# Z-Control on COVID-19-Exposed Patients in Quarantine

Authors: Nita H. Shah; Nisha Sheoran; Ekta Jayswal
Journal: International Journal of Differential Equations
 (2020)

---

## Abstract

In this paper, a mathematical model for diabetic or hypertensive patients exposed to COVID-19 is formulated along with a set of first-order nonlinear differential equations. The system is said to exhibit two equilibria, namely, exposure-free and endemic points. The reproduction number is obtained for each equilibrium point. Local stability conditions are derived for both equilibria, and global stability is studied for the endemic equilibrium point. This model is investigated along with Z-control in order to eliminate chaos and oscillation epidemiologically showing the importance of quarantine in the COVID-19 environment.


---

## Body

## 1. Introduction

COVID-19, an infectious disease with the first case reported in Wuhan city of China, has spread throughout the world. On 30 January 2020, the WHO declared its outbreak as a “public health emergency of international concern.” As on 2 April 2020, 08:02 GMT, it has caused 47,249 deaths worldwide and a total of 936,237 cases have been confirmed [1]. COVID-19 spreads through the contact of individuals with infected persons when they cough or sneeze. It is a respiratory disease with mild to moderate symptoms like dry cough, fever, and tiredness, and in more severe cases difficulty breathing [2]. Also, few individuals showing mild symptoms of the disease may recover themselves if they avoid contacting infected cases and maintain good hygiene. COVID-19 is a major health threat to those with a past medical history and also to those who are older than 60 years (elderly population). This was reported by Li et al. [3], who calculated the median age of 425 patients infected with COVID-19 in Wuhan, China, as 59 years and almost half of the patients were 60 years old.Governments throughout the world are taking several preventive measures to control the spread of the epidemic. Preventing the spread is the only way since no vaccine has been developed till date to fight the virus. Preventive measures include maintaining at least 1 meter distance from a person sneezing or coughing, washing hands regularly, and maintaining social distance.Various mathematical models have been developed so far to address various challenges in predicting the spread of COVID-19 disease. Batista [4] has used the basic SIR-model to find the actual size of the epidemic. Peng et al. [5] analysed the scenario of COVID-19 in China by formulating the SEIR dynamical system and have predicted that the situation will be under control at the beginning of April.Sun et al. [6] discussed various aspects of COVID-19 situation in China which helps understand the fatality rate and transmission rate of COVID-19 and control the epidemic spread. In the case of COVID-19 epidemic, exposure to disease plays a vital role in the spread of the disease. Rabajante [7] studied various models and concluded that with the basic reproduction number being 2 and considering the 14-day infectious period, if an infected person stays for more than 9 hours with others, he could infect others. If the exposure time is 18 hours, the model recommends full protection with more than 70% effectiveness to the attendees of the social gathering. In order to control the disease, the importance of travel quarantine or travel restriction, which delayed the progression of disease, in Wuhan was studied by Chinazzi et al. [8]. A similar result was also shown by Kucharski et al. [9]. They showed that with the air travel restriction in Wuhan, the daily reproduction number declined from 2.35 to 1.05. There is also a model evaluated by Tang et al. [10] and Tang et al. [11] in which they divided the subpopulation into quarantined and unquarantined classes to understand the transmission risk of the epidemic.From the studies so far, we have observed that quarantine or isolation plays a huge role in controlling the disease spread. Here, we have developed a model with Z-control applied to the quarantine class to obtain the required exposure state in order to make the system free of chaos. Few basic models of Z-control include prey-predator model by Alzahrani et al. [12] and epidemic model by Samanta [13].The construction of the paper is as follows: formulation and description of the mathematical model are given in Section2. In Section 3, the reproduction number is calculated for the state when there is no exposure to COVID-19 and for the endemic state. In Section 4, local stability and global stability are discussed. In Section 5, the model is constructed with Z-controller and ut is calculated. In Section 6, a numerical simulation is performed. Finally, the paper concludes with Section 7.


