# Control Strategies to Curtail Transmission of COVID-19

Authors: Nita H. Shah; Ankush H. Suthar; Ekta N. Jayswal
Journal: International Journal of Mathematics and Mathematical Sciences
 (2020)

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## Abstract

On 11 March 2020, the World Health Organization declared the outbreak of severe acute respiratory syndrome coronavirus 2 (SARS-Cov-2) a pandemic and a Public Health Emergency of International Concern. As of 29 March 2020, coronavirus disease 2019 (COVID-19) has affected 199 countries and territories, resulting in 683,536 positive cases and causing 32,139 deaths. The pandemic has the potential to become extremely destructive globally if not treated seriously. In this study, we propose a generalized SEIR model of COVID-19 to study the behaviour of its transmission under different control strategies. In the model, all possible cases of human-to-human transmission are considered and its reproduction number is formulated to analyse the accurate transmission dynamics of the coronavirus outbreak. Optimal control theory is applied to the model to demonstrate the impact of various intervention strategies, including voluntary quarantine, isolation of infected individuals, improving an individual's immunity, and hospitalization. In addition, the effect of control strategies on the model is analysed graphically by simulating the model numerically.


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## Body

## 1. Introduction

In December 2019, the pandemic outbreak of coronavirus disease 2019 (COVID-19) drew intense attention not only within China but internationally [1]. Doctors and scientists tested previously developed drugs to treat infected patients, but these have so far been unsuccessful. In March 2020, the World Health Organization (WHO) declared the outbreak of the highly contagious COVID-19 as a global pandemic [2]. To control human-to-human virus transmission, the central government of China together with local governments had tightened preventive measures. However, the virus had spread rapidly across most regions in China and in other countries and territories around the world. One major cause of the quick spread of COVID-19 was the lack of information and awareness about the virus during its early stages of infection. As of March 29, 2020, 13:29 GMT, COVID-19 has affected 199 countries and territories around the world with 683,536 confirmed cases, of which 32,139 have resulted in death and 25,422 are serious or critical (worldometers.info). However, there is still the possibility that the spread of this virus could become more intense and cause higher mortality. Chinese New Year celebrations accelerated the outbreak of COVID-19 throughout China, as many people travelled to their hometown or other destinations for the holiday.Symptoms of COVID-19 take at least 2 to 10 days to develop, which makes it difficult to isolate infected individuals during the initial stages of the infection. The major symptoms of COVID-19 include dry cough and high fever with difficulties in breathing [3]. The virus may spread in the environment through respiratory droplets from infected individuals when they cough or sneeze. A previously unaffected populace may become infected when they are exposed to the virus by touching infected surfaces or while breathing in an infected environment [4]. During the initial stages of the COVID-19 outbreak, such human transmission was commonly taking place because the public were unaware of these risk factors, and infected individuals were not isolated, and were therefore spreading the virus unknowingly to other individuals. Moreover, the risk factor of contamination remained very high since the virus can remain viable in the environment for several days in favourable conditions [5, 6]. Several studies have shown that the elderly, children, and those with serious diseases have lower immunity and tend to be seriously affected once they become infected [7, 8]. At this time, we still lack any proper treatment or vaccines as a cure for the disease. Hence, to control its transmission further, isolating infected individuals in special quarantine arrangements has been implemented in many of the countries affected. Despite these preventative strategies, transmission is still ongoing and the mortality rate due to the virus remains at a high level.To combat this situation, studies involving mathematical modelling play a crucial role in understanding the pandemic behaviour of infectious diseases. Bi et al. worked on stability and bifurcation of a mathematical model in the Zika virus. In addition, they established an optimal control problem including several popular disease intervention strategies which helped to mitigate against the Zika epidemics [9]. Zhao et al. modeled a vector-born visceral leishmaniasis disease transmission system, and in addition, they performed bifurcation analysis to support control conditions and analyse optimal control strategies to control zoonotic disease transmission [10]. Several studies have already been undertaken to analyse the COVID-19 transmission dynamics. Based on the database studies of COVID-19 outbreak from 31 December 2019 to 28 January 2020, Wu et al. introduced a SEIR model to estimate the spread of the disease nationally as well as globally [11]. Tang et al. proposed a compartmental model by dividing each group into two subpopulations, the quarantined and unquarantined. Moreover, they redesigned their previous model by using diagnosis and time-dependent contact rates and re-estimated the reproduction ratio to quantify the evolution in a better way [12, 13]. Peng et al. designed a generalized SEIR model that suitably incorporates the intrinsic impact of hidden exposed and infectious cases of COVID-19 [14]. Chen et al. presented the incubation period behaviour of a local outbreak of COVID-19 by constructing a dynamic system [15]. Khan and Atangana described brief details of interaction among bats, unknown hosts, humans, and infection reservoirs by formulating the mathematical results of the mathematical fractional model [16]. Chen et al. have also developed a Bats-Hosts-Reservoir-People transmission network model for simulating the potential transmission of COVID-19 [17]. Zhao et al. divided susceptible people from Wuhan city into different age groups and devel- oped a SEIARW model based on market-to-person and person-to-person transmission routes [18]. Zhong et al. constructed a mathematical model using epidemiological data and examined characteristics of the historical epidemic to make an early prediction of the COVID-19 outbreak in China [19]. Yang and Wang have studied the multiple transmission pathways in the infection dynamics and formed a mathematical model that describes the role of the environmental reservoir in the transmission and spread of COVID-19 [20].In this work, a COVID-19 model is constructed to study human-to-human transmission of the virus in Section2. Optimal control theory is introduced and applied to the model for development in Section 3, and in Section 4, the model is simulated numerically to observe the effect of control strategies on the model.


