# Mathematical Modelling and Optimal Control Strategies of a Multistrain COVID-19 Spread

Authors: Bouchaib Khajji; Lahoucine Boujallal; Omar Balatif; Mostafa Rachik
Journal: Journal of Applied Mathematics
 (2022)

---

## Abstract

In this paper, we propose a continuous mathematical model that describes the spread of multistrains COVID-19 virus among humans: susceptible, exposed, infected, quarantined, hospitalized, and recovered individuals. The positivity and boundedness of the system solution are provided in order to get the well posedness of the proposed model. Secondly, three controls are considered in our model to minimize the multistrain spread of the disease, namely, vaccination, security campaigns, social distancing measures, and diagnosis. Furthermore, the optimal control problem and related optimality conditions of the Pontryagin type are discussed with the objective to minimize the number of infected individuals. Finally, numerical simulations are performed in the case of two strains of COVID-19 and with four control strategies. By using the incremental cost-effectiveness ratio (ICER) method, we show that combining vaccination with diagnosis provides the most cost-effective strategy.


---

## Body

## 1. Introduction

In late 2019, coronavirus (COVID-19) outbreak occurred in the Chinese city of Wuhan and then spread to most countries of the world. The virus caused the infection and death of millions. Consequently, several countries have had to resort to the quarantine strategy, which is based on preventing the movement of people and animals from countries suffering from this disease. It also consists of establishing control points to prevent and regulate exit and entry in areas affected by the pandemic with the imposition of a health emergency in all countries of the world to limit the spread of COVID-19. After taking such measures, the number of infections and deaths decreased, but the quarantine strategy has greatly affected the global economy and made it shrink [1]. Domestic violence, unemployment, and psychological problems increased, especially for the elderly with disabilities and chronic diseases like diabetes, hypertension, and cardiovascular disease. These health problems are important risk factors for severity and mortality in people infected with COVID-19 [2].The COVID-19 virus mutated and induced a second strain, SARS-CoV-2 VOC, that appeared on December 14, 2020, initially in South-East England. The UK authorities informed the World Health Organization of a variant named by the United Kingdom as SARS-CoV-2 VOC 202012/01. The number of COVID-19 infections rose rapidly as well as hospital admissions in England in December after a decrease in infections in late November. Thus, an estimation of 1,122,000 people in England, or 1 in 50 people, was infected with COVID-19 in the first week of January 2021. This number accounts for 2.06% of the population and is more than twice the positivity rate recorded in the first week of December 2020, when 0.88% of the population were estimated to have the virus. The number of hospitalized people with COVID-19 also rose in the first week of January 2021 to 27.8 admissions per 100,000 people. This is almost twice the rate seen in the first week of December 2020, of 14 per 100,000 people, 8 January 2021 [3].Following England, South Africa’s national authorities announced on December 18 the outbreak of a new type of SARS-CoV-2 virus described by rapid spreading in three provinces of South Africa. South Africa gave the name 501Y.V2 to this variant, due to the N501Y mutation. SARS-CoV-2 VOC 2012/01 originated in the UK also has the N501Y mutation. Phylogenetic analysis showed that 501Y.V2 from South Africa is a different viral variant. In the third week of November 16, after conducting routine sequencing, South African health authorities concluded that this new variant of SARS-CoV-2 had largely supplanted other SARS-CoV-2 viruses existing in the Eastern Cape provinces, the Western Cape, and KwaZulu-Natal. While the genomic data demonstrated that the 501.V2 variant displaced other strains circulating in South Africa in a fast pace, preliminary studies indicate that the variant is associated with a higher viral load, which may refer to the possibility of increased transmissibility. Other factors having impact on transmissibility are also under investigation. Thorough studies are still needed to understand the new variant’s effect on transmission, clinical severity of infection, laboratory diagnosis, treatments, vaccinations, or public health preventive measures [3].The WHO confirmed that since it was first reported on Dec. 14, the mutated virus discovered by Britain has been announced in 50 countries, territories, and regions. The strain that was found in South Africa and first reported on December 18 exists up to now in 20 countries. The United Nations has expressed concern over the new variants like Alpha, Beta, Gamma, and Delta. These variants may have a serious effect on the immune response and need more investigation according to the organization. Indeed, the virus dissemination means it is more likely to mutate. Also, the high levels of transmission lead to the emergence of more strains [3].Since COVID-19 transmission continues and the number of infected people rises by the second strain, it was discovered that the virus affects all age groups. The new strain of COVID-19 happens to be detrimental for all people even if they are hospitalized due to the current pandemic. For example, data in Figure1 demonstrate that the likelihood of infections from the disease rise with the second strain in the UK and South Africa [4, 5].Figure 1
Schematic diagram of the total confirmed cases in the UK and South Africa.
(a)
Number of cases in South Africa(b)
Number of cases in UKConsequently, many mathematical studies were suggested to apprehend different factors that can lead to a disease transmission and to decide on the future control strategies to eradicate the disease. For example, in [6], the authors have introduced a new modelling approach based on the account of the age structure and uses fractional-order derivatives model, aiming to describe the transmission of COVID-19 and investigate the impact of some control strategies on its spread. The work done in [7] introduced a new approach of modelling based on a multiregion discrete time model that is composed of two groups: the human population and the animal population in different regions. It aimed to describe the spatial-temporal evolution of COVID-19 which appears in different geographical regions and showing the impact of one region on another. Moreover, they suggested several control strategies, including campaigns of awareness in a certain region, security campaigns, health measures to stop the movement of individuals from one region to another, and urging the individuals to join quarantine centers and the disposing of infected animals. In [8], the authors presented a mathematical model containing five compartments to depict the transmission evolution of the COVID-19 virus. The population is split into potential people, infected people without symptoms, people with serious complications, individuals under health surveillance and quarantined, and people who recovered from the disease. In addition, optimal control strategies were introduced, which include spreading awareness campaigns for citizens along with practical measures to limit the propagation of the virus including diagnosis of individuals, surveillance of airports, and imposing quarantine on infected people. Rahim et al. [9] have introduced a four compartments model for the description of the ongoing COVID-19. They have set up global and local dynamics for the established model. To decrease the number of infections and to minimize the cost of applying government control guidelines, an optimal control problem is formulated and solved in [10]. In [11], Ndii et al. proposed a deterministic and stochastic mathematical model to determine the probability of extinction of COVID-19 transmission. In addition, they calculated the proportion of undetected infected individuals and assessed the effectiveness of the following strategies: reducing the transmission rate and accelerating the detection time of infected persons. They found that the combination of the two strategies is effective in slowing outbreaks. They also found that accelerating the detection time of infected people without reducing the rate of transmission is not enough to slow epidemics.Generally, the existing models in the literature vary on the choice of the compartmental model, on the constraints enforced, or on the functional cost. These models are generally governed by ordinary equations [11–16], partial [17, 18] and stochastic differential equations [11, 19, 20]. More recently, fractional differential equations are increasingly used to model several phenomena.The above cited works, and the majority of models in the literature, did not take into consideration the possibility of multiple viral strain of COVID-19. However, multistrain models should be used to understand a mutated COVID-19 virus spreading into a population. Such models have been used in fewer studies, as in [21], where the authors investigated the global stability of two-strain SEIR model with two general incidence rates, and even for multistrain SEIR model with saturated incidence and treatment controls as in [22], which also deals with an optimal control problem. Local stability of multistrain SIR model with selective immunity by vaccination has been treated in [23]. The processes of mutation were also observed in many other infections such as tuberculosis [24–26]. Motivated by the relevance and scarcity of multistrain COVID-19 models, we intend to offer an extension of the classical models that describe the dynamics of COVID-19 with multistrains of infectious disease. Moreover, we propose some effective control strategies in order to minimize the infected individuals. More precisely, the main contributions of this paper are summarized as follows:
(i)
Evolving an epidemic model taking into account multistrain characteristics of COVID-19 and evaluating the impact of certain control strategies(ii)
Offering a developed SEIR model, where the population is divided inton+5 compartments. Particularly, Susceptible (S), Exposed (L), Infectious with i-th strain of COVID-19 Ii, where i=1,⋯,n, Hospitalized (H), Quarantined (Q), and Recovered (R) individuals(iii)
Seeking effective optimal control strategies that minimize the amount of infected population with symptoms of all strain types. To satisfy this objective, we introduce optimal control strategies related to three sorts of controls: the first control represents the density of susceptible individuals vaccinated per time unit; the second control represents the effort of the awareness programs, security campaigns, and social distancing measures to protect the susceptible individuals not to become infected. The third control is meant to encourage the infected individuals with strains of COVID-19 to join quarantine centers or go to hospitals(iv)
Providing numerical simulations in the case of two strains of COVID-19 along with a combination of the three control strategies. By using the incremental cost-effectiveness ratio (ICER) method, we prove that the combination of vaccination with diagnosis provides the most cost-effective strategyThe paper is organized as follows: In Section2, we introduce a novel mathematical model that illustrates the dynamics of the population and the transmission of the strains of COVID-19 disease as well as the existence and uniqueness of solution of the model. In Section 3, we design the optimal control terms by using Pontryagin’s maximum principle. While Section 4 discuss some numerical simulation results, and cost-effectiveness analysis is given in Section 5. Finally, we finish this paper in Section 6.