## 2. Mathematical Model

Mathematically, the compartmental model for the individuals already suffering from diabetes or hypertension who are at a higher risk of getting infected with COVID-19 is formulated in this section with three mutually exclusive disjoint classes. The classes are as follows:S: susceptible class, diagnosed with diabetes or hypertension; E: exposed class; and Q: quarantine class.Notations and parametric values used in the formulation of dynamical system are given in Table1.Table 1
Parametric definitions and their values.
NotationsDescriptionParametric valuesBBirth rate1Assumedβ1Transmission rate of individuals from susceptible class to exposed class0.9Calculatedβ2Force of infection0.01Assumedβ3Rate at which exposed individuals quarantine themselves0.80Calculatedβ4Rate at which susceptible individuals quarantine themselves0.60Calculateda,b,dHalf-saturation constants2, 10, 0.4Assumedc1,c2Conversion efficiency1, 2AssumedThe model considers the susceptible class as a population diagnosed with diabetes or suffering from hypertension. Here, recruitment of new individuals occurs at the rateB, and all the individuals in their respective compartments suffer death at the constant rate μ. We have considered the saturated incidence as β2SE/a+S, where a is the saturation constant and β2 is the force of infection as shown in Figure 1.Figure 1
Compartmental diagram of flow of individuals from one compartment to another compartment.Figure1 gives rise to the following system of nonlinear differential equations:(1)dSdt=BS−β1SE−β2SEa+S−β4SQb+S−μS2,dEdt=β1SE+β2SEa+S−β3EQd+E−μE,dQdt=c1β3EQd+E+c2β4SQb+S−μQ,with S>0,E>0,Q>0.EquatingdS/dt=0,dE/dt=0,dQ/dt=0 and solving, we get the following equilibrium points:(i)
Exposure-free equilibrium pointE1bμ/β4c2−μ,0,bc2Bβ4c2−bμ2−Bμ/β4c2−μ2.(ii)
