Table of contents:
2.1. The stochastic SIR epidemic model
4.1. Modeling the effects of the control measures
4.2. Model simulations with control and data
4.3. Model Validation and short-term prediction exercise with control measure
4.4. Growth rate and reproduction ratio
5.1. Model simulations and short term predictions
5.2. Cumulative cases and Incidences
5.3. Momentary growth rates (λ(t)) and momentary reproduction ratio (r(t))
5.4. Short term predictions of COVID-19 in the Basque Country
In December 2019, a new respiratory syndrome (COVID-19) caused by a new coronavirus (SARS-CoV-2), was identified in China  and spread rapidly around the globe. COVID-19 was declared a pandemic by the World Health Organization (WHO) in March, 2020 . Up to date, more than 40 million cases were confirmed with about 1.1 million deaths, with a global case fatality ratio (CFR) of approximate 3% .
Figure 1: COVID-19 pandemic figures. In a) bar chart race for the confirmed cases around the globe. The global COVID-19 deceased cases are shown in b) daily deceased cases and in c) total deceased cases.
In March 2020, a Multidisciplinary Task Force (so-called Basque Modelling Task Force, BMTF) was created to assist the Basque Health managers and the Basque Government during the COVID-19 responses. BMTF is a modeling team, working on different approaches, including stochastic processes, statistical methods and artificial intelligence. Members were collaborating taking into consideration all information provided by the public health frontline and using different available datasets in respect to the COVID-19 outbreak in the Basque Country. The objectives were, besides projections on the national health system’s necessities during the increased population demand on hospital admissions, the description of the epidemic in terms of disease spreading and control, as well as monitoring the disease transmission when the country lockdown was gradually lifted. All modeling approaches are complementary and are able to provide coherent results, assuring that the decisions made using the modeling results were sound and, in fact, adjusted to the current epidemiological data.
In this webpage we describe and present the results obtained by one of the modeling approaches developed within the BMTF, specifically using extended versions of the basic epidemiological SIR-type models, able to describe the dynamics observed for tested positive cases, hospitalizations, intensive care units (ICUs) admissions, deceased and finally the hospital discharges usd as a proxy for a proportion of recovered individuals. We start presenting the properties of the basic SIR epidemic model for infectious diseases , with the goal to introduce notation, terminology, and results that will be generalized in later sections on more advanced models describing the COVID19 epidemiology. Graphics will be updated every two weeks.
The SIR epidemic model divides the population into three classes: susceptible (S), Infected (I) and Recovered (R). It can be applied to infectious diseases where waning immunity can happen, and assuming that the transmission of the disease is contagious from person to person, the susceptibles become infected and infectious, are cured and become recovered. After a waning immunity period, the recovered individual can become susceptible again .
In the simple SIR epidemics without strain structure of the pathogens we have the following reaction scheme, for the possible transitions from one to another disease related state, susceptibles S, infected I and recovered R is shown in Fig. 2 a). For a host population of N individuals, with contact and infection rate β, recovery rate γ and waning immunity rate α, the dynamic model in terms of ordinary differential equations are shown in 2b), and the dynamical behavior for each variable is shown in Figure 2c).
The stochastic SIR epidemic is modeled as a time-continuous Markov process to capture population noise. The dynamics of the probability of integer infected and integer susceptibles, while the recovered follow from this due to constant population size, can be give as a master equation  in the following form:
This process can be simulated by the Gillespie algorithm giving stochastic realizations of infected and susceptibles in time [10,11]. The deterministic approach is obtained via the mean field approximation. For more details on the calculations see e.g. .
How does the program work: a short guide
Choose the parameters values for your simulation. Each susceptible individual is surrounded by 8 neighbors. "P" defines the probability of infection for each iteration. "Population Immunity" defines how many susceptible people exist in the population, and "Mortality" refers to the disease induced death probability.
- Color code: green cells represent susceptible individuals, red cells represent infected individuals, blue cells represent recovered and immune individuals, and black cells represent deceased individuals.
- Interpretation: when the simulation starts, susceptible individuals (green cells) become infected randomly and cells turn red. For each iteration, new infection might occur within the 8 neighbors, with probability "P" of infecting someone. For example, if P = 50%, on average each cell will infect 4 neighbors per iteration. Individuals are infected during the "Infected Period" and become recovered and immune (blue cells). For disease induced death (black cells), infected cells have a probability of "dying" at each iteration given by 1 - (1 - μ)^ (1 / p), where μ is the mortality rate and p is the infected period.
- Output: On the right hand side, model output in terms of time series, for each variable, susceptible, infected, recovered/immune and deceased.
We develop a basic SHAR-model including now a severity ratio η for susceptible individuals (S) developing severe disease and possibly being hospitalized H or (1- η ) for milder disease, including sub-clinical and eventually asymptomatic (A) infections, where mild infected A have different infectivity from severe hospitalized disease H, parametrized by a ratio Φ to be smaller or larger than 1 comparing to baseline infectivity rate β of the “hospitalized” H class and altered infectivity rate Φβ for mild/“symptomatic” A class. Hence we obtain a SHAR-type model
that needs to be further refined to describe COVID-19 dynamics, adjusting the modelling framework to the available empirical data.