## 2. Multistrains COVID-19 Model and Analysis

### 2.1. Description of the Model

In this section, we describe the dynamics of the population and the transmission of the strains of COVID-19 disease. The population is grouped inton+5 compartments: Susceptible S, Exposed L, Infected with ith strain Ii, Hospitalized H, Quarantined Q, and Recovered R. In Figure 2, we build a graphical representation of the proposed model.Figure 2
Illustrative design of then+5 infectious classes in the population.Then, the model can be governed by the followingn+5-ordinary differential equations:
(1a)dStdt=Λ−β¯StLtN−∑i=1nβiStIitN−μSt,dLtdt=β¯StLtN+∑i=1nβiStIitN−μ+∑i=1nγiLt,dIitdt=γiLt−μ+δi+αiIit,fori=1,2,.⋯⋯,n,dHtdt=∑i=1nσαiIit−μ+η1+ρ1Ht+η3Qt,dQtdt=∑i=1n1−σαiIit−μ+η2+η3+ρ2Qt,dRtdt=η1Ht+η2Qt−μRt,with initial states
(1b)S0≥0,L0≥0,Ii0≥0,H0≥0,Q0≥0,R0≥0.The compartmentS: is the group of susceptible persons who are not infected but can get sick. The population is born into this compartment at a recruitment rate presented by Λ and gets infected by an effective contact with L at a rate β¯, and with the infectious individuals Ii at βi rate. It is also decreased by a natural death at a rate μ. It is assumed that susceptible individuals can become infectious through effective contact with infected individuals in workplace or house.The compartmentL: is composed of the exposed who are carriers of the virus without symptoms or with low symptoms and can spread the virus and become infected with symptoms. It is increased by susceptible S who becomes exposed at the rates β¯,β1,β2,..⋯,βn, This group is decreased by natural death at rate μ and when the exposed become infectious with symptoms at rates γ1,γ2,..⋯and γn.The compartmentIi: contains infectious individuals with symptoms and with a ith strain of corona. It becomes larger as the amount of the infectious individuals with symptoms and with a i strain of corona increase by the rate γi (γi is a rate the exposed individuals who get infected with symptoms) and decreases when some of them recovered at a rate αi (αi is a rate of the infected individuals with symptoms who are hospitalized or quarantined). Furthermore, this class decreases by natural death rate μ and by δi; the death rate of infectious person with ith strains of COVID-19.The compartmentH: illustrates the individuals who join hospitals, where σαi denoted the recruitment rate of individuals and Ii who have joined hospitals. Some individuals in quarantine go to a hospital at a rate η3 (η3 is a rate of quarantined individuals who have been hospitalized) and decrease when they become recovered at a rate η1 (a rate of the hospitalized individuals who have recovered). Furthermore, H decreases by natural death μ and by ρ1 the death rate of infected individuals with COVID-19 in hospital.The compartmentQ: encompasses the individuals who have been quarantined, with ∑i=11−σαi is the recruitment rate of individuals who have infected with new strains of COVID-19. While η2 represents a rate of the quarantine individuals that recovered and η3 is a rate of the quarantine individuals that hospitalized. Also, the compartment Q decreases by natural death μ and by ρ2 the death rate of quarantined infected individuals.The compartmentR: consists of individuals group who recovered from COVID-19 virus, where parameters η1 and η2 are the recruitment rates of individuals (H and Q) who have been treated. Also, R is decreased by natural death μ.Thus, the total population at timet is presented by
(2)Nt=St+Lt+∑i=1nIit+Ht+Qt+Rt.