Endemic equilibrium pointEd∗S∗,E∗,Q∗S∗=r1,E∗=−dr1β4c2−μ−bμ/r1β3c1+β4c2−μ+bβ3c1−bμ,(2)S∗=r1,E∗=−dr1β4c2−μ−bμr1β3c1+β4c2−μ+bβ3c1−bμ,Q∗=r12β1abβ1c12μc1−c2b−a−ac12μ2a−b+abβ2c12μc2−c1β32+β1β3a2bβ1β4c1c2μc2−c1+abβ1c1μ22c1a−b−ac2−ac12c2β4μa−b−abβ2β4c1c2μc1−c2+c1μ2b3β1c2+2a2c1μ+abc1β2μ22c1−c2β2−abc1μ32c1−c2−b2c1c2μ2β2−μ+β1β4a2c1c2β4μ22μ−c2β4−c1μ2μ2−2β4c2μ−bβ1β4c2+abc22β4μ2c2β4−β1b+b2β1μ3c1+c2+bβ4c2μ2β2c1−c1μ−2c2μ−bμ3β2c1−c1μ−c2μ−b2c2μ2bβ1μ+c2β2β4−μc2β4+β2−μ+r1β3β4β3c1−c2abc12bβ1μb−a+Bβ1a+b+Bβ2+β2+μabc12μb−c1+Bac13μa−b+c1−c2a2bc1β12μ−c2β4bμ−B+Babc1c2β4β2−μ−ab2β1c1μ2bμ−β1d+b2c1μbβ13d+2aβ2μ−abc1μ22c1−c2bβ4c2+μ−aμ+bβ2c12μbβ1d+2Ba−β2c1−β2c2−c1μab2β4c1c2μ+2Bac12μβ4ac2+b2+b2β1c1dμβ2+c1μ+Bb2μc1c2β2−μ+Babc1μβ2c2+2c1μ+c2μ+β4c1−c2β2−μb2β1dμμ−β4−ab2μ2μ−c2β4+abβ1μ2bμb−a+dbβ1+μ−bμ2b2dβ12+a2μ2+aμ−β1dbβ1−β4c2+2μabβ4c2μ+Baβ4c2μbβ1−β4c2bc2−ac1−c1+c2Babμ2bβ1+μ+Bac1μ2ab−2aβ4c2+β2−μBbc2μbμ+β4c2−aβ4c1+Bμ2bc22β4c2+bβ1+ac1aμ+bβ2+β3β4β3c1−c2Bab2c12β1a−b+β2+c1−c2a2b2Bβ1c1β4c2−μ+a2bβ1c1dμc2β4−bβ1+abβ4c1c2μ2Ba−bβ1d+ab3β1c1μβ1d+2B+ab2β2c1Bβ4c2−2β1dμ+abβ2c1μβ4c2d−2Bb+b3β1c1dμβ2−μ−abc12dμBbβ4c2+2aβ1μ+a2c1c2dμ2bβ1+β4c1+ab2β1c1dμ23c1−2c2+2c1−c2abc1μ2Ba−b+dβ2−c12dμ3a2+b2+bc12dμ2aμ−bβ22+b2β2c1c2dμβ2−μ+bc12dμ22bβ2−β4c2+β4β4bβ4ac22μc1−c2baβ1+β2d+Ba+c1dμa−b+c1−c2bμc2ba2β1β1d+B−2abβ2bβ1d+aμB−Babβ2−μ−bβ22d+c2dμ2b22c1−c2aβ1+β2−a2bβ13c1−2c2−ac12aμ+3bβ2+ab2β2c2+3c1μ−b2c1μ+bc1−c2bdμ3β11−b2+β2b2a−b+b2c1+2c1−c2abdμ3aβ1+β2−dμ23c1−c2ab2β1μ−a2+b2c1+2a−bβ22bc1+b2β2β2c2+2μ+μb2β2c2+β1c1⋅c1μc1−c2μβ32+μ2c2−c1μβ3+μ3b3β1+c12μ2−ac1β1β4c1−c22+c1β2μc2−c1β32+2aβ4μc1−c22+2aμ22c1−c2β1+β2μ22c1−c2+β4c1c2μμ−β2+β2β4c22μ−2c1μ3β3−aμ2c1β4−2c2+μβ1−β4c2μ2−μ3μ−β2b2+c1c1−c2a2β4c1−c2+μβ1+ac1β2β4c22+c12−2ac12β2β4c2+μ2−ac1μβ2c2+β4c12−β2c1β32+c1−c22a2β4c1c2β4−2μ−a2μ22c1−c2β1+2ac1μ22μ+c1β4−β2+ac2β2c22β42+μ2+2ac2β2β4μ2c1−c2+aβ42c1c22μ−2β2−ac12c2β42μ−β2−ac1β4μ5c2μ+2c1β2β3+a2c2β42μc2−c1+a2μ2c1β4−2c2β4+μβ1+aβ4μc1μβ2+c2β4−c2β2−aμμ22μ+c1β4+β2c1c2β42−μ2−2ac2μ2β4c2β4−2μb+a2β4c12μc1−c2+a2c12μ2β32+2a2c1c2β42μc1−c2−2a2c1μ2c1β4−2c2β4−μβ3+a2μ2β43c2+c1−3a2β4c2μ3+a2β43c2μc1c2−2c1μ−c22+a2μ4−1,where(3)r1=Root ofβ3c1+β4c2−μZ3+aμ−Bβ3c1+β4c2−μ+bμβ3c1+μ+β1dβ4c1−c2+μZ2+β3c1−μabμ−Bb−Baβ3c1+β4c2−μ+aβ1d+β2dβ4c1−c2+μ+dμbβ1−β4c1Z+abBβ3c1+β4c1−μ+bdμaβ1+β2.


