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.

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 Figure 1 demonstrate that the likelihood of infections from the disease rise with the second strain in the UK and South Africa [4, 5].

(a) Number of cases in South Africa

(b) Number of cases in UK

(a) Number of cases in South Africa
(b) Number of cases in UK

Figure 1 

Schematic diagram of the total confirmed cases in the UK and South Africa.

Consequently, 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 into compartments. Particularly, Susceptible (), Exposed (), Infectious with -th strain of COVID-19 , where , Hospitalized (), Quarantined (), and Recovered () 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 strategy

The paper is organized as follows: In Section 2, 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 into compartments: Susceptible , Exposed , Infected with strain , Hospitalized , Quarantined , and Recovered . In Figure 2, we build a graphical representation of the proposed model.

Figure 2 

Illustrative design of the infectious classes in the population.

Then, the model can be governed by the following -ordinary differential equations: with initial states

The compartment : 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 and gets infected by an effective contact with at a rate , and with the infectious individuals at rate. It is also decreased by a natural death at a rate . It is assumed that susceptible individuals can become infectious through effective contact with infected individuals in workplace or house.

The compartment : 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 who becomes exposed at the rates ,,,, This group is decreased by natural death at rate and when the exposed become infectious with symptoms at rates and .

The compartment : contains infectious individuals with symptoms and with a strain of corona. It becomes larger as the amount of the infectious individuals with symptoms and with a strain of corona increase by the rate ( is a rate the exposed individuals who get infected with symptoms) and decreases when some of them recovered at a rate ( is a rate of the infected individuals with symptoms who are hospitalized or quarantined) Furthermore, this class decreases by natural death rate and by ; the death rate of infectious person with strains of COVID-19.

The compartment : illustrates the individuals who join hospitals, where denoted the recruitment rate of individuals and who have joined hospitals. Some individuals in quarantine go to a hospital at a rate ( is a rate of quarantined individuals who have been hospitalized) and decrease when they become recovered at a rate (a rate of the hospitalized individuals who have recovered). Furthermore, decreases by natural death and by the death rate of infected individuals with COVID-19 in hospital.

The compartment : encompasses the individuals who have been quarantined, with is the recruitment rate of individuals who have infected with new strains of COVID-19. While represents a rate of the quarantine individuals that recovered and is a rate of the quarantine individuals that hospitalized. Also, the compartment decreases by natural death and by the death rate of quarantined infected individuals.

The compartment : consists of individuals group who recovered from COVID-19 virus, where parameters and are the recruitment rates of individuals ( and ) who have been treated. Also, is decreased by natural death .

Thus, the total population at time is presented by

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

Proof. From the system (1a), we have: By taking it follows from Equation (3) that Multiplying Equation (5) by , we get which implies Integrating Equation (7), we have Therefore, the solution for all .
Similarly, from the other equations of the system (1a), it is easy to get the following inequalities: and for all . This completes the proof.

Lemma 2. For all initial condition (1b), the solution of system (1a) is bounded by the region

Proof. From the system (1a), we get Which implies that Hence, Thus,

Lemma 3. If the initial condition (1b) is satisfied, then the system (1a) admits a unique global solution in .

Proof. Let us express the model (1a) in the form where It can be seen easily that the first derivative of 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-19 for , during the time interval , and also minimize the cost spent of vaccination and awareness programs.

To achieve these objectives, we provide three control variables. The first control 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 provides the effort of the awareness programs, security campaigns, and social distancing measures to save the susceptible people () from the infection. Hence, the term is introduced to reduce the force of infections. The last control represents the encouraging and diagnosing the infected with strain of COVID-19 to join hospitals or quarantined. Then, the system under controls is given by the following system: with initial conditions

Now, we will introduce the optimal control problem, related to the following objective functional: where the parameters , , and are strictly positive cost coefficients. They are selected to weigh the relative importance of , , and at time and is the final time.

In other words, we look for optimal controls , , and in such a way for with is the set of admissible controls defined by

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, Let be the Hamiltonian defined, in time , by where is the right side of the system (21) of the component variable.

Theorem 4. If the couple is the solution of the system (19) related to an optimal control , then there exist adjoint functions , , , , , and satisfying the equations: with the transversality conditions at time Furthermore, for , the optimal controls , , and are given by:

Proof. The expression of the Hamiltonian is given as follows for . Thanks to Pontryagin’s maximum principle, the adjoint equations and transversality conditions can be obtained such that for . The expression of optimal controls , , and can be given via the optimality condition, That is Consequently, By the bounds, in , of the controls, it is easy to obtain , , and in the form of Equations (27), (28), and (29)), respectively.