## 2. COVID-19 Model Formulation

To analyse the human-to-human transmission dynamics of COVID-19, a compartmental model is constructed. The model consists of all possible human-to-human transmission dynamics of the virus. COVID-19 is highly contagious in nature, and cases have been seen in most of the countries around the world; hence, in the model, the susceptible population class is ignored and whole population is divided into five compartments: class of exposed individualsEt (individuals surrounded by infection but not yet infected), class of infected individuals by COVID-19 It, class of critically infected individuals by COVID-19 Ct, class of hospitalised individuals Ht, and class of dead individuals due to COVID-19 Dt. Human-to-human transmission dynamics of COVID-19 are described graphically in Figure 1. Parameters used in the model are described in Table 1.Figure 1
Transmission dynamics of COVID-19.Table 1
Parameters used in the model.
ParametersBBirth rate of class of exposed individuals0.80CalculatedμNatural death rate0.01Assumedβ1Transmission rate of individuals moving from exposed to infected class0.55Calculatedβ2Rate at which infected individuals go into severe condition or in critical condition0.40Calculatedβ3Rate at which critically infected individuals get hospitalised0.60Calculatedβ4Rate at which hospitalised individuals are not recovered and remain in critical condition0.80Calculatedβ5Mortality rate of critically infected individuals0.34Calculatedβ6Mortality rate of infected individuals0.30Calculatedβ7Rate at which infected dead body spreads infection0.35Assumedβ8Rate at which infected individuals get hospitalised0.30Calculatedβ9Rate at which hospitalised individuals get recovered and become exposed again0.35Assumedβ10Rate at which infected individuals recovered themselves due to strong immunity and again become exposed0.32AssumedUsing the above depiction, a dynamical system of set of nonlinear differential equations for the model is formulated as follows:(1)dEdt=B−β1EI+β7ED+β9H+β10EI−μE,dIdt=β1EI−β2I−β6I−β8I−β10EI−μI,dCdt=β2I−β5C−β3C+β4H−μC,dHdt=β3C−β4H+β8I−β9H−μH,dDdt=β5C+β6I−β7DE.Note that all the parameters used in this COVID-19 model are nonnegative. Consider the feasible region as follows:(2)Λ=E,I,C,H,D∈R+5:E+I+C+H+D≤Bμ.The regionΛ is positively invariant; all the solutions of system (1) remain in the feasible region (2).