## 3. Reproduction Number

n this paper, the reproduction number is defined as the number of secondary individuals who are asked to go for quarantine or social distancing due to a single quarantined susceptible individual exposed to COVID-19.For this model, the reproduction number for exposure-free equilibrium and that for endemic stage equilibrium are calculated. Using the next-generation matrix method by Diekmann et al. [14], we get the following Jacobian matrices:(4)F=β2Sa+S+β1S0β1E+β2Ea+S−β2SEa+S2c1β3Qd+E−c1β3EQd+E2c1β3Ed+E+c2β4Sb+Sc2β4Qb+S−c2β4SQb+S2000,V=β3Ed+E−β3EQd+E2+μβ3Ed+E00μ0β2Sa+S+β1Sβ4Sb+S−B+2μS+β2Ea+S−β2ESa+S2+β4Qb+S−β4QSb+S2+β1E.Reproduction numbersR0E1 and R0Ed∗ are defined as the spectral radius of FV−1 at E1 and Ed∗, respectively, which are numerically computed as 2.66 and 5.35.


## 4. Stability Analysis

In this section, local stability and global stability for the equilibrium points are studied.


### 4.1. Local Stability

Local stability is obtained for all of its equilibrium points by using the Routh-Hurwitz criterion [15].Theorem 1.
Exposure-free equilibrium pointE1 is said to be locally asymptotically stable if it satisfies β4c2−μβ1μd+c2β3B<μ+bβ2μ/aβ4c2−μ+bμdβ4c2−μ2/b+c2β3bμ2 and μbμ+B<c2Bβ4.Proof.
The elements of the Jacobian matrix of system (1) at E1 are(5)a11=B−Bc2β4−bμ2−Bμβ4c2−2μ2bβ4c2−μ,a12=−β2bμaβ4c2−μ+bμ−β1bμβ4c2−μ,a13=−μc2,a21=0,a22=bβ1μβ4c2−μ−bβ2μaβ4c2−μ+bμ+bc2β3Bβ4c2−bμ2−Bμdβ4c2−μ2−μ,a23=0,a31=Bc2β4−bμ2−Bμβ4,a32=c1c2bβ3Bβ4c2−bμ2−Bμdβ4c2−μ,a33=0.
The eigenvalues of this Jacobian matrix are(6)λ1=a22,λ2,3=12−a11±a112−4a13a31.
These eigenvalues are negative ifa22<0 and a13a31<0, i.e., β4c2−μβ1μd+c2β3B<μ+bβ2μ/aβ4c2−μ+bμdβ4c2−μ2/b+c2β3bμ2 and μbμ+B<c2Bβ4. Hence, satisfying the Routh–Hurwitz criterion, the system is locally asymptotically stable.Theorem 2.
Endemic point is said to be locally asymptotically stable if it satisfies(7)1B<af3E∗+bf5Q∗+β1E∗+2μS∗2β1S∗+df6Q∗<μ+f1S∗3c1f4E∗+c2f2S∗<μ4β1S∗−f1S∗+df6Q∗−μc1f4E∗+c2f2S∗−μ>c1df4f6E∗Q∗.Proof.
Let us consider the following Jacobian matrix of system (1) at Ed∗:(8)J∗=−b11−f1+β1S∗−S∗f2β1+af3E∗−b22f4E∗c2f5bQ∗c1f6dQ∗−b33,where b11=−B+af3E∗+bf5Q∗+β1E∗+2μS∗, b22=−β1S∗+f1S∗−df6Q∗+μ, b33=−c1f4E∗−c2f2S∗+μ, f1=β2/a+S, f2=β4/b+S, f3=β2/a+S2, f4=β3/d+E, f5=β4/b+S2, and f6=β3/d+E2.
The characteristic polynomial of (8) is given by(9)α3+ε2α2+ε1α+ε0,where ε0=c1df2f6af3+β1E∗Q∗S∗+β1+f1c2f4f5+b33af3+b33E∗S∗+bc2b22f2f5Q∗S∗+b11b22b33−c1df4f6E∗Q∗, ε1=bc2f2f5Q∗S∗+β1+f1af3+β1E∗S∗+b11b22+b11b33+b22b33−c1df4f6E∗Q∗, and ε2=b11+b22+b33.
The system has the real part of eigenvalues as negative, if it satisfiesb11>0,b22>0,b33>0 and b22b33>c1df4f6E∗Q∗ i.e., B<af3E∗+bf5Q∗+β1E∗+2μS∗, β1S∗+df6Q∗<μ+f1S∗, c1f4E∗+c2f2S∗<μ, β1S∗−f1S∗+df6Q∗−μc1f4E∗+c2f2S∗−μ>c1df4f6E∗Q∗.
Hence, satisfying the Routh–Hurwitz criterion, the endemic is said to be locally asymptotically stable.


### 4.2. Global Stability

In this section, global stability for the endemic equilibrium point is studied using the Lyapunov function.Theorem 3.
Endemic pointEd∗ is asymptotically globally stable.Proof.
Let us assume the Lyapunov functionL∗ as(10)L∗t=12S−S∗2+k112E−E∗2+k212Q−Q∗2,dL∗dt=S−S∗BS−β1SE−β2SEa+S−β4SQb+S−μS2+k1E−E∗β1SE+β2SEa+S−β3EQd+E−μE+k2Q−Q∗c1β3EQd+E+c2β4SQb+S−μQ=S−S∗2B−β1E−β2Ea+S−β4Qb+S−μS+S∗+k1E−E∗2β1S+β2Sa+S−β3Qd+E−μ+k2Q−Q∗2c1β3Ed+E+c2β4Sb+S−μ+S∗S−S∗B−μS∗−β2a+SS∗ES−S∗−SE∗E−E∗−β4b+SS∗QS−S∗−k2c2SQ∗Q−Q∗−β1S∗ES−S∗−k1SE∗E−E∗−β3d+Ek1QE∗E−E∗−k2c1Q∗EQ−Q∗−μk1E∗E−E∗+k2Q∗Q−Q∗.where dL∗/dt<0 when β1S+β2S/a+S<β3Q/d+E+μ and c1β3E/d+E+c2β4S/b+S<μ,B<μS∗. Hence, by LaSalle’s invariance principle [16], the endemic equilibrium point is globally stable.