### 2.2. Positivity and Boundedness of Solution

Motivated by the biological meaning of the mathematical model (1a), we prove, in this section, the positivity and boundedness of the solution of the system (1a) on a defined subset.Lemma 1.
Assume that condition (1b) is satisfied, then the solution of system (1a) is positive for any time t>0.Proof.
From the system (1a), we have:
(3)dStdt=Λ−β¯LtN+∑i=1nβiIitN+μSt.
By taking(4)Mt≐β¯LtN+∑i=1nβiIitN+μ,it follows from Equation (3) that
(5)dStdt+MtSt≥0.
Multiplying Equation (5) by exp∫0tMsds, we get
(6)dStdtexp∫0tMsds+MtStexp∫0tMsds≥0,which implies
(7)ddtStexp∫0tMsds≥0.
Integrating Equation (7), we have
(8)St≥S0exp−∫0tMsds.
Therefore, the solutionSt≥0 for all t≥0.
Similarly, from the other equations of the system (1a), it is easy to get the following inequalities:
(9)Lt≥L0exp−μ+∑i=1nγit≥0,Iit≥Ii0exp−μ+δi+αit≥0,fori=1,2,⋯,n(10)Ht≥H0exp−μ+η1+ρ1t≥0,(11)Qt≥Q0exp−μ+η2+η3+ρ2t≥0,and
(12)Rt≥R0exp−μt≥0,for all t≥0. This completes the proof.Lemma 2.
For all initial condition (1b), the solution of system (1a) is bounded by the region
(13)Ω=S,L,I1,I2,⋯,In,H,Q,R∈ℝ+n+5:S+L+∑i=1nIi+H+Q+R≤Λμ.Proof.
From the system (1a), we get
(14)dNtdt=Λ−μNt−∑i=1nIiδi−ρ1Ht−ρ2Qt.
Which implies that(15)dNtdt≤Λ−μNt.
Hence,(16)Nt≤Λμ+N0e−μt.
Thus,(17)limsupt⟶+∞Nt≤Λμ.Lemma 3.
If the initial condition (1b) is satisfied, then the system (1a) admits a unique global solution in ℝ+n+5.Proof.
Let us express the model (1a) in the form Y·=gY, where
(18)ccY·=dStdtdLtdtdIitdtdHtdtdQtdtdRtdt,gY=Λ−β¯StLtN−∑i=1nβiStIitN−μStβ¯StLtN+∑i=1nβiStIitN−μ+∑i=1nγiLtγiLt−μ+δi+αiIit∑i=1nσαiIit−μ+η1+ρ1Ht+η3Qt∑i=1n1−σαiIit−μ+η2+η3+ρ2Qtη1Ht+η2Qt−μRt.
It can be seen easily that the first derivative ofg is continuous, then it is locally Lipschitz. According to the fundamental existence and uniqueness theorem [27], Lemma 1 and 2, the system (1a) has a unique positive and bounded solution.


## 2.1. Description of the Model

In this section, we describe the dynamics of the population and the transmission of the strains of COVID-19 disease. The population is grouped inton+5 compartments: Susceptible S, Exposed L, Infected with ith strain Ii, Hospitalized H, Quarantined Q, and Recovered R. In Figure 2, we build a graphical representation of the proposed model.Figure 2
Illustrative design of then+5 infectious classes in the population.Then, the model can be governed by the followingn+5-ordinary differential equations:
(1a)dStdt=Λ−β¯StLtN−∑i=1nβiStIitN−μSt,dLtdt=β¯StLtN+∑i=1nβiStIitN−μ+∑i=1nγiLt,dIitdt=γiLt−μ+δi+αiIit,fori=1,2,.⋯⋯,n,dHtdt=∑i=1nσαiIit−μ+η1+ρ1Ht+η3Qt,dQtdt=∑i=1n1−σαiIit−μ+η2+η3+ρ2Qt,dRtdt=η1Ht+η2Qt−μRt,with initial states
(1b)S0≥0,L0≥0,Ii0≥0,H0≥0,Q0≥0,R0≥0.The compartmentS: is the group of susceptible persons who are not infected but can get sick. The population is born into this compartment at a recruitment rate presented by Λ and gets infected by an effective contact with L at a rate β¯, and with the infectious individuals Ii at βi rate. It is also decreased by a natural death at a rate μ. It is assumed that susceptible individuals can become infectious through effective contact with infected individuals in workplace or house.The compartmentL: is composed of the exposed who are carriers of the virus without symptoms or with low symptoms and can spread the virus and become infected with symptoms. It is increased by susceptible S who becomes exposed at the rates β¯,β1,β2,..⋯,βn, This group is decreased by natural death at rate μ and when the exposed become infectious with symptoms at rates γ1,γ2,..⋯and γn.The compartmentIi: contains infectious individuals with symptoms and with a ith strain of corona. It becomes larger as the amount of the infectious individuals with symptoms and with a i strain of corona increase by the rate γi (γi is a rate the exposed individuals who get infected with symptoms) and decreases when some of them recovered at a rate αi (αi is a rate of the infected individuals with symptoms who are hospitalized or quarantined). Furthermore, this class decreases by natural death rate μ and by δi; the death rate of infectious person with ith strains of COVID-19.The compartmentH: illustrates the individuals who join hospitals, where σαi denoted the recruitment rate of individuals and Ii who have joined hospitals. Some individuals in quarantine go to a hospital at a rate η3 (η3 is a rate of quarantined individuals who have been hospitalized) and decrease when they become recovered at a rate η1 (a rate of the hospitalized individuals who have recovered). Furthermore, H decreases by natural death μ and by ρ1 the death rate of infected individuals with COVID-19 in hospital.The compartmentQ: encompasses the individuals who have been quarantined, with ∑i=11−σαi is the recruitment rate of individuals who have infected with new strains of COVID-19. While η2 represents a rate of the quarantine individuals that recovered and η3 is a rate of the quarantine individuals that hospitalized. Also, the compartment Q decreases by natural death μ and by ρ2 the death rate of quarantined infected individuals.The compartmentR: consists of individuals group who recovered from COVID-19 virus, where parameters η1 and η2 are the recruitment rates of individuals (H and Q) who have been treated. Also, R is decreased by natural death μ.Thus, the total population at timet is presented by
(2)Nt=St+Lt+∑i=1nIit+Ht+Qt+Rt.


