Fuzzy-SIRD model: Forecasting COVID-19 death tolls considering governments intervention

Abstract

Modeling the trend of contagious diseases has particular importance for managing them and reducing the side effects on society. In this regard, researchers have proposed compartmental models for modeling the spread of diseases. However, these models suffer from a lack of adaptability to variations of parameters over time. This paper introduces a new Fuzzy Susceptible-Infectious-Recovered-Deceased (Fuzzy-SIRD) model for covering the weaknesses of the simple compartmental models. Due to the uncertainty in forecasting diseases, the proposed Fuzzy-SIRD model represents the government intervention as an interval type 2 Mamdani fuzzy logic system. Also, since society's response to government intervention is not a static reaction, the proposed model uses a first-order linear system to model its dynamics. In addition, this paper uses the Particle Swarm Optimization (PSO) algorithm for optimally selecting system parameters. The objective function of this optimization problem is the Root Mean Square Error (RMSE) of the system output for the deceased population in a specific time interval. This paper provides many simulations for modeling and predicting the death tolls caused by COVID-19 disease in seven countries and compares the results with the simple SIRD model. Based on the reported results, the proposed Fuzzy-SIRD model can reduce the root mean square error of predictions by more than 80% in the long-term scenarios, compared with the conventional SIRD model. The average reduction of RMSE for the short-term and long-term predictions are 45.83% and 72.56%, respectively. The results also show that the principle goal of the proposed modeling, i.e., creating a semantic relation between the basic reproduction number, government intervention, and society's response to interventions, has been well achieved. As the results approve, the proposed model is a suitable and adaptable alternative for conventional compartmental models.

Keywords: COVID-19 pandemic; Compartmental models; Epidemiologic time series; Fuzzy logic systems; Interval type 2 fuzzy logics.

Fig. 1

Block diagram representation of the Interval Type 2 Fuzzy Logic Systems .

Fig. 2

Block diagram representation of the proposed Fuzzy-SIRD model.

Fig. 3

The gaussian interval type 2 fuzzy sets with uncertain standard deviation (std.) values used for defining the fuzzy sub-system.

Fig. 4

Block diagram representation of the relation between the PSO algorithm and the proposed Fuzzy-SIRD model.

Fig. 5

The output plot of the fuzzy sub-system for different $\varepsilon$ values, $D^{sat}=100$, and $\frac{dD}{dt}$ in the interval $\left[-150,150\right]$.

Fig. 6

The 3D representation of the fuzzy sub-system’s output plane.

Fig. 7

The results achieved by Fuzzy-SIRD model in the learning and prediction phases, the corresponding error, infectious and recovered population, $R_e$, and output of the fuzzy system (fuzzy $R_e$) over the short time horizon for the US.

Fig. 8

The results achieved by SIRD model in the learning and prediction phases, infectious and recovered population over the short time horizon for the US.

Fig. 9

The results achieved by Fuzzy-SIRD model in the learning and prediction phase, the corresponding error, infectious and recovered population, $R_e$, and output of the fuzzy system (fuzzy $R_e$) over the long time horizon for the US.

Fig. 10

The results achieved by SIRD model in the learning and prediction phases, infectious and recovered population over the long time horizon for the US.

Fig. 11

The convergence diagram of the PSO algorithm (average of 100 runs) for identifying the parameters of the proposed system for the US in a 150-day time horizon.

Fig. 12

The results achieved by Fuzzy-SIRD model in the learning and prediction phase, the corresponding error, infectious and recovered population, $R_e$, and output of the fuzzy system (fuzzy $R_e$) over the short time horizon for Germany.

Fig. 13

The results achieved by SIRD model in the learning and prediction phases, infectious and recovered population over the short time horizon for Germany.

Fig. 14

The results achieved by Fuzzy-SIRD model in the learning and prediction phase, the corresponding error, infectious and recovered population, $R_e$, and output of the fuzzy system (fuzzy $R_e$) over the long time horizon for Germany.

Fig. 15

The results achieved by SIRD model in the learning and prediction phases, infectious and recovered population over the long time horizon for Germany.

Fig. 16

The results achieved by Fuzzy-SIRD model in the learning and prediction phase, the corresponding error, infectious and recovered population, $R_e$, and output of the fuzzy system (fuzzy $R_e$) over the short time horizon for Brazil.

Fig. 17

The results achieved by SIRD model in the learning and prediction phases, infectious and recovered population over the short time horizon for Brazil.

Fig. 18

The results achieved by Fuzzy-SIRD model in the learning and prediction phase, the corresponding error, infectious and recovered population, $R_e$, and output of the fuzzy system (fuzzy $R_e$) over the long time horizon for Brazil.

Fig. 19

The results achieved by SIRD model in the learning and prediction phases, infectious and recovered population over the long time horizon for Brazil.

Fig. 20

The results achieved by Fuzzy-SIRD model in the learning and prediction phase, the corresponding error, infectious and recovered population, $R_e$, and output of the fuzzy system (fuzzy $R_e$) over the short time horizon for Iran.

Fig. 21

The results achieved by SIRD model in the learning and prediction phases, infectious and recovered population over the short time horizon for Iran.

Fig. 22

The results achieved by Fuzzy-SIRD model in the learning and prediction phase, the corresponding error, infectious and recovered population, $R_e$, and output of the fuzzy system (fuzzy $R_e$) over the long time horizon for Iran.

Fig. 23

The results achieved by SIRD model in the learning and prediction phases, infectious and recovered population over the long time horizon for Iran.

Fig. 24

The results achieved by Fuzzy-SIRD model in the learning and prediction phase, the corresponding error, infectious and recovered population, $R_e$, and output of the fuzzy system (fuzzy $R_e$) over the short time horizon for Italy.

Fig. 25

The results achieved by SIRD model in the learning and prediction phases, infectious and recovered population over the short time horizon for Italy.

Fig. 26

The results achieved by Fuzzy-SIRD model in the learning and prediction phase, the corresponding error, infectious and recovered population, $R_e$, and output of the fuzzy system (fuzzy $R_e$) over the long time horizon for Italy.

Fig. 27

The results achieved by SIRD model in the learning and prediction phases, infectious and recovered population over the long time horizon for Italy.

Fig. 28

The results achieved by Fuzzy-SIRD model in the learning and prediction phase, the corresponding error, infectious and recovered population, $R_e$, and output of the fuzzy system (fuzzy $R_e$) over the short time horizon for Russia.

Fig. 29

The results achieved by SIRD model in the learning and prediction phases, infectious and recovered population over the short time horizon for Russia.

Fig. 30

The results achieved by Fuzzy-SIRD model in the learning and prediction phase, the corresponding error, infectious and recovered population, $R_e$, and output of the fuzzy system (fuzzy $R_e$) over the long time horizon for Russia.

Fig. 31

The results achieved by SIRD model in the learning and prediction phases, infectious and recovered population over the long time horizon for Russia.

Fig. 32

The results achieved by Fuzzy-SIRD model in the learning and prediction phase, the corresponding error, infectious and recovered population, $R_e$, and output of the fuzzy system (fuzzy $R_e$) over the short time horizon for UK.

Fig. 33

The results achieved by SIRD model in the learning and prediction phases, infectious and recovered population over the short time horizon for UK.

Fig. 34

The results achieved by Fuzzy-SIRD model in the learning and prediction phase, the corresponding error, infectious and recovered population, $R_e$, and output of the fuzzy system (fuzzy $R_e$) over the long time horizon for UK.

Fig. 35

The results achieved by SIRD model in the learning and prediction phases, infectious and recovered population over the long time horizon for UK.