Fractional optimal control problem for an age-structured model of COVID-19 transmission.

The aim of this study is to model the transmission of COVID-19 and investigate the impact of some control strategies on its spread. We propose an extension of the classical SEIR model, which takes into account the age structure and uses fractional-order derivatives to have a more realistic model. For each age group j the population is divided into seven classes namely susceptible S j , exposed E j , infected with high risk I h j , infected with low risk I l j , hospitalized H j , recovered with and without psychological complications R 1 j and R 2 j , respectively. In our model, we incorporate three control variables which represent: awareness campaigns, diagnosis and psychological follow-up. The purpose of our control strategies is protecting susceptible individuals from being infected, minimizing the number of infected individuals with high and low risk within a given age group j , as well as reducing the number of recovered individuals with psychological complications. Pontryagin's maximum principle is used to characterize the optimal controls and the optimality system is solved by an iterative method. Numerical simulations performed using Matlab, are provided to show the effectiveness of three control strategies and the effect of the order of fractional derivative on the efficiency of these control strategies. Using a cost-effectiveness analysis method, our results show that combining awareness with diagnosis is the most effective strategy. To the best of our knowledge, this work is the first that propose a framework on the control of COVID-19 transmission based on a multi-age model with Caputo time-fractional derivative.

Introduction

Appeared in Wuhan, China, on December 2019, the Coronavirus disease 2019, known as COVID-19, is one of the most dangerous pandemics that radically impacts our lives. This new disease develops as a viral zoonosis caused by a novel strain of corona virus called SARS-CoV-2 that affects the lower respiratory tract. According to clinical analysis, the most common symptoms at COVID-19 onset were fever, cough, and myalgia or fatigue [1]; while the less common symptoms were dizziness, headache, pharyngalgia, nausea, diarrhoea, and abdominal pain [2]. Despite the majority of cases have spontaneously resolved, some have developed various fatal complications including organ failure, septic shock, pulmonary edema, severe pneumonia, and Acute Respiratory Distress Syndrome [3]. Within one month of the first reported cases, the virus has spread rapidly throughout the world, and health system in several countries was exposed to a high risk. As of this writing, the COVID-19 has affected 218 countries and territories [4] causing 61,036,793 confirmed cases and a total of 1,433,316 deaths [5]. Nowadays, COVID-19 has become a global problem causing not only considerable health damage but also serious economic and social consequences. Due to the lack of a specific treatment for COVID-19, urgent measures have been taken that aimed at reducing exposure and contain the danger of this disease as the risk factor of contamination is very high. Among these measures we recall: improvement of personal and public hygiene, intense media coverage, social distancing, quarantine, lockdown and imposing the wearing of masks. To predict the effect of such measures, policy makers and public health authorities need effective tools to guide their efforts. In this context, mathematical modelling plays a crucial role to understand the pandemic behaviour of the COVID-19 disease and evaluate the effectiveness of the control strategies.

There are several mathematical modelling studies that have emerged which aim at understanding the COVID-19 dynamics. For example, Khajji et al [6] have devised a new modelling approach based on a multi-regions discrete time model that consists 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 emerges in different geographical regions and showing the influence of one region on another. Authors proposed several control strategies, including awareness campaigns in a given region, security campaigns, health measures to prevent the movement of individuals from one region to another and encouraging the individuals to join quarantine centers and the disposal of infected animals. Kouidere et al. [7], proposed a mathematical model with five compartments to describe transmission process of the COVID-19 virus. The population is divided into potential people, infected people without symptoms, people with serious complications as well as those under health surveillance and quarantine and people who recovered from the disease. In addition, optimal control strategies were proposed, which consist of conducting awareness campaigns for citizens along with practical measures to reduce the spread of the virus including diagnosis of individuals, surveillance of airports and imposing quarantine on infected people. Chen et al. [8] developed a Bats-Hosts-Reservoir-People transmission network model for simulating the potential transmission of SARS-CoV-2 from the infection source (probably be bats) to the human infection. The basic reproduction number () is calculated based on data published in a literature. The main result of this study showed that the transmissibility of SARS-CoV-2 might be higher than MERS in the Middle East countries, similar to SARS, but lower than MERS in the Republic of Korea. Tang et al. [9] investigate the effectiveness of quarantine and isolation strategy on the trend of the COVID-19 epidemics in the final phase of the current outbreak in China. A novel model in line with the epidemics process and control measures was proposed, utilizing multisource datasets including cumulative number of reported, death, quarantined and suspected cases. Mishra et al. [10] considered a mathematical model of COVID-19 disease with susceptible, exposed, infected, asymptotic, quarantine/isolation and recovered classes. They performed the elasticity and sensitivity analysis and investigate the global stability of their model. Mandal et al. [11] proposed a mathematical model with quarantine class and some governmental intervention measures namely lock-down, media coverage on social distancing, and improvement of public hygiene. To reduce the number of infected individuals as well as to minimize the cost of implementing government control measures, an optimal control problem is formulated and solved. For more examples of works that addresses the problem of modelling and controlling the COVID-19 transmission, we refer to [12], [13], [14], [15], [16].