## 2.2. Positivity and Boundedness of Solution

Motivated by the biological meaning of the mathematical model (1a), we prove, in this section, the positivity and boundedness of the solution of the system (1a) on a defined subset.Lemma 1.
Assume that condition (1b) is satisfied, then the solution of system (1a) is positive for any time t>0.Proof.
From the system (1a), we have:
(3)dStdt=Λ−β¯LtN+∑i=1nβiIitN+μSt.
By taking(4)Mt≐β¯LtN+∑i=1nβiIitN+μ,it follows from Equation (3) that
(5)dStdt+MtSt≥0.
Multiplying Equation (5) by exp∫0tMsds, we get
(6)dStdtexp∫0tMsds+MtStexp∫0tMsds≥0,which implies
(7)ddtStexp∫0tMsds≥0.
Integrating Equation (7), we have
(8)St≥S0exp−∫0tMsds.
Therefore, the solutionSt≥0 for all t≥0.
Similarly, from the other equations of the system (1a), it is easy to get the following inequalities:
(9)Lt≥L0exp−μ+∑i=1nγit≥0,Iit≥Ii0exp−μ+δi+αit≥0,fori=1,2,⋯,n(10)Ht≥H0exp−μ+η1+ρ1t≥0,(11)Qt≥Q0exp−μ+η2+η3+ρ2t≥0,and
(12)Rt≥R0exp−μt≥0,for all t≥0. This completes the proof.Lemma 2.
For all initial condition (1b), the solution of system (1a) is bounded by the region
(13)Ω=S,L,I1,I2,⋯,In,H,Q,R∈ℝ+n+5:S+L+∑i=1nIi+H+Q+R≤Λμ.Proof.
From the system (1a), we get
(14)dNtdt=Λ−μNt−∑i=1nIiδi−ρ1Ht−ρ2Qt.
Which implies that(15)dNtdt≤Λ−μNt.
Hence,(16)Nt≤Λμ+N0e−μt.
Thus,(17)limsupt⟶+∞Nt≤Λμ.Lemma 3.
If the initial condition (1b) is satisfied, then the system (1a) admits a unique global solution in ℝ+n+5.Proof.
Let us express the model (1a) in the form Y·=gY, where
(18)ccY·=dStdtdLtdtdIitdtdHtdtdQtdtdRtdt,gY=Λ−β¯StLtN−∑i=1nβiStIitN−μStβ¯StLtN+∑i=1nβiStIitN−μ+∑i=1nγiLtγiLt−μ+δi+αiIit∑i=1nσαiIit−μ+η1+ρ1Ht+η3Qt∑i=1n1−σαiIit−μ+η2+η3+ρ2Qtη1Ht+η2Qt−μRt.
It can be seen easily that the first derivative ofg is continuous, then it is locally Lipschitz. According to the fundamental existence and uniqueness theorem [27], Lemma 1 and 2, the system (1a) has a unique positive and bounded solution.


## 3. Optimal Control Design

Worldwide, many countries suffer from the pandemic propagation of COVID-19, which result in deaths, infection, and health and economic damage and as consequence, it impacts negatively the individuals and societies. For these reasons, we propose some control strategies which will contribute to reduce the amount of infectious people with strains of COVID-19Iit for i∈1,2,⋯,n, during the time interval t0,tf, and also minimize the cost spent of vaccination and awareness programs.To achieve these objectives, we provide three control variables. The first controlu denotes the effort of the vaccination of susceptible individuals by assuming that all susceptible vaccinated are transferred directly to the removed class. The second control v provides the effort of the awareness programs, security campaigns, and social distancing measures to save the susceptible people (S) from the infection. Hence, the term 1−v is introduced to reduce the force of infections. The last control wi represents the encouraging and diagnosing the infected with ith strain of COVID-19 to join hospitals or quarantined. Then, the system under controls is given by the following system:
(19)dStdt=Λ−β¯1−vtStLtN−∑i=1n1−vtβiStIitN−μSt−utSt,dLtdt=β¯1−vtStLtN+∑i=1n1−vtβiStIitN−μ+∑i=1nγiLt,dIitdt=γiLt−μ+δi+αiIit−witIit,fori=1,2,⋯,n,dHtdt=∑i=1nσαiIit−μ+η1+ρ1Ht+η3Qt+χwitIit,dQtdt=∑i=1n1−σαiIit−μ+η2+η3+ρ2Qt+1−χwitIit,dRtdt=η1Ht+η2Qt−μRt+utSt,with initial conditions
(20)S0≥0,L0≥0,Ii0≥0,H0≥0,Q0≥0,R0≥0.Now, we will introduce the optimal control problem, related to the following objective functional:(21)Ju,v,wi=∑i=1nIitf+∫t0tf∑i=1nIit+A12u2t+A22v2t+∑i=1nBi2wi2tdt,fori=1,2,⋯,nwhere the parameters A1, A2, and Bi are strictly positive cost coefficients. They are selected to weigh the relative importance of u, v, and wi at time t and tf is the final time.In other words, we look for optimal controlsu, v, and wi in such a way
(22)Ju,v,wi=minu,v,wi∈UadJu,v,wi,for i=1,2,⋯,n, with Uad is the set of admissible controls defined by
(23)Uad=ut,vt,wit:0≤ut≤1,0≤vt≤1,0≤wit≤1,fori=1,2,⋯,n,andt∈t0,tf.