### 2.1. Equilibrium Points

By solving system (1), we get two equilibrium points:(i)
Disease-free equilibrium point:(3)E0=Bμ,0,0,0,0.(ii)
Endemic equilibrium point:(4)E∗=β2+β6+β8+μβ1−β10,iμq1,cμq1,hμq1,dμβ7q2,where(5)i=β5+μBβ4β1−β10−β4μβ2+β6−β8+μβ5+μβ4μ+β9+μβ3+β5Bβ1−Bβ10−β2μ−μβ6+β8β3+β5+μ−Bμβ1+β10+μ2β2+β3+β5+μ,c=β4+β9+μBβ2β1−β10β22μ−μβ4β8+μβ2+β8−β4β8Bβ1−Bβ10−β6μ−μβ2β9+μ1+β6+β8,h=β8β5+μBβ1−Bβ10−β2μ−β8μ2β6+μ−μβ8β8+μβ3+β5−μβ6β8β3+β5−Bβ1β2β3+β8−Bβ10β3β2+β8−μβ2β3β2+β6−μβ8β2β3+μ,d=β9+μBβ6β1−β10−μβ62−β8+μβ2β5μ+β6μβ3+β5+β6+μ−β2β5μβ2+β6−β6μβ2β3+μ+β3β6+Bβ1−β10β3β6+β2β5+β4β6β5+μBβ1−β10−μβ2+β8−μβ4β6+μβ2β5+β5β6+β6μ−β8+μβ4β5β8μ−β4β5β2+β6μβ8+β2+Bβ10+β1,q1=β1−β10β2β4+β9+μ+β8+β9+μβ3+β5+β4β5+β8+μ+μβ9+μ+β2β3+β8μ,q2=μβ7β2+β6+β8β2β3+β2β4+β8+μ+β4β5+β8+μ+β8+β9+μβ3+β5+μ.


### 2.2. Basic Reproduction Number

The basic reproduction numberR0 for the model can be established using the next-generation matrix method [21, 22]. The basic reproduction number R0 is obtained as the spectral radius of matrix FV−1 at the disease-free equilibrium point, where F and V are as follows:(6)F=0Bβ10μ00Bβ7μ0Bβ1μ000000000000000000,V=μBβ1μ0−β900Bβ10μ+β8+β6+β2+μ0000−β2β3+β5+μ−β400−β8−β3β9+β4+μ00−β6−β50Bβ7μ.The defined effective basic reproduction number is(7)R0=Bβ1Bβ10+μβ2+β6+β8+μ.


## 2.1. Equilibrium Points

By solving system (1), we get two equilibrium points:(i)
Disease-free equilibrium point:(3)E0=Bμ,0,0,0,0.(ii)
Endemic equilibrium point:(4)E∗=β2+β6+β8+μβ1−β10,iμq1,cμq1,hμq1,dμβ7q2,where(5)i=β5+μBβ4β1−β10−β4μβ2+β6−β8+μβ5+μβ4μ+β9+μβ3+β5Bβ1−Bβ10−β2μ−μβ6+β8β3+β5+μ−Bμβ1+β10+μ2β2+β3+β5+μ,c=β4+β9+μBβ2β1−β10β22μ−μβ4β8+μβ2+β8−β4β8Bβ1−Bβ10−β6μ−μβ2β9+μ1+β6+β8,h=β8β5+μBβ1−Bβ10−β2μ−β8μ2β6+μ−μβ8β8+μβ3+β5−μβ6β8β3+β5−Bβ1β2β3+β8−Bβ10β3β2+β8−μβ2β3β2+β6−μβ8β2β3+μ,d=β9+μBβ6β1−β10−μβ62−β8+μβ2β5μ+β6μβ3+β5+β6+μ−β2β5μβ2+β6−β6μβ2β3+μ+β3β6+Bβ1−β10β3β6+β2β5+β4β6β5+μBβ1−β10−μβ2+β8−μβ4β6+μβ2β5+β5β6+β6μ−β8+μβ4β5β8μ−β4β5β2+β6μβ8+β2+Bβ10+β1,q1=β1−β10β2β4+β9+μ+β8+β9+μβ3+β5+β4β5+β8+μ+μβ9+μ+β2β3+β8μ,q2=μβ7β2+β6+β8β2β3+β2β4+β8+μ+β4β5+β8+μ+β8+β9+μβ3+β5+μ.


## 2.2. Basic Reproduction Number

The basic reproduction numberR0 for the model can be established using the next-generation matrix method [21, 22]. The basic reproduction number R0 is obtained as the spectral radius of matrix FV−1 at the disease-free equilibrium point, where F and V are as follows:(6)F=0Bβ10μ00Bβ7μ0Bβ1μ000000000000000000,V=μBβ1μ0−β900Bβ10μ+β8+β6+β2+μ0000−β2β3+β5+μ−β400−β8−β3β9+β4+μ00−β6−β50Bβ7μ.The defined effective basic reproduction number is(7)R0=Bβ1Bβ10+μβ2+β6+β8+μ.