Among the models used for epidemic analysis of COVID-19, the majority was formulated using ODE’s, while others were based on fractional calculus. Recall that the fractional calculus is applied in different directions of physics, mathematical biology, fluid mechanics, electrochemistry, signal processing, viscoelasticity, finance and in many others (see for instance [17], [18], [19], [20], [21], [22], [23], [24]). Fractional derivatives were used in the literature to monitor the effect of memory on the system dynamics by replacing the normal derivative arrangement with a fractional derivative arrangement. In epidemic modelling, fractional derivatives and fractional integrals are important aspects, because the effect of memory plays an important role in the spread of the disease. The presence of memory effects on past events will affect the spread of the disease in the future so that the disease can be controlled in the future, and the distance of the memory effect indicates the date of the disease spread. Thus, the effects of memory on the spread of infectious diseases can be verified using fractional derivatives [25], [26], [27], [28], [29]. The fractional calculus adds an extra dimension in the study of dynamics of epidemiological models, especially for COVID-19 pandemic. Therefore, the fractional version of many epidemic models has been investigated as in Khan and Atangana [21], Akman Yıldız [24], Naik et al. [30], Higazy [31], Zhang et al. [32].

As the transmission of COVID-19 is still ongoing and the number of infected people increase, it was found that the virus infects specific age groups. According to experts at Harvard’s Chan School of Public Health, based in Boston, Massachusetts, the new coronavirus appears to be dangerous for elderly people as they are the most hospitalized due the current pandemic. For example, data in Fig. 1 (a)–(d) [33], [34], [35], [36], show that the likelihood of dying from the disease increases with age. The population can be divided into three age groups which are as follows: the first age group is between the date of birth and 20 years, the second age group is between 20 and 40 years old and the third age group is older than 40 years. Similar data from other countries can be found in Sta [37], 38].

Fig. 1

COVID-19 death rate per age group.

Motivated by the fact that the local derivative does not take into account the time scale and does not precisely reproduce the nonlocal, frequency and history dependent proprieties of the epidemic [39] and in order to have a more realistic model that takes into account the time scale, we propose a model with fractional derivatives which provide a useful tool for modelling real dynamics of COVID-19 pandemic. Furthermore, since age is an important factor in COVID-19 epidemic, we propose an age-structured model that describes the dynamics of COVID-19 among different age groups which can be used for evaluating the effectiveness of some control strategies. To the best of our knowledge, this work is the first to combine the fractional calculus and the age structure to model the dynamics of COVID-19 pandemic and solve an optimal control based on this model.

The main contributions of this work are as follows:

Developing an epidemic model with fractional derivative equations to better capture the dynamics of COVID-19 disease and evaluate the impact of control strategies.

Highlighting the role of the age factor in the spread of COVID-19 by dividing the population into age groups.

Proposing an extension of the classical SEIR model with additional compartments, where (S), (E), (I), and (R) stand for Susceptible, Exposed, Infectious, and Recovered people respectively. In our model, for each group the population is divided into seven different classes, namely: Susceptible Infected without symptoms symptomatic infected with high risk symptomatic infected with low risk hospitalized individuals recovered individuals with psychological complications and recovered individuals without psychological complications .