### 3.1. Design of Optimal Controls

In this section, we will use the Pontryagin maximum principle [18, 28] to get the expression of optimal controls and in order to minimize the number of infectious individuals and the cost of control strategies.Now, LetH^ be the Hamiltonian defined, in time t, by
(24)H^t=∑i=1nIit+A12u2t+A22v2t+∑i=1nBi2wi2t+∑k=1n+5λkfk,where fk is the right side of the system (21) of the kth component variable.Theorem 4.
If the coupleS∗,L∗,I1∗,I2∗⋯,In∗,H∗,Q∗,R∗ is the solution of the system (19) related to an optimal control u∗,v∗,wi∗∈Uad, then there exist adjoint functions λ1, λ2, λi+2, λn+3, λn+4, and λn+5 satisfying the equations:
(25)λ̇1=−∂H^∂S=λ1−λ2β¯LtN+∑i=1nβiIitN+λ1−λn+5ut+μλ1,λ̇2=−∂H^∂L=λ1−λ2β¯1−vtStN+λ2∑i=1nγi−γiλi+2+μλ2,λ̇i+2=−∂H^∂Ii=−1+λ1−λ2∑i=1nβiStN+γiλi+2−λn+3∑i=1nσαi−λi+4∑i=1n1−σαi+λi+2−χλn+3−1−χλn+4wi+μ+δiλi+2,fori=1,2,⋯,nλ̇n+3=−∂H^∂H=η1λn+3−λn+5+μ+ρ1λn+3,λ̇n+4=−∂H^∂Q=η3λn+4−λn+3+η2λn+4−λn+5+μ+ρ2λn+4,λ̇n+5=−∂H^∂R=μλn+5,with the transversality conditions at time tf(26)λ1tf=λ2tf=λn+3tf=λn+4tf=λn+5tf=0andλi+2tf=1.
Furthermore, fort∈t0;tf, the optimal controls u∗t, v∗t, and wi∗t are given by:
(27)u∗t=min1,max0,λ1−λn+5StA1,(28)v∗t=min1,max0,λ1−λ2β¯StEt/N+∑i=1nβiStIit/NA2,(29)wi∗t=min1,max0,λi+2−χλi+3−1−χλi+4IitBi.Proof.
The expression of the HamiltonianH^ is given as follows
(30)H^t=∑i=1nIit+A12u2t+A22v2t+∑i=1nBi2wi2t+λ1Λ−β¯1−vtStLtN−∑i=1n1−vtβiStIitN−μSt−utSt+λ2β¯1−vtStLtN+∑i=1n1−vtβiStIitN−μ+∑i=1nγiLt+λi+2γiLt−μ+δi+αiIit−witIit+λn+3∑i=1nσαiIit−μ+η1+ρ1Ht+η3Qt+χwitIit+λn+4∑i=1n1−σαiIit−μ+η2+η3+ρ2Qt+1−χwitIit+λn+5η1Ht+η2Qt−μRt+utSt,for t∈t0;tf. Thanks to Pontryagin’s maximum principle, the adjoint equations and transversality conditions can be obtained such that
(31)λ̇1=−∂H^∂S,λ1tf=0,λ̇2=−∂H^∂E,λ2tf=0,λ̇i+2=−∂H^∂Ii,λi+2tf=1,λ̇n+3=−∂H^∂H,λn+3tf=0,λ̇n+4=−∂H^∂Q,λn+4tf=0,λ̇n+5=−∂H^∂R,λn+5tf=0,for t∈t0;tf. The expression of optimal controls u∗t, v∗t, and wi∗t can be given via the optimality condition,
(32)∂H^∂u=0,∂H^∂v=0,and∂H^∂wi=0.
That is(33)∂H^∂u=A1ut+λi+5−λ1St=0,∂H∂v=A2vt+λ2−λ1β¯StLtN+∑i=1nβiStIitN=0,∂H∂wi=Biwit+−λi+2+χλi+3+1−χλi+4Iit=0.
Consequently,(34)u∗t=min1,max0,λ1−λi+5StA1,v∗t=min1,max0,λ1−λ2β¯StLt/N+∑i=1nβiStIit/NA2,wi∗t=min1,max0,λi+2−χλi+3−1−χλi+4IitBi.
By the bounds, inUad, of the controls, it is easy to obtain u∗t, v∗t, and wi∗t in the form of Equations (27), (28), and (29)), respectively.


