. and C.S.A.; and funding acquisition, J.L.V.N. and P.P.-E. All authors have read and agreed to the published version of the manuscript.  
This publication has been prepared with the support of CONACYT and UAA, within the grant PINV20-40. The contents of this publication are the sole responsibility of the authors and cannot be taken to reflect the opinion of CONACYT.  
Publicly available datasets from the Ministry of Public Health and Social Welfare (MSPyBS: Ministerio de Salud Publica y Bienestar Social) of Paraguay were analyzed in this study. This data can be found here: [[https://public.tableau.com/app/profile/mspbs/viz/COVID19](https://public.tableau.com/app/profile/mspbs/viz/COVID19) PY-Registros/REGISTRODIARIO]([https://public.tableau.com/app/profile/mspbs/viz/COVID19](https://public.tableau.com/app/profile/mspbs/viz/COVID19) PY-Registros/REGISTRODIARIO). Date accessed: 6 June 2021.  
The authors declare no conflict of interest.