Interval type-2 Fuzzy control and stochastic modeling of COVID-19 spread based on vaccination and social distancing rates

Abstract

Background and objective

Besides efforts on vaccine discovery, robust and intuitive government policies could also significantly influence the pandemic state. However, such policies require realistic virus spread models, and the major works on COVID-19 to date have been only case-specific and use deterministic models. Additionally, when a disease affects large portions of the population, countries develop extensive infrastructures to contain the condition that should adapt continuously and extend the healthcare system's capabilities. An accurate mathematical model that reasonably addresses these complex treatment/population dynamics and their corresponding environmental uncertainties is necessary for making appropriate and robust strategic decisions.

Methods

Here, we propose an interval type-2 fuzzy stochastic modeling and control strategy to deal with the realistic uncertainties of pandemics and manage the size of the infected population. For this purpose, we first modify a previously established COVID-19 model with definite parameters to a Stochastic SEIAR (S²EIAR) approach with uncertain parameters and variables. Next, we propose to use normalized inputs, rather than the usual parameter settings in the previous case-specific studies, hence offering a more generalized control structure. Furthermore, we examine the proposed genetic algorithm-optimized fuzzy system in two scenarios. The first scenario aims to keep infected cases below a certain threshold, while the second addresses the changing healthcare capacities. Finally, we examine the proposed controller on stochasticity and disturbance in parameters, population sizes, social distance, and vaccination rate.

Results

The results show the robustness and efficiency of the proposed method in the presence of up to 1% noise and 50% disturbance in tracking the desired size of the infected population. The proposed method is compared to Proportional Derivative (PD), Proportional Integral Derivative (PID), and type-1 fuzzy controllers. In the first scenario, both fuzzy controllers perform more smoothly despite PD and PID controllers reaching a lower mean squared error (MSE). Meanwhile, the proposed controller outperforms PD, PID, and the type-1 fuzzy controller for the MSE and decision policies for the second scenario.

Conclusions

The proposed approach explains how we should decide on social distancing and vaccination rate policies during pandemics against the prevalent uncertainties in disease detection and reporting.

Nomenclature

Variables

S

Susceptible people

E

Exposed people

I

Infected people

A

Asymptomatic people

R

Recovered people

Parameters

β

The rate of susceptible people

ε

Exposed people's infectivity reduction

q

Contact reduction by isolation

δ

Asymptomatic people's infectivity reduction

ρ

Infection fraction of exposed people

κ

The infection rate of exposed people

z

The recovery fraction of asymptotic people

η

The recovery rate of asymptotic people

α

The recovery rate of infected people

f

The recovery fraction of infected people

1. Introduction

The COVID-19 outbreak started in December 2019 in Wuhan, China, and soon became a worldwide pandemic [1]. Numerous studies have been performed so far on different aspects of COVID-19 spread such as its detection and treatment strategies [2], mortality prediction [3] and disease detection [4] from X-Ray images, face mask improper wearing detection by deep-learning methods [5], and COVID-19 time series prediction [6,7]. There are also efforts on population modeling for policy planning purposes. Modeling the disease growth trend, considering the influential factors in changing this trend, and examining people's behavior in the community are the first steps in predicting growth stages. Such mathematical modeling could also develop appropriate policies in the face of unforeseen conditions of the COVID-19 pandemic. However, due to real-world uncertainties, a realistic mathematical model of pandemics and a robust control policy to respond to dynamic changes are highly challenging.

Various mathematical models have been developed to model and analyze COVID-19 spreading [8,9] and community behavior [1]. Most of them are devoted to introducing mathematical models for COVID-19 spreading. In [10], a model containing suspected, asymptomatic, unreported symptomatic, reported symptomatic, and recovered groups are developed. It is assumed that unreported deterministic population changes with a fixed rate from infected individuals. In [11], a Susceptible Infectious Dead Recovered (SIDR) model is fitted to the various data cases. In [12], a simulation is performed for the Susceptible Infectious Recovered (SIR) people considering variational infection rates and contact per day. In [13], stochasticity is inserted into the state of each individual (suspected, infected, and recovered) in the population.

Moreover, several works are devoted to analyzing the community's behavior to consider social distancing. In [14], the idea of the social distancing effect on the disease dynamic is explored by game theory. In [15], a community's social behavior against social distancing and vaccination during the COVID-19 spread is analyzed using game theory. In [16], a stochastic Susceptible Exposed Infectious Recovered (Re-infected) Deceased-based Social Distancing (SEIR(R)D-SD) model is presented considering random changes of S, E, I, and D. In [17], it is shown that social distancing is a more efficient way of infection rather than lockdown.

Besides the mentioned deterministic models, several works integrate the stochasticity of variables to address real-world daily life effects and reach beneficial information about the suspected, infected, and death trends. In [18], a Stochastic Susceptible Infected Vaccinated Removed (SIVR) model is analyzed under the variable stochasticity. This work shows that white noise remarkably affects COVID-19 controllability. Also, the stochastic SEIR models in [17] and [19] include random variables. In [20], a Susceptible Infected Recovered Protected Isolated (SIRCX) model is presented where the daily COVID-19 cases include stochastic components. In [21], a SIR model with stochastic variables of S, I, and R is presented. In [21], a Susceptible Exposed Quarantine Infected Recovered (SEQIR) model with stochastic variables is investigated.

Several studies have introduced stochastic models in which only model parameters are stochastic. For example, in [22], the parameters of an SEIR model, including seasonality and stochastic infection rates, are optimized by Particle Swarm Optimization (PSO). In [23], the SIR model is extended to deterministic and delayed stochastic models, where the parameters are perturbed in the stochastic model. Parameter perturbations are also applied for a SIR model [24], a SIDR model [25], and an SEIR model [26]. In [27], an uncertain SEIAR model is introduced, including human uncertainty factors. The uncertain parameters are then estimated using the number of COVID-19 cases.