## 4.1. Local Stability

Local stability is obtained for all of its equilibrium points by using the Routh-Hurwitz criterion [15].Theorem 1.
Exposure-free equilibrium pointE1 is said to be locally asymptotically stable if it satisfies β4c2−μβ1μd+c2β3B<μ+bβ2μ/aβ4c2−μ+bμdβ4c2−μ2/b+c2β3bμ2 and μbμ+B<c2Bβ4.Proof.
The elements of the Jacobian matrix of system (1) at E1 are(5)a11=B−Bc2β4−bμ2−Bμβ4c2−2μ2bβ4c2−μ,a12=−β2bμaβ4c2−μ+bμ−β1bμβ4c2−μ,a13=−μc2,a21=0,a22=bβ1μβ4c2−μ−bβ2μaβ4c2−μ+bμ+bc2β3Bβ4c2−bμ2−Bμdβ4c2−μ2−μ,a23=0,a31=Bc2β4−bμ2−Bμβ4,a32=c1c2bβ3Bβ4c2−bμ2−Bμdβ4c2−μ,a33=0.
The eigenvalues of this Jacobian matrix are(6)λ1=a22,λ2,3=12−a11±a112−4a13a31.
These eigenvalues are negative ifa22<0 and a13a31<0, i.e., β4c2−μβ1μd+c2β3B<μ+bβ2μ/aβ4c2−μ+bμdβ4c2−μ2/b+c2β3bμ2 and μbμ+B<c2Bβ4. Hence, satisfying the Routh–Hurwitz criterion, the system is locally asymptotically stable.Theorem 2.
Endemic point is said to be locally asymptotically stable if it satisfies(7)1B<af3E∗+bf5Q∗+β1E∗+2μS∗2β1S∗+df6Q∗<μ+f1S∗3c1f4E∗+c2f2S∗<μ4β1S∗−f1S∗+df6Q∗−μc1f4E∗+c2f2S∗−μ>c1df4f6E∗Q∗.Proof.
Let us consider the following Jacobian matrix of system (1) at Ed∗:(8)J∗=−b11−f1+β1S∗−S∗f2β1+af3E∗−b22f4E∗c2f5bQ∗c1f6dQ∗−b33,where b11=−B+af3E∗+bf5Q∗+β1E∗+2μS∗, b22=−β1S∗+f1S∗−df6Q∗+μ, b33=−c1f4E∗−c2f2S∗+μ, f1=β2/a+S, f2=β4/b+S, f3=β2/a+S2, f4=β3/d+E, f5=β4/b+S2, and f6=β3/d+E2.
The characteristic polynomial of (8) is given by(9)α3+ε2α2+ε1α+ε0,where ε0=c1df2f6af3+β1E∗Q∗S∗+β1+f1c2f4f5+b33af3+b33E∗S∗+bc2b22f2f5Q∗S∗+b11b22b33−c1df4f6E∗Q∗, ε1=bc2f2f5Q∗S∗+β1+f1af3+β1E∗S∗+b11b22+b11b33+b22b33−c1df4f6E∗Q∗, and ε2=b11+b22+b33.
The system has the real part of eigenvalues as negative, if it satisfiesb11>0,b22>0,b33>0 and b22b33>c1df4f6E∗Q∗ i.e., B<af3E∗+bf5Q∗+β1E∗+2μS∗, β1S∗+df6Q∗<μ+f1S∗, c1f4E∗+c2f2S∗<μ, β1S∗−f1S∗+df6Q∗−μc1f4E∗+c2f2S∗−μ>c1df4f6E∗Q∗.
Hence, satisfying the Routh–Hurwitz criterion, the endemic is said to be locally asymptotically stable.