## 3.1. Design of Optimal Controls

In this section, we will use the Pontryagin maximum principle [18, 28] to get the expression of optimal controls and in order to minimize the number of infectious individuals and the cost of control strategies.Now, LetH^ be the Hamiltonian defined, in time t, by
(24)H^t=∑i=1nIit+A12u2t+A22v2t+∑i=1nBi2wi2t+∑k=1n+5λkfk,where fk is the right side of the system (21) of the kth component variable.Theorem 4.
If the coupleS∗,L∗,I1∗,I2∗⋯,In∗,H∗,Q∗,R∗ is the solution of the system (19) related to an optimal control u∗,v∗,wi∗∈Uad, then there exist adjoint functions λ1, λ2, λi+2, λn+3, λn+4, and λn+5 satisfying the equations:
(25)λ̇1=−∂H^∂S=λ1−λ2β¯LtN+∑i=1nβiIitN+λ1−λn+5ut+μλ1,λ̇2=−∂H^∂L=λ1−λ2β¯1−vtStN+λ2∑i=1nγi−γiλi+2+μλ2,λ̇i+2=−∂H^∂Ii=−1+λ1−λ2∑i=1nβiStN+γiλi+2−λn+3∑i=1nσαi−λi+4∑i=1n1−σαi+λi+2−χλn+3−1−χλn+4wi+μ+δiλi+2,fori=1,2,⋯,nλ̇n+3=−∂H^∂H=η1λn+3−λn+5+μ+ρ1λn+3,λ̇n+4=−∂H^∂Q=η3λn+4−λn+3+η2λn+4−λn+5+μ+ρ2λn+4,λ̇n+5=−∂H^∂R=μλn+5,with the transversality conditions at time tf(26)λ1tf=λ2tf=λn+3tf=λn+4tf=λn+5tf=0andλi+2tf=1.
Furthermore, fort∈t0;tf, the optimal controls u∗t, v∗t, and wi∗t are given by:
(27)u∗t=min1,max0,λ1−λn+5StA1,(28)v∗t=min1,max0,λ1−λ2β¯StEt/N+∑i=1nβiStIit/NA2,(29)wi∗t=min1,max0,λi+2−χλi+3−1−χλi+4IitBi.Proof.
The expression of the HamiltonianH^ is given as follows
(30)H^t=∑i=1nIit+A12u2t+A22v2t+∑i=1nBi2wi2t+λ1Λ−β¯1−vtStLtN−∑i=1n1−vtβiStIitN−μSt−utSt+λ2β¯1−vtStLtN+∑i=1n1−vtβiStIitN−μ+∑i=1nγiLt+λi+2γiLt−μ+δi+αiIit−witIit+λn+3∑i=1nσαiIit−μ+η1+ρ1Ht+η3Qt+χwitIit+λn+4∑i=1n1−σαiIit−μ+η2+η3+ρ2Qt+1−χwitIit+λn+5η1Ht+η2Qt−μRt+utSt,for t∈t0;tf. Thanks to Pontryagin’s maximum principle, the adjoint equations and transversality conditions can be obtained such that
(31)λ̇1=−∂H^∂S,λ1tf=0,λ̇2=−∂H^∂E,λ2tf=0,λ̇i+2=−∂H^∂Ii,λi+2tf=1,λ̇n+3=−∂H^∂H,λn+3tf=0,λ̇n+4=−∂H^∂Q,λn+4tf=0,λ̇n+5=−∂H^∂R,λn+5tf=0,for t∈t0;tf. The expression of optimal controls u∗t, v∗t, and wi∗t can be given via the optimality condition,
(32)∂H^∂u=0,∂H^∂v=0,and∂H^∂wi=0.
That is(33)∂H^∂u=A1ut+λi+5−λ1St=0,∂H∂v=A2vt+λ2−λ1β¯StLtN+∑i=1nβiStIitN=0,∂H∂wi=Biwit+−λi+2+χλi+3+1−χλi+4Iit=0.
Consequently,(34)u∗t=min1,max0,λ1−λi+5StA1,v∗t=min1,max0,λ1−λ2β¯StLt/N+∑i=1nβiStIit/NA2,wi∗t=min1,max0,λi+2−χλi+3−1−χλi+4IitBi.
By the bounds, inUad, of the controls, it is easy to obtain u∗t, v∗t, and wi∗t in the form of Equations (27), (28), and (29)), respectively.


## 4. Numerical Simulation and Discussion

Now, we treat numerically the optimal control problem of theSLI1I2HQR model. Here, we obtain the optimality system from the state and adjoint equations. The optimality system is solved by an iterative method. Using an initial guess for the control variables, ut, vt, and wit with (i =1,2), the state variables S, L, I1, I2, H, Q, and R are solved forward and the adjoint variables λi for i=1,2,3,4,5,6,7 are solved backwards at times step k=t0 and k=tf. If the new values of the state and adjoint variables differ from the previous values, the new values are used to update uk, vk, and wi,k (i =1,2) and the process is repeated until the system converges.Based on real data of COVID-19 confirmed cases of the UK, the parameter estimation is made for the first 100 days from 10 Sep 2020 to 10 Jan 2021 (available online at [29]). The total population of the UK is approximately N=66.460.344 and the life expectancy of the UK [30] for the year 2020 is 81. Hence, the natural death rate is calculated to be μ=1/81∗365/per day. Furthermore, at the onset of the pandemic in the UK, only one symptomatic case was confirmed, no individual was confirmed asymptomatic, no hospitalized, and no recovery cases. Consequently, assuming initial number of exposed cases to be 300, the initial number of population is assumed N0=66.460.344 where Nt=St+Lt+I1t+I2t+Ht+Qt+Rt. The parameter Λ is calculated from the relation Λ=μNt and hence, Λ=8.175.322/day, σ=1/14/day, β¯=0.3,β1=0.4, β2=0.7, γ1=0.001, γ2=0.0019, α1=0.04, α2=0.03, η1=0.002, η2=0.001,η3=0.002., δ1=0.0002., δ2=0.0002., ρ1=0.01., ρ2=0.01.Some numerical simulations are presented here in order to illustrate the theoretical results and take into account three control strategies. Then, we consider system (1) with the parameters mentioned above and with initial conditions S0=66460000, L0=300, I1,0=20, I2,0=20, H0=3, Q0=1, and R0=0.