Furthermore, recently several works have recommended the optimal level of vaccination in different phases of the COVID-19 pandemic [28]. In [29], an optimal controller is applied to the suspected, infected, and viral COVID-19 model. This paper examines two scenarios to address the minimization of the infected and viral group populations and antiviral drug usage in the least time. In [30], a COVID-19 epidemic model is developed considering migrating and impulsive traveling, and an optimal vaccination policy is then designed accordingly. In [31], a T-S fuzzy mathematical model is used for COVID-19 modeling, and linear quadratic regulator and mixed H₂−H_∞ optimization techniques are employed to control the disease's spread. In [32], non-pharmaceutical interventions are used to manage the sensitive parameters and design optimal control of COVID-19. In [33], vaccine administration is determined using two mono- and multi-objective optimal control strategies. Finally, in [34], the daily vaccination and testing rates are used to assess vaccination strategies by solving optimal control problems.

Several other works have performed controllers for the COVID-19 spread control. In [35], the authors reform the SEIR model [36] for control design and propose a robust controller to reduce the number of individuals exposed to the virus or infected by it. They also use a derivative-free optimization algorithm, the Most Valuable Player (MVP), to tune the control gains. With the same epidemiological model, a controller is designed via sliding-mode reference conditioning techniques to keep infected people below a predefined threshold in [37]. In [38], an SEIR model is developed for observing the trends in COVID-19 deaths due to limitations of the Intensive Care Unit (ICU) bed use. Authors in [39] focus on reducing the ICU occupancy rate. They approximate the Susceptible Exposed Infected Recovered Decreased (SEIRD) model [40] to a linear system with varying parameters and control lockdowns at five different levels. In [41], a model based on Susceptible Exposed Infected Hospitalized Recovered people (SEIHR) is developed and used to design a stable direct adaptive controller to mitigate the outbreak of COVID-19. In [41], a PID controller is applied to control the infection rate. In [42], the devised approach is composed of three main units: a Long Short-Term Memory (LSTM) neural network that estimates parameters of a discrete-time epidemiological model, an extended Kalman filter as a state observer, and a sliding-mode controller to stabilize the model and decrease the spread of the virus. Authors in [43] consider the Susceptible Quarantined Exposed Infected Asymptomatic Recovered (SQEIAR) model and its impulsive version. The aim is to find optimal controllers that minimize an objective function containing the model states.

There are also other studies on controlling COVID-19 diseases. In [44], populations of Susceptible Infected Quarantine Recovered (SIQR) individuals are used to model COVID-19. The Trust Region method is used to solve the nonlinear optimization problem of finding the parameters of SIQR. A feedback controller is designed based on the Lyapunov stability theory to guarantee the prescribed degree of exponential decay in the number of infections. In [45], a Susceptible-Infected Removed Contagious Quarantined-Threatened Healed Extinct (SIRCQTHE) model is designed for predicting the dynamics of COVID-19. Then, a stochastic nonlinear model predictive controller is introduced to determine the restrictions on the mobility of the social groups. In [46], a mathematical model of COVID-19 is presented based on the model in [47], and its equilibriums are derived. Then, an optimal controller is added to improve its performance.

Fuzzy systems are among the intelligent paradigms that incorporate human knowledge into the control design. They have been applied to various control and decision-making problems [48], [49], [50]. They have been also incorporated for controlling COVID-19 disease. For instance, in [51] and [52], COVID-19 is modeled by Susceptible Exposed Infected Asymptomatic Recovered (SEIAR) and Susceptible Infected Diagnosed Ailing Recognized Threatened Healed Extinct (SIDARTHE) individuals, respectively. In both works, an Adaptive-Network-based Fuzzy Inference System (ANFIS) is then presented for controlling the number of infected and suspected cases by vaccination and isolation. The ANFIS control system is optimized using the genetic algorithm. In [53], the COVID-19 epidemic is modeled by fractional order chaotic models. The models are then stabilized using appropriate fuzzy logic controllers. In [54], COVID-19 prediction is investigated by MATLAB Simulink. Both unsupervised and supervised learning strategies are used to construct a predictive-control system. The supervised learning models are fuzzy PID and wavelet neural-network PID learning. Finally, a fuzzy fractional controller is designed in [55] for controlling nonlinear systems, and as an example is applied to control the number of infected people in COVID-19 spread.

Type-2 fuzzy systems have three-dimensional fuzzy sets with fuzzy membership grades. When exposed to uncertainties in words, rules, or even data, they can outperform their type-1 counterparts [56]. They have been used in different applications and show superiority over type-1 fuzzy systems [50,57]. Different methods are also used for their parameter optimization such as genetic algorithm [58], particle swarm optimization [59], and Shark Smell Metaheuristic algorithm [48,60]. Hence, the type-2 fuzzy systems are expected to better handle uncertainties in COVID-19 modeling and control. However, they have not been incorporated to control COVID-19 infected population yet. Few studies have used type-2 fuzzy systems for COVID-19 time-series prediction. The works [6] and [7] incorporate type-2 fuzzy systems with LSTM and ensemble neural networks, respectively, for handling uncertainties in the COVID-19 time series prediction. Type-2 fuzzy systems have also been used in evaluating the risk of COVID-19 medical waste transportation [61], vaccination selection [57], and predicting the spreading behavior of COVID-19 [62].

Here, we aim to present a decision-making model based on type-2 fuzzy logic systems for controlling the number of infected people according to the vaccination rate and social distance policies. First, we employ the deterministic SEIAR model in [51] and upgrade it to a stochastic SEIAR (S²EIAR) model that includes parameters and variables’ uncertainties. Second, an interval type-2 fuzzy system is designed to robustly determine both the vaccination and social distancing rates for each day to control the number of infected individuals. The genetic algorithm is used to optimize the parameters of the type-2 fuzzy system. The type-2 fuzzy systems are chosen because of their higher abilities in handling uncertainties compared to the ordinary type-1 fuzzy systems. We incorporate normalized infected populations, which facilitates the extension of the proposed method to various regions and countries. Then, we design new and more realistic scenarios compared with previous studies. To the best of our knowledge, none of the earlier works consider the uncertainty in their models and solely try to form mathematical equations to analyze the overall dynamic of COVID-19. Here, we propose to include uncertainty in the model variables and parameters. Additionally, the work is among the pioneering works in incorporating fuzzy logic into the COVID-19 spread control.