## 3. Optimal Control Theory

Control measures can play a significant role in controlling the epidemic of COVID-19 at a certain level. In this control theory, five control variables are used as five possible control strategies. Since the virus is highly contagious, it quickly infects any people who come into contact with an infected individual. To avoid this situation, we have includedu1 as a control variable for self-quarantine of exposed individuals and u2 as a control variable for isolation of infected individuals. Moreover, to minimise the mortality rate of COVID-19, u3 control variable is included which helps to reduce critically infected cases by taking extra medical care of infected individuals. u4 and u5 control variables are included to improve hospitalization facilities for infected and critically infected individuals, respectively. The purpose of this study of control theory is to protect people from the outbreak by applying controls or treatment in each stage. The objective function for the required scenario is(8)Jci,Ω=∫0TA1E2+A2I2+A3C2+A4H2+A5D2+w1u12+w2u22+w3u32+w4u42+w5u52dt,where Ω denotes the feasible region for the set of compartmental variables, A1,A2,A3,A4,andA5 denote the nonnegative weight constants for compartments E,I,C,H,andD, respectively. w1,w2,w3,w4, and w5 are the weight constants for each control ui, where i=1,2,…,5, respectively. The modified model is given in Figure 2.Figure 2
COVID-19 model with control variables.Hence, the modified system for Figure2 is stated as follows:(9)dEdt=B−β1EI+β7ED+β9H+β10EI−u1E+u2I−μE,dIdt=β1EI−β2I−β6I−β8I−β10EI+u1E−u2I−u5I−u3I−μI,dCdt=β2I−β5C−β3C+β4H+u3I−u4C−μC,dHdt=β3C−β4H+β8I−β9H+u5I+u4C−μH,dDdt=β5C+β6I−β7DE.The weight parameterswi, i=1,2,…,5, are constants applied on the control variable, respectively, from which the optimal condition is normalized. Now, every value of control variables is calculated from t=0 to t=T such that(10)Juit=minJui∗,Ω/ui∈ϕ,i=1,2,…,5,where ϕ is a smooth function on the interval 0,1. The optimal effect is found by using the results of Fleming and Lions [23]. We use Pontryagin’s Maximum Principle to derive the optimal amount of control [24]. The associated Lagrangian function with adjoint variables λ1,λ2,λ3,λ4,andλ5 is given by(11)LΩ,A=A1E2+A2I2+A3C2+A4H2+A5D2+w1u12+w2u22+w3u32+w4u42+w5u52+λ1B−β1EI+β7ED+β9H+β10EI−μE+λ2β1EI−β2I−β6I−β8I−β10EI−μI+λ3β2I−β5C−β3C+β4H−μC+λ4β3C−β4H+β8I−β9H−μH+λ5β5C+β6I−β7DE.The partial derivatives of the Lagrangian function with respect to each variable of the compartment give the adjoint equation variablesAi=λ1,λ2,λ3,λ4,λ5 corresponding to the system which are as follows:(12)λ1•=−∂L∂E=−2A1E+λ1−λ2β1I−λ1−λ2β10I−λ1β7D−λ1μ+λ5β7D+λ1−λ2u1,λ2•=−∂L∂I=−2A2I+λ1−λ2β1E−β10E−u2+λ2−λ3β2+u3+λ2−λ4β8+u5+λ2−λ5β6+λ2μ,λ3•=−∂L∂C=−2A3C+λ3−λ4β3+u4+λ3−λ5β5+λ3μ,λ4•=−∂L∂H=−2A4H−λ1−λ4β9−λ3−λ4β4+λ4μ,λ5•=−∂L∂D=−2A5D−λ1−λ5β7E.Hence, the optimal controls are given by(13)u1=λ1−λ2E2w1,u2=λ2−λ1I2w2,u3=λ2−λ3I2w3,u4=λ3−λ4C2w4,u5=λ2−λ4I2w5.In addition, the optimal conditions are given as follows:(14)u1∗=maxa1,minb1,λ1−λ2E2w1,u2∗=maxa2,minb2,λ2−λ1I2w2,u3∗=maxa3,minb3,λ2−λ3I2w3,u4∗=maxa4,minb4,λ3−λ4C2w4,u5∗=maxa5,minb5,λ2−λ4I2w5.This calculation provides analytical behaviour of optimal control on the system. Numerical interpretation of optimal control theory is simulated in the next section.