We formulate a fractional optimal control problem aiming to find the optimal strategies to minimize the number of symptomatic infected individuals with high risk and with low risk and recovered individuals with psychological complications. In order to achieve this purpose, we use optimal control strategies associated with three types of controls: the first represents awareness campaigns and security campaigns with the same age groups or different age groups that aim at protecting individuals from being infected by the virus. The second represents the effort to encourage diagnosing the high-risk infected individuals with symptoms to go into hospitals. The third represents the effort of treatment and psychological follow-up for recovered individuals with psychological complications.

Provides numerical simulation and a cost-effectiveness analysis of possible combinations of the three control measures. Our goal is to show the impact of our controls in reducing the number of infected people, to investigate the role of the fractional order on the efficiency of the control strategies and decide what the most effective strategy according to cost-effectiveness analysis is.

The rest of the paper is organized as follows. In Section 2, we give some basis definitions and proprieties on fractional order integral and derivative. For the fractional order differentiation, we will use the Caputo definition due to its convenience for initial conditions of the differential equations. Section 3, is devoted to constructing our multi-age mathematical model using a detailed epidemiological background of the parameters. In Section 4, we present the optimal control problem based on the proposed model and we characterize the optimal control terms using Pontryagin’s maximum principle. Numerical simulations and cost-effectiveness analysis are given in Section 5. Finally, we conclude the paper in Section 6.

Definitions and preliminaries

In this section we collect the conventional definitions of the mathematical formulation of fractional optimal control model. There are many types of fractional derivatives but the most used ones in mathematical modelling and engineering applications are Riemann–Liouville derivative and Caputo derivative (see [17], [18], [40], [41]). Regarding this work, we develop the optimization model with Caputo fractional derivatives.

The left Caputo fractional derivative is defined as follows:The right Caputo fractional derivative is defined as follows:where is the Euler gamma function, is a time dependent function and is the order of the derivative

Consider the following fractional differential system (FDS):with and is a n-dimensional state vector, is a m-dimensional control vector, and is a n-dimensional vector-valued function with Here is the terminal time of the control process.

Suppose that the performance index is given bywhere and are two arbitrary functions.

To find the optimal control we follow the traditional approach and define a modified performance index aswhere is the Lagrange multiplier also known as a costate or an adjoint variable. Taking variation of Eq. (5), we obtainwhere and are the variation of , and consistent with the specified terminal condition.

Minimization of (and hence minimization of ) requires that the coefficients of and in Eq. (7) be zero. This leads to

Model presentation

We propose a continuous-time model as a fractional order and an age groups that describes the dynamics of COVID-19 within an age-structured population using fractional order differential equations. The population is divided into homogeneous age groups, for th () group the population can be divided into sub-age groups jwhich represent (children, adults and elderly), where is the total number of the groups: The immunity of the human body varies from one person to another according to age, and it is stronger in healthy people whose ages are around puberty. After the period of adolescence, the human immunity decreases automatically as a person gets older. Consequently, caution must be taken when contacting family, especially the elderly, because their bodies do not respond to viruses as efficiently as young people. Therefore, the population in our model can be divided into three age groups. The first age group is from birth to 20 years, the second age group is from 20 years to 40 years, and the third age group is more than 40 year.

The and are the number of infectious classes with age groups . We note that the transmission of COVID-19 disease takes place through contact between these groups.

For each age group we note the number of the human population: susceptible infected without symptoms ; infected with high risk and with symptoms infected with low risk and with symptoms hospitalized individuals recovered with psychological complications and recovered without psychological complications as functions of time . The total population is denoted by with .

The following system describes the propagation of the disease in a given population with age groups :where is the Caputo fractional derivative and and are the given initial states.