The main contributions of this paper are hence summarized below:

• 1To the best of the authors’ knowledge, this is the first work to prepare a more realistic model by including randomness in both the model parameters and variables (S,  E,  I,  A,  R). However, the previous works often consider uncertainty only in one of its parameters or variables.
• 2An interval type-2 fuzzy system is proposed to robustly control the infected population, deal with the above randomness, and prepare an interpretation for control strategies of COVID-19 spread. But, previous studies mainly focused on type-1 fuzzy systems to handle uncertainties of parameters or variables lacking higher uncertainty handling of type-2 fuzzy systems [56].
• 3The controller is general and works based on the normalized infected population to the overall population. Therefore, it could be implemented in various countries, unlike previous case-specific controllers.
• 4A novel scenario is examined, allowing countries to plan for handling the pandemic based on their future infrastructure developments, compared to traditional scenarios that define fixed control goals (for example, infection rate should not exceed a value). We define the desired infection curve based on the capability of hospitals and the country's infrastructure. The fuzzy controller then manipulates the vaccination and social distance to track the desired curve. Besides, several experiments are performed to study the performance of the proposed controller under two scenarios with different values of parameter disturbances.

The paper's organization is as follows: Section 2 provides the details of the S²EIAR model and introduces the interval type-2 fuzzy system for deciding the vaccination rate and social distancing. Section 3 prepares the simulation results. Finally, discussions are drawn in Section 4.

2. Proposed methods

In this section, we first introduce the proposed uncertainty-embedded mathematical modeling of COVID-19 spread. The model is then used to optimize the proposed type-2 fuzzy controller using a genetic algorithm.

2.1. S²EIAR model

Here, we modify the SEIAR epidemiological model in [51] to include parameter and variable uncertainties and present a more realistic Stochastic SEIAR (S²EIAR) model as the mathematical model for the COVID-19 spread analysis. All the parameters, i.e., β, ε, q, δ, ρ, κ, z, η, α, and f, and variables, i.e., S, E, I, A, and R are not fixed and have uncertainties by adding small random numbers to their mean value. More specifically, the vectors $\underset{\_}{x}\left[k\right]={\left[S,E,I,A,R\right]}^T$and $\underset{\_}{\pi }={\left[\beta ,\epsilon ,q,\delta ,\rho ,\kappa ,z,\eta ,\alpha ,f\right]}^T$are perturbed by small random numbers as follows:

$$\begin{array}{ccc}\hfill \tilde{x}\left[k\right]& =& \underset{\_}{x}\left[k\right]+\mathfrak{N}\underset{\_}{n}\times \underset{\_}{x}\left[k\right],\hfill \\ \hfill \tilde{\pi }\left[k\right]& =& \underset{\_}{\pi }+\mathfrak{M}\underset{\_}{m}\times \underset{\_}{\pi },\hfill \end{array}$$ (1)

where $\underset{\_}{n}\in {\mathbb{R}}^5$ and $\underset{\_}{m}\in {\mathbb{R}}^{10}$ with its’ elements belonging to $\mathcal{N}\left(0,1\right)$, standing for uncertainty in real-world measurements in which $\mathcal{N}\left(0,1\right)$ is the normal distribution with mean 0 and variance 1, and ×  denotes the element-by-element product. The $\mathfrak{N}$ and $\mathfrak{M}$ are additive perturbation constants. As shown in Fig. 1 , the overall structure seems the same as the SEIAR model in [51].

Fig. 1.

The S²EIAR model diagram.

The main difference between S²EIAR and the SEIAR model is the uncertain variables and parameters. The above is used to model COVID-19, and then an intelligent, intuitive uncertainty-handler controller using fuzzy logic is designed to prevent the undesired dispersion of COVID-19. Also, we change the controller input from [I[k], S[k]] in [51] to [e(t), Δe[k]] that we discuss further in Section 3. The continuous dynamics of the proposed S²EIAR model is as follows:

$$\begin{array}{ccc}\hfill S\left[k+1\right]& =& \tilde{S}\left[k\right]-\tilde{\beta }\left[k\right]\left(\tilde{\epsilon }\left[k\right]+\left(1-\tilde{q}\left[k\right]\right)+\tilde{\delta }\left[k\right]\right)\tilde{S}\left[k\right]\hfill \\ & & -\tilde{S}\left[k\right]U_V\left(e\left[k\right],\Delta e\left[k\right]\right),\hfill \end{array}$$ (2)

$$\begin{array}{ccc}\hfill E\left[k+1\right]& =& \tilde{E}\left[k\right]+\tilde{\beta }\left[k\right]\left(\tilde{\epsilon }\left[k\right]\tilde{E}\left[k\right]+\left(1-\tilde{q}\left[k\right]\right)\tilde{I}\left[k\right]+\tilde{\delta }\left[k\right]\tilde{A}\left[k\right]\right)\tilde{S}\left[k\right]\hfill \\ & & -\tilde{\kappa }\left[k\right]\tilde{E}\left[k\right],\hfill \end{array}$$ (3)

$$\begin{array}{ccc}\hfill I\left[k+1\right]& =& \tilde{I}\left[k\right]+\tilde{\rho }\left[k\right]\tilde{\kappa }\left[k\right]\tilde{E}\left[k\right]+\left(1-\tilde{z}\left[k\right]\right)\tilde{\eta }\left[k\right]\tilde{A}\left[k\right]-\tilde{\alpha }\left[k\right]\tilde{I}\left[k\right]\hfill \\ & & -\tilde{I}\left[k\right]U_I\left(e\left[k\right],\Delta e\left[k\right]\right),\hfill \end{array}$$ (4)

$$A\left[k+1\right]=\tilde{A}\left[k\right]+\left(1-\tilde{\rho }\left[k\right]\right)\tilde{\kappa }\left[k\right]\tilde{E}\left[k\right]-\tilde{\eta }\left[k\right]\tilde{A}\left[k\right],$$ (5)

