Study with the support of Future Healthcare

The study entitled **“The Multi-Compartment SI(RD) Model with Regime Switching: An Application to COVID-19 Pandemic”** was published in the journal Symmetry (2021), no. 13, pp. 2427 ( https://www.mdpi.com/2073-8994/13/12/2427 ) with financial support from Future Healthcare.

The main results of the work are as follows:

- there was a reduction in the number of deaths associated with infection when an infection-mitigating measure was imposed, in this case, confinement;
- the model returns the number of deaths associated with infection in remarkable agreement with the observed data where there is a 2% error in excess of this observed data;
- for the asymptomatic or undiagnosed, the most plausible value of the number of contacts – the average number of new infections that an infected person generates – is 50% of the number of contacts of a diagnosed one;
- for the asymptomatic or undiagnosed, the most plausible mortality rate – caused by the disease – is zero;
- the most plausible value for the duration of the disease is 7.5 days.

Of the study’s two fundamental contributions, the first, from the perspective of usefulness to the work community, is the detailed study of the first wave of the epidemic of COVID-19 in Portugal in the year 2020. The study aims to explain the data observed in two temporal series: the number of cases and lethality, i.e. the number of deaths associated with the disease in those infected. To do so, we use a compartment model corresponding to classes in the total population – namely: susceptible to be infected, infected, recovered, and dead – a model which, we emphasize, includes the class of the dead associated with the disease; besides this additional class that we introduce to a classic model – the SIR model studied by many authors, among which we highlight Kermack and McKendric in 1927 – the model explored in this work allows that the conditions conditioning the change between classes of the elements of the population may change at a given moment. The change in these conditions, which in the case of our study occurs with the imposition of confinement, leads to a change in the model. The models of the type considered have parameters that govern the evolution of the changes of the elements of the population between classes, for example, the mortality rate among the infected. In the study these parameters are considered constant before the confinement, but from the date of confinement they change to other values that then remain constant during the period in which there is confinement. To be able to explore the model, the parameters have to be estimated from the observed data – data of the evolution of the number of cases and lethality – in each of the periods, before and after the confinement. The work introduces a method for estimating the parameters that is highly effective for the type of observed data available.

The second essential contribution is the introduction and study of a model of the evolution of a population that faces an epidemic and is divided into classes – or compartments – namely: susceptible, infected, recovered, and dead; the model introduced has an original component which is that of the regime shift. The model is a differential equation model that describes the evolution over time of the proportions of the different classes of the elements of the population in which, in a given regime, the coefficients are constant and a shift of regime occurs when the coefficients change. The shift of regime corresponds, in the case considered, to a sudden change in the conditions that affect the population’s reactions to infection; specifically, confinement, by substantially reducing contacts between members of the population, also reduced infections; there is a parameter in the model that describes the intensity of infections that, naturally, must be adjusted with confinement. The examination of this model undertaken in the study consists essentially in showing that it is a “well-posed” problem – that the system of differential equations has a solution and that this solution is unique; a result of this type is essential when, as is the case in the study, the solutions of the system of equations are calculated numerically, since it is impossible to verify that functions that can only be calculated numerically, with the use of appropriate software, are solutions of a system of differential equations. If the system of differential equations of the model admits a unique solution, provided that the process of discretization of the differential equations leads to a succession of functions that converge to a solution, we can be sure that the result obtained corresponds to an approximation of the unique solution.

In the light of these results, we can conclude that the model explored in this work allows an adequate description of the epidemic period under consideration, demonstrating the influence of confinement on the reduction of mortality associated with the disease; it also allows conclusions to be drawn about unobserved quantities in the period, such as mortality among the infected who were not diagnosed or were asymptomatic, the average duration of the disease, and the mortality rate among the infected who were not diagnosed or were asymptomatic. As a final consideration, we underline that the model studied is very useful for retrospective studies, but it does not allow us to predict the future evolution of the epidemic; the reasons for this inability to allow predictions are, essentially, that each new phase of the epidemic occurs in a population that will have a different reaction to the infection from the reaction of the population in the previous phase of the epidemic, given that a significant portion of the population has already been infected by the virus; additionally, each new phase of the epidemic occurs with a variant of the virus from the previous episode, a variant that interacts differently with the population – it is generally observed in viral infections of this type, recurring several times in a year, that the new variant of the virus is more contagious and less lethal than the previous variant.

More details of this study are available at https://www.mdpi.com/2073-8994/13/12/2427

**Acknowledgments:** This work was published with financial support from Future HealthCare. The authors would like to thank Future HealthCare for their interest in the development of LTC in Portugal and for the essential financial support for the publication of this work.