### 4.1. Strategy 1: Vaccinating Susceptible Individuals and Protecting Them from Contacting Infected Individuals

To realize this objective, we apply only the controlsu and v, i.e., vaccinating susceptible individuals having priority who represent individuals with chronic diseases and frontline workers as well as protecting them from contacting infected individuals.In Figure3(a), it is observed that there is a significant decrease in the number of infected individuals with the first strain of COVID-19 with control compared to a situation where there is no control which induces a decrease from 1,32.106 to 6,93.105 at the end of the implementation of the proposed control strategy. Figure 3(b) shows that the number of the infected individuals with second strain of COVID-19 decreased from 3,05.106 (without control) to 1,54.106 (with control) at the end of the implementation of the proposed control. These changes are important but not sufficient; it is for this reason that we must also add other controls.Figure 3
The number of infected individuals with both first (a) and second (b) strain when controlsu and v are applied (dashed line), and without controls (solid line).
(a)(b)


### 4.2. Strategy 2: Vaccinating and Protecting Susceptible Individuals from Contacting Infected Individuals and Encouraging the Symptomatic Infected Individuals with the First Strain of COVID-19 to Go to Hospitals or Do Self-Quarantine

To realize this objective, we apply only the controlu, v, and w1, i.e., vaccinating susceptible individuals having priority who represent individuals with chronic diseases and frontline workers, protecting them from contacting infected individuals, and encouraging the symptomatic infected individuals with the first strain of COVID-19 to go to hospitals or do self-quarantine.Figure4 shows that the number of the infected individuals with the first strain of COVID-19 decreases from the value 1,32.106 (without controls) to 2,206.105 (with controls) at the end of the implementation of the proposed control strategy. As a result, the strategy set before has been achieved.Figure 4
The number of infected individuals with the first strain when controlsu, v, and w1 are applied (dashed line), and without controls (solid line).


### 4.3. Strategy 3: Vaccinating and Protecting Susceptible Individuals from Contacting Infected Individuals and Encouraging the Symptomatic Infected Individuals with the Second Strain of COVID-19 to Go to Hospitals or Do Self-Quarantine

To realize this objective, we apply only the controlu, v, and w2, i.e., vaccinating susceptible individuals having priority who represent individuals with chronic diseases and frontline workers, protecting them from contacting infected individuals, and encouraging the symptomatic infected individuals with the second strain of COVID-19 to go to hospitals or do self-quarantine.Figure5 shows that the number of the infected individuals with the second strain of COVID-19 decreases from 3,05.106 (without controls) to 4,44.105 (with controls) at the end of the implementation of the proposed control strategy. As a result, the strategy set before has been achieved.Figure 5
The number of infected individuals with second strain when controlsu, v, and w2 are applied (dashed line) and without control (solid line).


### 4.4. Strategy 4: Vaccinating Susceptible Individuals and Encouraging the Symptomatic Infected Individuals with the First and the Second Strain of COVID-19 to Go to Hospitals or Do Self-Quarantine

To realize this objective, we apply only the controlu, w1, and w2, i.e., vaccinating susceptible individuals having priority who represent the individuals with chronic diseases and encouraging the symptomatic infected individuals with the first and the second strain of COVID-19 to go to hospitals or do self-quarantine.Figure6(a) shows that the number of the infected individuals with first strain of COVID-19 decreases from 1,32.106 (without controls) to 2,565.105 (with controls) at the end of the implementation of the proposed control strategy. Figure 6(b) shows that the number of the infected individuals with second strain of COVID-19 decreases from 3,05.106 (without controls) to 5,25.105 (with controls) at the end of the implementation of the proposed control strategy. As a result, the strategy set before has been achieved.Figure 6
The number of infected individuals with both first (a) and second (b) strain when controlsu, w1, and w2 are applied (dashed line) and without control (solid line).
(a)(b)


## 4.1. Strategy 1: Vaccinating Susceptible Individuals and Protecting Them from Contacting Infected Individuals

To realize this objective, we apply only the controlsu and v, i.e., vaccinating susceptible individuals having priority who represent individuals with chronic diseases and frontline workers as well as protecting them from contacting infected individuals.In Figure3(a), it is observed that there is a significant decrease in the number of infected individuals with the first strain of COVID-19 with control compared to a situation where there is no control which induces a decrease from 1,32.106 to 6,93.105 at the end of the implementation of the proposed control strategy. Figure 3(b) shows that the number of the infected individuals with second strain of COVID-19 decreased from 3,05.106 (without control) to 1,54.106 (with control) at the end of the implementation of the proposed control. These changes are important but not sufficient; it is for this reason that we must also add other controls.Figure 3
The number of infected individuals with both first (a) and second (b) strain when controlsu and v are applied (dashed line), and without controls (solid line).
(a)(b)


## 4.2. Strategy 2: Vaccinating and Protecting Susceptible Individuals from Contacting Infected Individuals and Encouraging the Symptomatic Infected Individuals with the First Strain of COVID-19 to Go to Hospitals or Do Self-Quarantine

To realize this objective, we apply only the controlu, v, and w1, i.e., vaccinating susceptible individuals having priority who represent individuals with chronic diseases and frontline workers, protecting them from contacting infected individuals, and encouraging the symptomatic infected individuals with the first strain of COVID-19 to go to hospitals or do self-quarantine.Figure4 shows that the number of the infected individuals with the first strain of COVID-19 decreases from the value 1,32.106 (without controls) to 2,206.105 (with controls) at the end of the implementation of the proposed control strategy. As a result, the strategy set before has been achieved.Figure 4
The number of infected individuals with the first strain when controlsu, v, and w1 are applied (dashed line), and without controls (solid line).


## 4.3. Strategy 3: Vaccinating and Protecting Susceptible Individuals from Contacting Infected Individuals and Encouraging the Symptomatic Infected Individuals with the Second Strain of COVID-19 to Go to Hospitals or Do Self-Quarantine

To realize this objective, we apply only the controlu, v, and w2, i.e., vaccinating susceptible individuals having priority who represent individuals with chronic diseases and frontline workers, protecting them from contacting infected individuals, and encouraging the symptomatic infected individuals with the second strain of COVID-19 to go to hospitals or do self-quarantine.Figure5 shows that the number of the infected individuals with the second strain of COVID-19 decreases from 3,05.106 (without controls) to 4,44.105 (with controls) at the end of the implementation of the proposed control strategy. As a result, the strategy set before has been achieved.Figure 5
The number of infected individuals with second strain when controlsu, v, and w2 are applied (dashed line) and without control (solid line).