$$R\left[k+1\right]=\tilde{R}\left[k\right]+\tilde{z}\left[k\right]\tilde{\eta }\left[k\right]\tilde{A}\left[k\right]+\tilde{f}\left[k\right]\tilde{\alpha }\left[k\right]\tilde{I}\left[k\right],$$ (6)

$$N\left[k\right]=\tilde{S}\left[k\right]+\tilde{E}\left[k\right]+\tilde{I}\left[k\right]+\tilde{A}\left[k\right]+\tilde{R}\left[k\right],$$ (7)

where S[k] indicates the susceptible (uninfected), E[k] is the exposed (infected but not yet infectious), I[k] is infected (symptomatic), A[k] is the asymptomatic (infected without symptoms), and R[k] shows the recovered population. U_V(e[k],Δe[k]) is the vaccination rate, U_I(e[k],Δe[k]) is the isolation rate, and $e\left[k\right]=(I_d\left[k\right]-\tilde{I}\left[k\right])/N\left[k\right]$ in which I_d[k] is the desired infected population at time t.

2.2. Control of COVID-19

We use the interval type-2 Mamdani fuzzy inference system as a controller in which the error, e[k], and the error change, Δe[k], are applied to the fuzzy system. The fuzzy controller then determines the vaccination rate and social distancing policies to be applied to the community by the government. Hence, the fuzzy rules are presented as:

$$\text{If}e\left[k\right]\text{is}{\tilde{A}}_1^l\text{and}\Delta e\left[k\right]\text{is}{\tilde{A}}_2^l,\text{then}{U}_V\text{is}{\tilde{B}}_1^{l}and{U}_I\text{is}{\tilde{B}}_2^l,$$ (8)

where ${\tilde{A}}_1^l$ and ${\tilde{A}}_2^l$ are the interval type-2 fuzzy sets of the antecedent of the fuzzy system, and ${\tilde{B}}_1^l$ and ${\tilde{B}}_2^l$ are the interval type-2 fuzzy sets of the consequent of the l^th fuzzy rule with l = 1, …, M. The proposed fuzzy controller uses a singleton fuzzifier, minimum Mamdani inference engine, and Karnik-Mendel type-reduction. The overall structure is then attained as:

$$\begin{array}{c}\hfill \begin{array}{ccc}\hfill {\mathrm{F}}_{\mathcal{l}}\left(\underset{\_}{\mathrm{e}}|\mathrm{W}\right)& =& {\displaystyle \frac{{\sum }_{\mathrm{l}=1}^{\mathcal{L}}{\overline{\mathrm{W}}}_{\mathcal{l}}^{\mathrm{l}}{\overline{\mathrm{f}}}^{\mathrm{l}}+{\sum }_{\mathrm{l}=\mathcal{L}+1}^{\mathrm{M}}{\overline{\mathrm{W}}}_{\mathcal{l}}^{\mathrm{l}}{\underset{\_}{\mathrm{f}}}^{\mathrm{l}}}{{\sum }_{\mathrm{l}=1}^{\mathcal{L}}{\overline{\mathrm{f}}}^{\mathrm{l}}+{\sum }_{\mathrm{l}=\mathcal{L}+1}^{\mathrm{M}}{\underset{\_}{\mathrm{f}}}^{\mathrm{l}}}}={\mathrm{W}}_{\mathcal{l}}^{\mathrm{T}}{\varphi }_{\mathcal{l}},\hfill \\ \hfill {\mathrm{F}}_{\mathcal{r}}\left(\underset{\_}{\mathrm{e}}|\mathrm{W}\right)& =& {\displaystyle \frac{{\sum }_{\mathrm{l}=1}^{\mathcal{R}}{\overline{\mathrm{W}}}_{\mathcal{r}}^{\mathrm{l}}{\underset{\_}{\mathrm{f}}}^{\mathrm{l}}+{\sum }_{\mathrm{l}=\mathcal{R}+1}^{\mathrm{M}}{\overline{\mathrm{W}}}_{\mathcal{r}}^{\mathrm{l}}{\overline{\mathrm{f}}}^{\mathrm{l}}}{{\sum }_{\mathrm{l}=1}^{\mathcal{R}}{\underset{\_}{\mathrm{f}}}^{\mathrm{l}}+{\sum }_{\mathrm{l}=\mathcal{R}+1}^{\mathrm{M}}{\overline{\mathrm{f}}}^{\mathrm{l}}}}={\mathrm{W}}_{\mathcal{r}}^{\mathrm{T}}{\varphi }_{\mathcal{r}},\hfill \\ \hfill \mathrm{F}\left(\underset{\_}{\mathrm{e}}|\mathrm{W}\right)& =& {\displaystyle \frac{{\mathrm{F}}_{\mathcal{l}}\left(\underset{\_}{\mathrm{e}}|\mathrm{W}\right)+{\mathrm{F}}_{\mathcal{r}}\left(\underset{\_}{\mathrm{e}}|\mathrm{W}\right)}{2}}=\frac{1}{2}\left({\mathrm{W}}^{\mathrm{T}}\varphi \right),\hfill \end{array}\end{array}$$ (9)