## 4.2. Global Stability

In this section, global stability for the endemic equilibrium point is studied using the Lyapunov function.Theorem 3.
Endemic pointEd∗ is asymptotically globally stable.Proof.
Let us assume the Lyapunov functionL∗ as(10)L∗t=12S−S∗2+k112E−E∗2+k212Q−Q∗2,dL∗dt=S−S∗BS−β1SE−β2SEa+S−β4SQb+S−μS2+k1E−E∗β1SE+β2SEa+S−β3EQd+E−μE+k2Q−Q∗c1β3EQd+E+c2β4SQb+S−μQ=S−S∗2B−β1E−β2Ea+S−β4Qb+S−μS+S∗+k1E−E∗2β1S+β2Sa+S−β3Qd+E−μ+k2Q−Q∗2c1β3Ed+E+c2β4Sb+S−μ+S∗S−S∗B−μS∗−β2a+SS∗ES−S∗−SE∗E−E∗−β4b+SS∗QS−S∗−k2c2SQ∗Q−Q∗−β1S∗ES−S∗−k1SE∗E−E∗−β3d+Ek1QE∗E−E∗−k2c1Q∗EQ−Q∗−μk1E∗E−E∗+k2Q∗Q−Q∗.where dL∗/dt<0 when β1S+β2S/a+S<β3Q/d+E+μ and c1β3E/d+E+c2β4S/b+S<μ,B<μS∗. Hence, by LaSalle’s invariance principle [16], the endemic equilibrium point is globally stable.


## 5. Z-Control

In this section, Z-control is applied on the abovementioned model. The basic idea of Z-control approach is adopted from [17–20]. Z-control plays a vital role in reducing chaos and/or oscillation and thus in achieving the stability of the model. The Z-control approach is used to derive an expression for the system input say ut, so that the actual system output yt will be forced to achieve the desired output ydt. Moreover, Z-control is more useful if the difference between the actual output of the system and its desired output is negligible, namely, if et=yt−ydt⟶0 as t⟶∞. This objective is achieved by forcing error function et to converge exponentially to zero when it satisfies the following equation:(11)e˙t=−λet,forλ>0.This equation is also known as the design formula andλ is the design parameter which is strictly positive indicating the convergence rate. The Z-control approach is based on the following steps. Firstly, define the error function and, secondly, use the design formula whenever required to obtain the explicit expression of ut.Now, by applying indirect Z-controller to the quarantine population, system (1) is rewritten as(12)dSdt=BS−β1SE−β2SEa+S−β4SQb+S−μS2,dEdt=β1SE+β2SEa+S−β3EQd+E−μE,dQdt=c1β3EQd+E+c2β4SQb+S−μQ−utQ,where ut is the indirect control variable for quarantine population in measuring the dynamics of the model. The purpose is to adjust the effectiveness of quarantine due to exposure to COVID-19, i.e., Et⟶Edt.Let us considerEd=Edt be the desired stage for the population exposed to COVID-19 and defining the first error function as e1=Et−Edt.Now using the design formula of the Z-control method, we have(13)E˙t−E˙dt=−λEt−Edt,where Q˙t is not explicitly included in equation (13); therefore, in the view of equation (12), ut cannot be derived explicitly from (13). Hence, the second error function is considered while applying Z-controller for the second time.The second error function is defined as(14)e2=e˙1+λe1=E˙t−E˙dt+λEt−Edt.But,(15)e˙2t=−λe2t,⟹E¨t−E¨dt+λE˙t−E˙dt=−λE˙t−E˙dt+λEt−Edt.⟹β1S˙E+β1SE˙+β2SE˙a+S+β2S˙Ea+S−β2S˙SE˙a+S2−β3EQ˙d+E−β3E˙Qd+E+β3EE˙Qd+E2−μE˙−E¨dt+λE˙t−E˙dt=−λE˙t−E˙dt+λEt−Edt.⟹Q˙=d+Eβ3Eβ1S˙E+β1SE˙+β2SE˙a+S+β2S˙Ea+S−β2S˙SE˙a+S2−β3E˙Qd+E+β3EE˙Qd+E2−μE˙−E¨dt+λE˙t−E˙dt+λE˙t−E˙dt+λEt−Edt.We finally obtain the expression for Z-controller as(16)⇒ut=c1β3Ed+E+c2β4Sb+S−μ−d+Eβ3EQβ1S˙E+β1SE˙+β2SE˙a+S+β2S˙Ea+S−β2S˙SE˙a+S2−β3E˙Qd+E+β3EE˙Qd+E2−μE˙−E¨dt+λE˙t−E˙dt+λE˙t−E˙dt+λEt−Edt.Theorem 4.
Starting from the positive initial stateS0,E0,Q0T, the error function e1 of model (12) equipped with Z-controller (16) converges to zero exponentially for a continuously differentiable and bounded desired state Edt.Proof.
According to (16), we consider e2=e˙1+λe1, and substituting it in e˙2t=−λe2t, we get(17)e¨1+2λe˙1+λ2e1=0.
The solution of this second-order differential equation is calculated as(18)e1=a1+a2texp−λt,∀t≥0,where a1 and a2 are arbitrary constants. Using the initial value condition, we obtain a1=E0−Ed0,a2=E˙0−E˙d0+λE0−Ed0.
According to Lemma 1 in [21], ∃c¯>0 and λ¯>0 such that e1≤c¯exp−λt. Thus, the tracking error exponentially converges to zero for λ¯>0, which implies that the population exposed to COVID-19 converges exponentially to its desired state.