The susceptible population represents the susceptible individuals in the age group who are not infected including all population groups (children, adults and the elderly). The susceptible population is increased by the recruitment of individuals at a rate and decreased by an effective contact with firstly the exposed individuals at a rate ( represents the transmission rate of susceptible individuals to exposed individuals), secondly with the symptomatic infected with high risk at a rate represents the rate of contact between susceptible individuals and symptomatic infected with high risk), thirdly with the low-risk infected without symptoms at a rate ( which represents the rate of contact with the susceptible individuals and low-risk infected humans with symptoms), fourthly with exposed age at a rate ( represents the rate of contact between the susceptible individuals age and exposed individuals age ), fifthly with symptomatic infected individuals with high risk and with age at a rate ( represents the rate of contact between the susceptible individuals with age and symptomatic infected individuals with high risk and with age ) and sixthly with symptomatic infected individuals with low risk and with age at a rate ( represents the rate of contact between the susceptible individuals with age and symptomatic infected individuals with low risk and with age ). Finally, the susceptible population suffer natural mortality (at a rate ) in th age groups.

The exposed are the individuals carrying the disease without symptoms and belonging to the age group j. This compartment is increased at the rates and . The number of some exposed individuals decreases when they become symptomatic infected individuals with high risk at a rate represents the transmission rate of exposed individuals moving to symptomatic infected individuals with high risk), symptomatic infected individuals with low risk at a rate represents the transmission rate of exposed individuals moving to symptomatic infected individuals with low risk). It is decreased by natural death (at a rate ).

The symptomatic infected individuals with high risk are the individuals who have high risk infection with symptoms in jth age group by the Corona disease (The number of comorbidities is a predictor of mortality in COVID-19). In addition to diabetes, the other common comorbidities were hypertension, cardiovascular disease and lung disease. People with diabetes, indeed, are a high-risk group with a severe disease that makes it difficult to be treated from COVID19, with an increased requirement of hospitalisation. Patients with those diseases need special care to reduce the risk of fatalities among them and they should follow prevention advice given by the authorities thoroughly to avoid infection with COVID-19. Notably, diabetes was also a risk factor for severe disease and mortality in the previous SARS, MERS (Middle East respiratory syndrome) coronavirus infections and the severe influenza AH1N1 pandemic in 2009). The infected population with high risk and with symptoms is increased by the exposed individuals at a rate and decreased by the infected individuals with high- risk and with symptoms who join into hospitals at a rate represents the transmission rate from infected with symptoms who join the group of hospitalized). It is decreased by natural death (at the rate ) and is the death rate of the infected population with high risk and with symptoms.

The symptomatic infected individuals with low risk are the individuals who have low risk infection with symptoms in jth age group by the Corona disease. The population of the symptomatic infected individuals with low risk is increased by the exposed individuals at a rate and decreased by the rates (infected individuals with symptoms become individuals with psychological complications) and (infected individuals with symptoms become recovered individuals with psychological complications) and (infected individuals with symptoms become recovered individuals without psychological complications). It is decreased by natural death (at the rate ).

The hospitalized are the individuals who join the group of hospitalized with age group . The hospitalized population is increased by the symptomatic infected individuals with high risk (at a rate ) and decreased by (hospitalized individuals who become recovered individuals with psychological complications) and (hospitalized individuals who become recovered individuals without psychological complications). It is decreased by natural death (at the rate ) and the death rate of hospitalized individuals.

The recovered with psychological complications represents the recovered individuals with psychological complications. It is also increased when some individuals become recovered at a rates and . The number of the recovered individuals with psychological complications of this population decreases by recovered individuals who do not have any psychological complications at a rate It is decreased by natural death (at the rate ).

The recovered individuals without psychological complications represent the recovered individuals who do not have any psychological complications. It is also increased when the individuals with psychological complications become recovered without psychological complications at the rates and It is decreased by natural death (at the rate ).

The variables , and are the numbers of the individuals in the seven classes at time and with age groups respectively.

The graphical representation of the proposed model is shown in Fig. 2 .

Fig. 2

Schematic diagram of the seven infectious classes in the model.

The total population size at time is denoted by .