where $\underset{\_}{e}={\left[e\left[k\right],\Delta e\left[k\right]\right]}^T$, ${\mu }_{{\tilde{A}}_1^l}\left(e\left[k\right]\right)$ and ${\mu }_{{\tilde{A}}_2^l}\left(\Delta e\left[k\right]\right)$ indicate the membership functions for the input to the sets ${\tilde{A}}_1^l$ and ${\tilde{A}}_2^l$, respectively. Also, ${\underset{\_}{f}}^l=\min \left({\underset{\_}{\mu }}_{A_1^l}\left(e\left[k\right]\right),{\underset{\_}{\mu }}_{A_2^l}\left(\Delta e\left[k\right]\right)\right)$, ${\overline{f}}^l=\min \left({\overline{\mu }}_{A_1^l}\left(e\left[k\right]\right),{\overline{\mu }}_{A_2^l}\left(\Delta e\left[k\right]\right)\right)$, where ${\underset{\_}{\mu }}_{A_1^l}\left(e\left[k\right]\right)$ and ${\underset{\_}{\mu }}_{A_2^l}\left(\Delta e\left[k\right]\right)$ indicate the lower membership functions of ${\mu }_{{\tilde{A}}_1^l}\left(e\left[k\right]\right)$ and ${\mu }_{{\tilde{A}}_2^l}\left(\Delta e\left[k\right]\right)$, respectively; and ${\overline{\mu }}_{A_1^l}\left(e\left[k\right]\right)$ and${\overline{\mu }}_{A_2^l}\left(\Delta e\left[k\right]\right)$ show the upper membership functions of ${\mu }_{{\tilde{A}}_1^l}\left(e\left[k\right]\right)$ and ${\mu }_{{\tilde{A}}_2^l}\left(\Delta e\left[k\right]\right)$, respectively. The terms ${\overline{W}}_{\mathcal{l}}^l$ and ${\overline{W}}_{\mathcal{r}}^l$ are the center of the interval sets in the consequent part of the l^th rule, $\mathcal{R}$ and $\mathcal{L}$ are switching points and satisfy ${\overline{W}}_{\mathcal{l}}^{\mathcal{L}}\le y_l\le {\overline{W}}_{\mathcal{l}}^{\mathcal{L}+1}$ and ${\overline{W}}_{\mathcal{r}}^{\mathcal{R}}\le y_r\le {\overline{W}}_{\mathcal{r}}^{\mathcal{R}+1}$, $W_{\mathcal{l}}^T=\left[{\overline{W}}_{\mathcal{l}}^1,\dots ,{\overline{W}}_{\mathcal{l}}^M\right]$, and $W_{\mathcal{r}}^T=\left[{\overline{W}}_{\mathcal{r}}^1,\dots ,{\overline{W}}_{\mathcal{r}}^M\right]$. Also, $\varphi =\left[{\varphi }_{\mathcal{l}},{\varphi }_{\mathcal{r}}\right]$ is the vector of fuzzy basis functions with${\varphi }_{\mathcal{l}}={\left[{\phi }_{\mathcal{l}}^1,{\phi }_{\mathcal{l}}^2,\dots ,{\phi }_{\mathcal{l}}^M\right]}^T$, ${\varphi }_{\mathcal{r}}={\left[{\phi }_{\mathcal{r}}^1,{\phi }_{\mathcal{r}}^2,\dots ,{\phi }_{\mathcal{r}}^M\right]}^T$, ${\phi }_{\mathcal{l}}^i=\left[{\overline{f}}^i/D_{\mathcal{l}},{\underset{\_}{f}}^i/D_{\mathcal{l}}\right]$, $D_{\mathcal{l}}={\sum }_{i=1}^{\mathcal{L}}{\overline{f}}^i+{\sum }_{i=\mathcal{L}+1}^M{\underset{\_}{f}}^i$, ${\begin{array}{c}\phi \end{array}}_{\mathcal{r}}^i=\left[{\underset{\_}{f}}^i/D_{\mathcal{r}},{\overline{f}}^i/D_{\mathcal{r}}\right]$, $D_{\mathcal{r}}={\sum }_{i=1}^{\mathcal{R}}{\underset{\_}{f}}^i+{\sum }_{i=\mathcal{R}+1}^M{\overline{f}}^i$, and $W={\left[W_{\mathcal{l}},W_{\mathcal{r}}\right]}^T$. Interval type-2 Mamdani fuzzy system toolbox of MATLAB2022 is used to simulate the proposed controller.

Fig. 2 shows the structure of the proposed type-2 fuzzy controller. In this structure, the difference between the desired value of infected and the currently infected individuals is divided by the total population. This scaling eliminates the need to reset the controller for different populations because the desired infected percent is the same in most governments. For this purpose, we need to normalize the infected population according to the current total population at each time step. Therefore, both the total population and infected people should be fed-backed to the controller. The proposed controller is optimized by a genetic algorithm (Algorithms 1 and 2).

Fig. 2.

Structure of type-2 fuzzy controller for S²EIAR model. The dashed line shows the optimization of the fuzzy system by genetic algorithm.

Algorithm 1.

Optimization procedure of the fuzzy controller for S²EIAR model.

Input	Fuzzy system ($\mathcal{F}$), Max generation ($\mathcal{G}$), Population size ($\mathcal{P}$), Mutation rate ($\mathcal{M}$), Crossover rate ($\mathcal{C}$).
Output	Optimized fuzzy system (${\mathcal{F}}_o$).
Initialization	Set model parameter, Set SEIR populations, Generate feasible solutions randomly.
Do
	Cost Evaluation
	Parent selection
	Crossover operation
	Mutation operation
While stop conditions are met.

Algorithm 2.

Cost function.

Input	Candidate fuzzy system (${\mathcal{F}}_c$), S2EIAR Model, Scenario, Number of simulation days ($\mathcal{D}$).
Output	Cost
For k in $\mathcal{D}$
	Simulation based on yesterday's variables.
	$\underset{\_}{e}\left[k\right]$ calculation
	Control signal generation: $U_I,U_V={\mathcal{F}}_c\left(\underset{\_}{e}\left[k\right]\right)$
End for
Return	MSE (Id, $\tilde{I}$)

3. Results

In this section, the performance of the proposed controller is investigated under two scenarios, and its capabilities are compared with PD, PID, and type-1 fuzzy controllers. The type-1 fuzzy controller has a structure and parameters similar to the proposed method. The standard values of the state variables are provided as S[0] = 0.9, I[0] = 0.1, N[0] = 1, and E[0] = A[0] = R[0] = 0. The model parameters are taken from [51,63], as shown in Table I . In the following, more details about scenarios are provided. The parameters $\mathfrak{N}$ and $\mathfrak{M}$ are considered 0.01 in simulations.

Table I.

standard values of the state variables and model parameters.

