A simple but complex enough θ -SIR type model to be used with COVID-19 real data. Application to the case of Italy

Abstract

Since the start of the COVID-19 pandemic in China many models have appeared in the literature, trying to simulate its dynamics. Focusing on modeling the biological and sociological mechanisms which influence the disease spread, the basic reference example is the SIR model. However, it is too simple to be able to model those mechanisms (including the three main types of control measures: social distancing, contact tracing and health system measures) to fit real data and to simulate possible future scenarios. A question, then, arises: how much and how do we need to complexify a SIR model? We develop a $\theta$ -SEIHQRD model, which may be the simplest one satisfying the mentioned requirements for arbitrary territories and can be simplified in particular cases. We show its very good performance in the Italian case and study different future scenarios.

Keywords: θ -SEIQHRD model; COVID-19; Effective reproduction number; Numerical simulation; Parameter estimation; SARS-CoV-2.

Fig. 1

Diagram summarizing the model for COVID-19 given by system (1).

Fig. 2

Comparison of some of the outputs of the simulation run with $\omega =1.4555\text{\%}$ (Est.) and the Italian official reported data (Rep.). Top-left: Cumulative number of cases and deaths. Top-right: People in hospital. Bottom-left: People in quarantine. Bottom-right: Cumulative number of recovered people.

Fig. 3

Comparison of some of the outputs of the simulation run with $\omega =1.4555\text{\%}$ (Est.) and, when available, the Italian official reported data (Rep.). Top-left: New detected cases per day. Top-right: New detected deaths per day. Bottom-Left: Function modeling the social distancing measures. Bottom-Right: Effective reproduction number and the contribution of each infectious state (i.e. $E$, $I$, $I_{\mathrm{u}}$, $H_R$ and $H_D$) to this number. The vertical line corresponds to the first date when R${}_e<1$.

Fig. 4

Some of the outputs of the simulation run with $\omega =1.4555\text{\%},\omega =0.9145\text{\%}$ and 3.9559% for the Italian case. Top-left: Effective reproduction number. Top-right: Function modeling the social distancing measures. Bottom-left: Cumulative number of total cases (including undetected cases). Bottom-right: $\theta \left(t\right)$.

Fig. 5

Outputs obtained when simulating possible future scenarios (using ${\kappa }_8={\kappa }_2=62.0015$). Top-left: New detected people when $m_8=0.25$ and $\omega =0$.9145%, 1.4555% or 3.9559%, Top-right: Effective reproduction number when $\omega =1.4555\text{\%}$ and $m_8=0.15$, 0.25 or 0.4, Bottom-left: New detected cases per day, when $\omega =1.4555\text{\%}$ and $m_8=0.15$, 0.25 or 0.4, Bottom-right: New detected deaths per day, when $\omega =1.4555\text{\%}$ and $m_8=0.15$, 0.25 or 0.4.

Fig. 6

Outputs obtained when simulating possible future scenarios when $\omega =1.4555\text{\%}$. Top-left: Function modeling the social distancing measures. Top-right: Effective reproduction number. Bottom-left: New detected cases per day. Bottom-right: New detected deaths per day.