## 6. Numerical Simulation

In this section, simulation is developed to study the importance of quarantine for the population suffering from diabetes or hypertension in the COVID-19 environment. Data given in Table1 have been used for the numerical simulation.Figure2 shows the plotting of susceptible class, exposed class, and quarantine class. It depicts the model completely indicating the movement of individuals from one compartment to another compartment.Figure 2
Transmission of COVID-19 in the diabetic/hypertension population.Figure3 is plotted with respect to all the compartments, namely, susceptible, exposed, and quarantine at the endemic equilibrium. It shows the periodic nature of system (1).Figure 3
Scatter diagram.Figure4 shows that the individuals exposed to COVID-19 are quarantining themselves or are maintaining social distance to avoid the transmission of the disease.Figure 4
Behaviour of population exposed to COVID-19.Figure5 shows the cycle’s oscillation for the endemic equilibrium point at β3=1.8 without applying Z-controller. With further increase in β3, the system becomes more chaotic in nature showing its instability. For better understanding of the periodic oscillation and chaotic nature, we draw a bifurcation diagram taking β3 as the bifurcating parameter.Figure 5
Oscillations atβ3=1.8.
(a)(b)(c)(d)Figure6 shows the periodicity along each compartment with respect to β3. Here, the maximum and the minimum values are plotted with blue and red, respectively. Here, it is observed that if the population is exposed to COVID-19, more and more individuals undergo quarantine increasing the periodicity, which further turns into chaos. In order to make the system stable and free from chaos or periodicity, we apply Z-control to system (1).Figure 6
Bifurcation diagram.
(a)(b)(c)By implementing Z-control as explained in Section5 to system (1), one can observe from Figure 7 that the system has been made free from chaos or periodicity, making the system stable. Here, Ed=0.001, i.e., the desired value of Ed. We can see that Figure 5 gives us the uncontrolled system with unstable dynamics and Figure 7 gives us the controlled system with stable dynamics. Figure 7 is interpreted as the susceptible individuals, before exposure to COVID-19, quarantine or isolate themselves to avoid contact with the disease.Figure 7
Implementation of Z-control.
(a)(b)(c)Figure8 shows the tracking errors which have been used in the design formula. Figure 9 depicts the behaviour of the control variable applied in the Z-control model. It shows that initially 20% control is required and later on, as the exposure to disease further increases, control to be applied also increases, i.e., a greater number of individuals are to be quarantined.Figure 8
Error function.Figure 9
Control variable.


## 7. Conclusion

This paper considers a mathematical model for the population suffering from diabetes or hypertension exposed to COVID-19. Here, we have calculated the reproduction number when there is no exposure to COVID-19 which turns out to be 2.66, indicating that a single individual asks atleast two individuals to go for quarantine. In a similar manner, the reproduction number in the endemic stage is calculated as 5.33, indicating that at least 5 individuals quarantine themselves when they are exposed to the disease. Next, we have derived the conditions for the system to be locally stable with the help of the Routh–Hurwitz criterion. Global stability is also studied for the endemic equilibrium point. In this paper, we have investigated Z-control in order to control the disease dynamics. Here, the control is applied to the quarantine class in order to obtain the desired exposure in attaining the stability of the system. At the end, the numerical simulation is established indicating the importance of quarantine which helps reduce the chaos of the system. If individuals quarantine themselves at the onset of COVID-19, it keeps them safe, which is the only way to protect oneself as there is no vaccine till date.



---

*Source: 7876146-2020-06-19.xml*