Parameter	Initial value	Parameter	Initial value
κ	0.54	f	0.965
ρ	0.1	ε	0
η	0.3	δ	1
z	0.02	q	0.5
α	0.3	β	1

3.1. Scenario 1

Hospital capacity is one of the most critical factors in controlling COVID-19 disease, which means that with an excessive increase in infected cases, the quality of public health is impaired, and the number of deaths increases. Therefore, in the first scenario, we aim to control the disease so that the number of patients is not higher than the critical level. Based on this, a constraint can be formulated as follows:

$$\tilde{I}\left[k\right]<I_{max},$$ (10)

where I_max is the critical level authorities provide to ensure sufficient healthcare system response.

3.2. Scenario 2

Due to the healthcare system's capacity, factors such as slowing the disease progression and flattening the infection curve are necessary. In other words, increasing the healthcare system's ability is a time-consuming process, leading to the increased importance of proper planning and implementing protocols. In [17], a SIR model for the COVID-19 outbreak is conducted under reinfection and limited medical resources scenarios. Despite such analysis, the disease progress must be controlled to provide sufficient time for protocol implementation. Therefore, in the second scenario, the intention is to design a controller that follows the curve desired by health officials.

As mentioned in Section 2, the controller inputs ([I[k], S[k]]) are replaced with [e[k], Δe[k]]. Previous works tried to control COVID-19 spread in optimal manners with no desired solution, while here, both scenarios consider the desired infected population according to the healthcare capacity. Therefore, a control policy based on the tracking error seems more reasonable when the desired values are available.

There are two Gaussian Membership Functions (MFs) in each dimension; hence the number of fuzzy rules is 2² = 4 for the proposed controller. The parameters of the input and output Gaussian MFs are initialized randomly in [-0.25,0.25] and [0,0.5], respectively. Then, the genetic algorithm with a population size of 1000 is used to optimize the fuzzy controller's parameters through 1000 generations.

The total simulation time is considered sixty days for both scenarios. The total population N is initially set to one, which may decrease over time because of the number of casualties. Furthermore, researchers have been monitoring the spread of different variants of coronavirus, such as Omicron, globally to understand its transmissibility better [64]. Thus, the robustness of the fuzzy controller should be studied in the face of different variants of COVID-19. For this purpose, in addition to the values given in Table I, we have tested the robustness of the model for κ = 0.25,  0.75, and q = 0.25,  0.75 without reoptimizing the controller.

There are different types of strategies for controlling a pandemic. In COVID-19, social distancing and vaccination are two essential factors in planning a strategy for controlling the pandemic. Fig. 3 shows the surface of the type-2 fuzzy system for both scenarios. Compared to the type-1 fuzzy controller (please see the Appendix), the type-2 fuzzy system has a smoother surface because of better uncertainty handling. As can be seen, the type-2 fuzzy system behaves differently in the face of these two scenarios; in other words, the goal in each scenario directly affects the control policy.

Fig. 3.

The control surface of the proposed type-2 fuzzy controller for the first and second scenarios. The goal in the first scenario is to prevent reaching the critical level; on the other hand, the second scenario follows the disease progression curve.

In the first scenario, as the error rate rises, the social distancing and vaccination rates increase; however, the error rate does not have a tangible impact on these two variables in the second scenario. In the first scenario, the vaccination rate increases by error reduction, while increasing the absolute value of the error changes the vaccination rate significantly in the second scenario.

In addition, as shown in Fig. 3, the vaccination and social distancing rates are inversely related in the second scenario, while they have synergistic relation in the first scenario. That is the increase of one variable results in the rise of the other. Also, at least one vaccination or social distancing control strategy is applied in almost all the errors and error rates for the second scenario. However, the fuzzy controller does not employ these two factors in several error and error rate values for the first scenario.

Fig. 4 depicts the infected population size (controlled, desired, and uncontrolled), vaccination rate, and isolation rate over time for different q and κ. As shown, the best performance is achieved for the standard parameter values in the first scenario (q = 0.5 and κ = 0.54). By increasing the isolation rate q, the objective for the first scenario is achieved by not reaching the maximum value of the critical level. On the other hand, with the growth of κ, the results may slightly cross the critical level that indicates the robustness of the controller.

Fig. 4.

Comparing the infected population size (controlled, desired, and uncontrolled), vaccination rate, and isolation rate over sixty days with different q and κ in the first scenario for the proposed type-2 fuzzy controller.

Fig. 5 shows the results for the second scenario. As is shown, the controller can follow the given curve with different parameter values. The rise in q or reduction of κ causes the under-control infected population to follow the given curve better. In all simulations, vaccination and social distancing values remain constant after passing the disease period due to the non-zero infection population caused by the model uncertainties.

Fig. 5.

Simulation results of the second scenario for various values of q and κ over sixty days for the proposed type-2 fuzzy controller.

Furthermore, the numeric analysis of these two scenarios can help the officials to choose effective actions. Therefore, Table II shows the Mean Squared Error (MSE) of PD, PID, type-1, and type-2 fuzzy controllers for both scenarios with different parameter values. In Scenario 1, despite classic controllers (PD and PID) showing lower MSE, comparing Figs 4,5, A1, and A3 (in Appendix) shows that fuzzy controllers perform smoothly in various situations and have the least limit passing, while the classic controllers aim to follow the critical level I_max. Thus, the smooth behavior of the fuzzy controllers is more desirable in real-world situations where governments aim to gently control the pandemic. Also, the results indicate that type-1 and type-2 fuzzy controllers have comparable results in Scenario 1, while the proposed type-2 fuzzy controller outperforms all three controllers (classic ones and the type-1 counterpart) in Scenario 2 by reaching significantly lower MSE values.

Table II.

MSE for scenarios with different values of q and κ

κ	q	MSE*
		1st Scenario				2nd Scenario
		PD	PID	Type-1 Fuzzy	Type-2 Fuzzy	PD	PID	Type-1 Fuzzy	Type-2 Fuzzy
0.25	0.25	5.6	5.7	6.1	6.1	0.7	1.1	0.25	0.26
0.25	0.5	5.5	5.6	6.4	6.2	0.7	1.1	0.43	0.28
0.25	0.75	5.5	5.6	6.7	6.3	0.6	1.2	0.73	0.32
0.54	0.25	5.7	5.8	5.7	6.1	0.8	1.1	0.28	0.26
0.54	0.5	5.6	5.8	6.1	6.1	0.8	1.1	0.31	0.26
0.54	0.75	5.6	5.7	6.2	6.2	0.7	1.2	0.42	0.28
0.75	0.25	5.7	5.9	6.1	6.1	0.9	1.1	0.50	0.29
0.75	0.5	5.7	5.8	6.0	6.2	0.8	1.1	0.26	0.26
0.75	0.75	5.7	5.8	6.4	6.2	0.8	1.2	0.26	0.26

^⁎multiplied by 1000.