## 4. Numerical Simulation

In this section, the COVID-19 model is simulated numerically, wherein the parametric values for simulation are taken from the recent pandemic outbreak of coronavirus (https://www.who.int/emergencies/diseases/novel-coronavirus-2019/situation-reports). The initial condition taken to plot variations in all the compartments is y0E,I,C,H,D=8,4,0.12,1,5.Figure3 represents the variations in all the compartments of the COVID-19 model with respect to time. In addition, the pandemic behaviour of the COVID-19 outbreak can be clearly seen here. We can say that a large population of exposed individuals become infected within the first week. Moreover, the critically infected cases and hospitalization cases also increase with time. Furthermore, it clearly shows that after one week, the mortality rate is also increased.Figure 3
Variations in all compartments.Figure4(a) shows variation in each compartment with and without control variables. It is observed that the COVID-19 outbreak can be controlled up to a certain level in seven weeks after applying control strategies. Meanwhile, Figures 4(b) and 4(c) represent the number of infected and critically-infected individuals decreasing; this then leads to a decrease in the hospitalized class as shown in Figure 4(d). Thus, reduction in the mortality rate under the effect of control strategies can be clearly observed Figure 4(e).Figure 4
Variation in compartments with and without optimal controls.
(a)(b)(c)(d)(e)Variation in the intensity of control strategies with time is shown in Figure5. Moreover, the figure also demonstrates which control can be applied and at what intensity to control a COVID-19 outbreak in seven weeks. The range of all the four controls u1, u2, u3, and u4 is −0.7481,0.4776, −1.733,0.8913, −7.409,1.04, and 0.0158,0.6338, respectively. The high fluctuation in the u3 control variable at an initial stage suggests that it is very important to control infected individuals moving to the critical stage to reduce the mortality rate due to COVID-19. In addition, this could be achieved easily if an infected individual gets proper vaccination for the disease. Since effective vaccination is not available for COVID-19, proper care of infected individuals must be taken to improve their immunity so that their body becomes capable of fighting the virus and they do not reach the critical stage. Moreover, fluctuation in the u2 control variable suggests that it is very important to isolate or quarantine infected individuals to control this pandemic outbreak.Figure 5
Variation in control variable with time.The separate effects of control variables on each compartment can be observed in Figure6. From this figure, we can interpret that the u2 control variable is highly effective for stabilizing the epidemic situation. Figure 6(b) shows that the population class of infected individuals is lowest under the influence of the u5 control variable which suggests that rapid hospitalization of infected individuals is an effective measure for reducing infected cases of COVID-19. Figure 6(c) suggests that in order to reduce critical cases of COVID-19,, we should first implement controls to prevent infected individuals from becoming critically infected, and that we should improve hospitalization and medical facilities for these critically infected individuals to save their lives. Figure 6(e) shows that the mortality rate due to COVID-19 can be reduced effectively within three weeks of outbreak by applying u1, u2, and u3 control strategies. This means that self-quarantine for an exposed individual, isolation of an infected individual, and reducing critical cases by taking extra care of infected individuals are likely to be effective strategies to control further transmission of COVID-19.Figure 6
Variation in each compartment under individual effect of control variables.
(a)(b)(c)(d)(e)Figure7 represents the change in the objective function under influence of all the controls. The combined effect of all the controls indicates a desirable effect on the model.Figure 7
Change in objective function with time.Figure8 demonstrates the scatter diagram representing the chaotic situation created during a COVID-19 outbreak. Figures 8(a)–8(c) show periodic mortality from classes of exposed individuals, infected individuals, and critically infected individuals with respect to time (in weeks), respectively, when under hospitalization. By comparing Figures 8(a) and 8(b), it can be interpreted that the mortality ratio in the class of infected individuals is higher and much faster than that for the class of exposed individuals. The chaotic Figure 8(c) diagram represents a very high mortality rate for critically infected individuals. Hence, in the absence of an effective vaccination for COVID-19, it becomes a challenging situation to cure critically infected individuals.Figure 8
Scatter diagram for COVID-19 outbreak.
(a)(b)(c)


## 5. Conclusion

In this study, a compartmental model is constructed to examine transmission of COVID-19 within the human pop- ulation. Moreover, the basic reproduction number is formulated to calculate the threshold value of the disease. In order to develop strategies to counteract COVID-19 epidemics, optimal control theory is applied to the model. In addition, in order to advance this control theory, five control variables are introduced in the model in the form of control strategies. These strategies include self-quarantine of exposed individuals, isolation of infected individuals, taking extra care of infected individuals to reduce critical cases of COVID-19, and increased hospitalization facilities for infected and critically infected individuals. Individual and combined effects of these control variables on all the compartments are observed and examined graphically by simulating the COVID-19 model. Numerical simulation of the model indicates that quarantine and better medical treatment of infected individuals can reduce the number of critically-infected cases, which will further reduce the transmission risk and mortality rate.



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