## 4.4. Strategy 4: Vaccinating Susceptible Individuals and Encouraging the Symptomatic Infected Individuals with the First and the Second Strain of COVID-19 to Go to Hospitals or Do Self-Quarantine

To realize this objective, we apply only the controlu, w1, and w2, i.e., vaccinating susceptible individuals having priority who represent the individuals with chronic diseases and encouraging the symptomatic infected individuals with the first and the second strain of COVID-19 to go to hospitals or do self-quarantine.Figure6(a) shows that the number of the infected individuals with first strain of COVID-19 decreases from 1,32.106 (without controls) to 2,565.105 (with controls) at the end of the implementation of the proposed control strategy. Figure 6(b) shows that the number of the infected individuals with second strain of COVID-19 decreases from 3,05.106 (without controls) to 5,25.105 (with controls) at the end of the implementation of the proposed control strategy. As a result, the strategy set before has been achieved.Figure 6
The number of infected individuals with both first (a) and second (b) strain when controlsu, w1, and w2 are applied (dashed line) and without control (solid line).
(a)(b)


## 5. Cost-Effectiveness Analysis

In this section, we analyze the cost-effectiveness of the previous four strategies by comparing between them to determine the most cost-effective strategy. Following the method as applied in several studies [31, 32], we evaluate the costs using the incremental cost-effectiveness ratio ICER. This ratio used to compare the differences between the costs and health outcomes of two competing intervention strategies.TheICER is defined as the quotient of the difference in costs in strategies i and j, by the difference in infected averted in strategies i and ji,j∈1,2,3,4.Given two competing strategiesE and F, where strategy F has higher effectiveness than strategy ETAF>TAE, the ICER values are calculated as follows:
(35)ICERE=TCETAE,ICERF=TCF−TCETAF−TAE.The total costs (TC) and the total cases averted (TA) are defined during a given period, for strategy i, with i=1,2,3,4, by:
(36)TCi=∫0tfC1utSt+C2vtSt+C3w1tI1t+C4w2tI2tdt,TAi=∫0tfEt+I1t+I2t−E∗t+I1∗t+I2∗tdt,where C1, C2, C3, and C4 correspond to the person unit cost of the four possible interventions, while E∗,I1∗,I2∗ is the optimal solution associated to the optimal control u∗,v∗,w1∗,w2∗.Using the simulation resultsC1=C2=C3=C4=1, we ranked in Table 1 our control strategies in order of increased numbers of averted infections.Table 1
Total costs and total averted infections for strategies 1-4.
StrategyTotal averted infections (TA)Total cost (TC)41,6829.1092,9307.10711,9557.1093,3107.10822,3679.1097,0051.10832,408.1097,0632.108Strategy4 is compared with strategy 1 with respect to increased effectiveness and in reference to Table 1. So:
(37)ICER4=TC4TA4=2,9307.1071,6829.109=0,017,ICER1=TC1−TC4TA1−TA4=3,3107.108−2,9307.1071,9557.109−1,6829.109=1,106.Strategy 1 is less effective than strategy 4, sinceICER4<ICER1. Hence, strategy 1 is eliminated from the set of alternatives.Next, we compare strategy 4 with strategy 2. TheICER values for strategy 4 and strategy 2 are calculated below:
(38)ICER4=TC4TA4=2,9307.1071,6829.109=0,017,ICER2=TC2−TC4TA2−TA4=7,0051.108−2,9307.1072,3679.109−1,6829.109=0,979.Strategy 4 is less effective than strategy 2, sinceICER2<ICER4. Consequently, strategy 4 is omitted from the set of alternatives.Now, we compare strategy 4 with strategy 3. TheICER values for strategy 4 and strategy 3 are calculated below:
(39)ICER4=TC4TA4=2,9307.1071,6829.109=0,017,ICER3=TC3−TC4TA3−TA4=7,0632.108−2,9307.1072,408.109−1,6829.109=0,933.Then, strategy 3 is less effective than strategy 4, sinceICER4<ICER3, Thereafter, strategy 3 is excluded from the set of alternatives.Hence, we conclude that strategy 4 (vaccinating susceptible people and encouraging the diagnosis of the symptomatic infected individuals with the first and the second strain of COVID-19 to go to hospitals or do self-quarantine) is the most effective strategy.


## 6. Conclusion

This work proposed a novel deterministic model of COVID-19 taking into consideration multistrain characteristics of the virus and optimal control measures in such a way to minimize the number of infected individuals. Therefore, unlike previous works in the literature, we have taken into account the impact of vaccination, hospitalization, quarantine, and social distancing on multistrains of COVID-19 propagation.Reducing contact, disinfecting, and imposing quarantine and hospitalization all have substantial influence on the dynamics of COVID-19 propagation and can greatly reduce the spread of multistrain COVID-19. Thus, it is crucial to urge people to take the vaccine and go to hospitals or do self-quarantine. For these reasons, we formulated an optimal control problem based on the proposed model to which we included three control measures: representing vaccination, awareness programs (security campaigns and/or social distancing measures), and diagnosing the infected individuals to join hospital or do self-quarantine. By using the Pontryagin maximum principle, we obtained the expression of the optimal controls which minimize the amount of infectious individuals and the cost of control strategies.Numerical simulations are carried out in the case of two-strain COVID-19 and with four different control strategies. The optimal strategies including vaccination of susceptible individuals, protection of individuals from being infected with the virus, security campaigns and health measures to prevent the contact of susceptible individuals with infected individuals, and diagnosing the infected with first and second straini to join hospital or do self-quarantine (strategies 1, 2, 3, and 4) showed significant differences in the number of infections with the first strain and with the second strain, quarantined and hospitalized individuals compared to a situation where no controls are applied. The optimal strategies used have positive effects in decreasing the number of the infected individuals with the first strain and the second strain and also on increasing the number of quarantined individuals. Furthermore, by using the incremental cost-effectiveness ratio method, we show that strategy 4 (vaccinating susceptible individuals and diagnosing the symptomatic infected individuals with the first and the second strain of COVID-19 to go to hospitals or do self-quarantine) is the most effective one.



---

*Source: 9071890-2022-07-16.xml*