Fig. A1.

Comparing the infected population size (controlled, desired, and uncontrolled), vaccination rate, and isolation rate over sixty days with different q and κ in the first scenario for the competing PD controller

Fig. A3.

Comparing the infected population size (controlled, desired, and uncontrolled), vaccination rate, and isolation rate over sixty days with different q and κ in the first scenario for the competing PID controller.

4. Discussion

This paper investigates the COVID-19 pandemic control using a fuzzy control policy based on a novel SEIAR model. The proposed S²EIAR includes stochastic behaviors to provide more realistic information for the post-modeling decision stage. The proposed type-2 fuzzy controller then uses this stochastic model to manage the spread of the disease. The inputs to the controller are normalized to provide a general control structure, and the fuzzy control parameters are optimized using a genetic algorithm. A more realistic scenario is also tested to validate the fuzzy controller's performance.

Our work considers uncertainty in the information during the disease reporting process rather than disease biological spreading (Fig. 6 green part). In other words, several works deal with the infection rate of exposed people to the disease [52] while we focus on the uncertainty of afterward reporting (after infection). This latter uncertainty often happens because of a widespread infection, poor detection technologies, and inadequate access to all infected people for proper reporting. In this regard, a possible direction of future works could be to extend the model by simultaneously considering both mentioned uncertainties.

Fig. 6.

The contribution of the proposed method from practical prospectives. Green, orange, and blue blocks show modeling, control, and examination stages respectively.

Furthermore, we seek a system to simultaneously determine the vaccination and social distancing rates as investigated in [51,52]. In contrast, the system should be potentially able to deal with the uncertainty existing in pandemics such as COVID-19. Besides uncertainty handling, interpretability is another important feature clinicians need to trust in the system decisions. Specifically, they should know how the system decides and which input is important at a given time, considering uncertain data. Here we address both of them (uncertainty handling and interpretation) in a type-2 fuzzy system which is also robust to uncertainty for controlling vaccination and social distancing rates (Fig. 6 orange part).

Moreover, when a disease affects a large part of the population, countries develop infrastructures to control the disease. Therefore, they continuously extend their capabilities. In other words, they are able to cure or protect more people from infection rather than early days. Here, we also examine the proposed method in a real-world scenario when countries’ hospitalization and health vary over time, by considering a desired infection curve. In contrast, [41] satisfies susceptible and hospitalized goals through media campaigns and therapeutic effects (Fig. 6 Blue part). As a future study, the capacity limitations could be addressed separately under more complex strategies.

The main priority goal in various works, including ours, is limited to healthcare issues. However, this is not the only issue that arises during such pandemics. Specifically, governments have to also deal with critical emerging problems from different domains, such as economic and social. In other words, many objectives from various domains should be addressed instead of a single health-oriented objective. This work provides a proper basis to include any objective by optimization of the controller using genetic algorithms. For further investigation, the controller could be optimized using many-objective evolutionary algorithms on various concerns.

Finally, the results show better performance by the type-2 fuzzy controller compared to the competing type-1 fuzzy approach with similar parameters and design. Here we consider a desired infected population rate without addressing the constraints on actual vaccination rate and social distancing. A promising research hence would also consider limits on the input.

Future works could extend the proposed approach to include multiple outputs such as the hospitalized population rate, the impact of national and religious ceremonies, and the spread behavior in different countries and cultures. These elements currently appear as disturbances for the model, but they drastically influence the infection rate from its nominal values. Furthermore, we could also investigate other optimization techniques, such as reinforcement learning to address temporal learning and model adaptation.

Data availability

All data to reproduce the results of this research are provided in the manuscript.

Code availability

The codes of the proposed method are accessible at: https://github.com/salehiali1374/Interval-Type-2-Fuzzy-Control-and-Stochastic-Modeling-of-COVID-19-Spread.git

CRediT authorship contribution statement

H. Rafiei: Conceptualization, Methodology, Formal analysis, Software, Writing – original draft. A. Salehi: Conceptualization, Methodology, Formal analysis, Software, Writing – original draft. F. Baghbani: Supervision, Conceptualization, Methodology, Formal analysis, Writing – review & editing, Resources. P. Parsa: Conceptualization, Methodology, Writing – original draft, Writing – review & editing. M.-R. Akbarzadeh-T．: Supervision, Conceptualization, Methodology, Formal analysis, Writing – review & editing, Resources.

Declaration of Competing Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Appendix: Ablation study

A.1. The PD controller

For better comparison, here a PD controller is designed to control the number of infected people. Since the model is multi-output, we use two separate controllers for social distancing and vaccinations as follows:

$$\begin{array}{c}\hfill \begin{array}{c}{\mathrm{U}}_{\mathrm{i}}=[{\mathrm{P}}_{\mathrm{c}1},{\mathrm{D}}_{\mathrm{c}1}]{[\mathrm{e}\left[\mathrm{k}\right],\Delta \mathrm{e}\left[\mathrm{k}\right]]}^{\mathrm{T}},\\ {\mathrm{U}}_{\mathrm{v}}=[{\mathrm{P}}_{\mathrm{c}2},{\mathrm{D}}_{\mathrm{c}2}]{[\mathrm{e}\left[\mathrm{k}\right],\Delta \mathrm{e}\left[\mathrm{k}\right]]}^{\mathrm{T}},\end{array}\end{array}$$ (A.1)

where P _c1, D _c1, P _c2, and D _c1 are parameters of two disjoint controllers. The same optimization procedure by the genetic algorithm is used to concurrently determine the P and D coefficients of the controllers for standard values (κ = 0.54, q = 0.5). Then, we examine the optimized controller for various scenarios like Section 3. Fig. A1, Fig. A2 show the results of the PD controller. As shown, the PD controller has a lower variance for different parameter settings in the first scenario, while it has rough behavior compared to the fuzzy controllers.

Fig. A2.

Comparing the infected population size (controlled, desired, and uncontrolled), vaccination rate, and isolation rate over sixty days with different q and κ in the second scenario for the competing PD controller.

A.2. The PID controller

Furthermore, we examine the feasibility of a PID controller, optimized also by genetic algorithm, under various scenarios. Here, two parallel separate controllers are used as follows:

$$\begin{array}{c}\hfill \begin{array}{c}{\mathrm{U}}_{\mathrm{i}}=[{\mathrm{P}}_{\mathrm{c}1},{\mathrm{I}}_{\mathrm{c}1},{\mathrm{D}}_{\mathrm{c}1}]{[\mathrm{e}\left[\mathrm{k}\right],\mathrm{e}\left[\mathrm{k}\right]+\mathrm{e}[\mathrm{k}-1],\Delta \mathrm{e}\left[\mathrm{k}\right]]}^{\mathrm{T}},\\ {\mathrm{U}}_{\mathrm{v}}=[{\mathrm{P}}_{\mathrm{c}2},{\mathrm{I}}_{\mathrm{c}2},{\mathrm{D}}_{\mathrm{c}2}]\left[\mathrm{e},\left[\mathrm{k}\right],,,\mathrm{e},\left[\mathrm{k}\right],+,\mathrm{e}\right[\mathrm{k}-1{\left],,\Delta ,\mathrm{e},\left[\mathrm{k}\right]\right]}^{\mathrm{T}},\end{array}\end{array}$$ (A.2)

where P _c1, I _c1, D _c1, P _c2, I _c1, D _c1 are parameters of two disjoint controllers. Fig. A3, Fig. A4 show the results of the PID controller. As shown, contrary to the PD controller, the designed PID controller has a high variance of behavior in different execution of the algorithm for the first scenario. Also, it couldn't follow the constrained given in both scenarios as well.

Fig. A4.

Comparing the infected population size (controlled, desired, and uncontrolled), vaccination rate, and isolation rate over sixty days with different q and κ in the second scenario for the competing PID controller

A.3. The type-1 fuzzy controller

A type-1 fuzzy controller is also designed to better verify the capabilities of the proposed type-2 fuzzy controller. The structure of the type-1 fuzzy system is designed similar to the type-2 fuzzy controller. The only difference is the use of type-1 fuzzy sets. Therefore, the fuzzy rules are constructed as follows,

$$\text{If}e\left[k\right]\text{is}{A}_1^l\text{and}\Delta e\left[k\right]\text{is}{A}_2^l,\text{then}{U}_V\text{is}{B}_1^l\text{and}{U}_I\text{is}{B}_2^l,$$ (A.3)

where $A_1^l$ and $A_2^l$ are the type-1 fuzzy sets of the antecedent of the fuzzy system, and $B_1^l$ and $B_2^l$ are the type-1 fuzzy sets of the consequent of the l^th fuzzy rule with l = 1, …, M. Using the singleton fuzzifier, minimum Mamdani inference engine, and Centroid defuzzifier, the overall structure is attained as:

$$F\left(\underset{\_}{e}|W\right)=\frac{{\sum }_{l=1}^M{\overline{W}}^l\min \left({\mu }_{A_1^l}\left(e\left[k\right]\right),{\mu }_{A_2^l}\left(\Delta e\left[k\right]\right)\right)}{{\sum }_{l=1}^M\min \left({\mu }_{A_1^l}\left(e\left[k\right]\right),{\mu }_{A_2^l}\left(\Delta e\left[k\right]\right)\right)}=W^T\varphi ,$$ (A.4)

where ${\mu }_{A_1^l}\left(e\left[k\right]\right)$ and ${\mu }_{A_2^l}\left(\Delta e\left[k\right]\right)$ indicate the membership function for the inputs to the sets $A_1^l$ and $A_2^l$, ϕ = [φ₁,…, φ_M]^T is the vector of fuzzy basis functions with

${\phi }_l\left({\underset{\_}{x}}_i\right)={\mu }_{A_1^l}\left(e\left[k\right]\right){\mu }_{A_2^l}\left(e\left[k\right]\right)/\sum _{l=1}^M{\mu }_{A_1^l}\left(e\left[k\right]\right){\mu }_{A_2^l}\left(e\left[k\right]\right)$, ${\overline{W}}^l$ is the vector of the center of $\left[B_1^l{B}_2^l\right]$, and $W^T=\left[{\overline{W}}^1,\dots ,{\overline{W}}^M\right]$.

Similar to the type-2 fuzzy controller, Fig. A5 depicts the surface of the type-1 fuzzy system for both scenarios (A and B in Section 3). As shown, the type-1 fuzzy controller has rough vaccination and social distancing in the face of error and error rates. Fig. A6 depicts the infected population size (controlled, desired, and uncontrolled), vaccination rate, and isolation rate over time for different q and κ in the first scenario. With the increase of isolation rate q, the objective is achieved by not reaching the maximum value of the critical level, similar to the type-2 fuzzy system in the first scenario. Additionally, the results may slightly cross the critical level with the growth of κ, indicating the controller's robustness. Fig. A7 shows the results for the second scenario. As shown, the increase in q or decrease of κ often causes the under-control infected population to follow the given curve better, much like the type-2 fuzzy system.

Fig. A5.

The surface of the fuzzy system for the first and second scenarios. The goal in the first scenario is to prevent reaching the critical level; on the other hand, the second scenario follows the disease progression curve.

Fig. A6.

Comparing the infected population size (controlled, desired, and uncontrolled), vaccination rate, and isolation rate over sixty days with different q and κ in the first scenario for the competing type-1 fuzzy controller.

Fig. A7.

Simulation results of the second scenario for various values of q and κ over sixty days for the competing type-1 fuzzy controller.