Fractional optimal control problem

Problem formulation

At the present time, there is no treatment to fight the new Corona disease 2019. As a result, many countries adopted some strategies such as: quarantine, putting the affected with symptoms in hospitals inside each region, reducing contact with the susceptible individuals and infected individuals, closing most regions, states and cities. However, these strategies have negative effects on the individual and society as a whole. These negative effects can be psychological complications, fear of death and economic damage for the individuals and companies etc. Therefore, there are other strategies that have advantages on the individual and society economically and psychologically, which are social coexistence with disease, social distancing, diagnosis of the infected people in hospitals and treatment and psychological follow-up for recovered individuals and those who have psychological complications.

The strategy of control that we proposed aims at minimizing the number of the symptomatic infected individuals with high risk the symptomatic infected individuals with low risk and the recovered individuals with psychological complications during the time step to and also to minimize the cost spent in awareness and security campaigns, treatment and psychological follow-up.

In the model (12), we include three controls and for The first control represents the effort of awareness and security campaigns that aim at sensitizing people about social distancing and imposing it as well as introducing health measures to protect individuals from being infected by the virus with same age or from different age groups. Hence, the term is used to reduce the force of infections. The control represents the effort to diagnose the infected individuals with symptoms in hospitals in the jth age group. Finally, measures the effort of treatment and psychological follow-up for recovered people with some psychological complications. The aim here is to reduce the number of individuals with psychological complications. So, the controlled mathematical system is given by the following system of differential equations.where and are the given initial states.

Then, the problem is to minimize the objective functionalwhere the parameters and are the cost coefficients with age and is the final time.

In other words, we seek the optimal controls and such thatwhere is the set of admissible controls defined by

The main point in FOCPs is to find the optimal controls which minimizes the following objective function:subject to the constraint,

where represent the right side of the system (13), and where the initial conditions are and

The following expression defines a modified objective function:where is the following Hamiltonian

From (17) and (19), we can derive the following

with the transversality conditions at time

Eqs. (20) and (21) describe the necessary conditions in terms of a Hamiltonian for the FOCP defined previously. These conditions result in a set of fractional differential equations, in terms of the variables state , controls and Lagrange multiplier , to be solved analytical or numerically or even both.

Characterization of optimal controls

We apply the Pontryagin’s Maximum Principle [42], [43], [44], [45], [46], [47]. The key idea is introducing the adjoint function to attach the system of differential equations to the objective functional resulting in the formation of a function called the Hamiltonian. This principle converts the problem of finding the control to optimize the objective functional subject to the state differential equations with initial condition to find the control to optimize Hamiltonian pointwise (with respect to the control).

Now, we have the Hamiltonian in time defined bywhere is the right side of the system of differential Eq. (4) of the state variable.

Given an optimal control, and solutions,,andof corresponding state system(13), there exists adjoint functions,,andsatisfyingwith the transversality conditions at time

Furthermore, forthe optimal controlsandare given by

The Hamiltonian at time step in age groups is given by

For the adjoint equations and transversality conditions can be obtained by using Pontryagin’s Maximum Principle given in Ahmed et al. [26], Higazy [31], Sta [34], 35], Doh [36], Sta [37] such that

For the optimal controls and can be solved from the optimality condition, that is

We have

However, the control attached to this case will be eliminated and removed. By the bounds in of the controls, it is easy to obtain and in the form of (25), (26) and (27). □

Numerical simulations and discussion

In this section, we shall solve numerically the optimal control problem for our as a fractional order and age groups model. Here, we obtain the optimality system from the state and adjoint equations. The proposed optimal control strategy is obtained by solving the optimal system which consists of seven differential equations and boundary conditions. The optimality system can be solved by using an iterative method. Using an initial guess for the control variables, and the state variables and are solved forward and the adjoint variables for are solved backwards at times step and  =  . If the new values of the state and adjoint variables differ from the previous values, the new values are used to update and and the process is repeated until the system converges.

This model is valid for all age groups of humans, but the simulation will be limited to the study of a model that is composed of two age groups (): adults and elderly people . Here, we assume that the adult age group includes active young people, who are more likely to work outside the home and hang out with more people. Therefore, they tend to get infected with the coronavirus and pass it on to other age groups of young people, children and the elderly in their family or friends at work.

We present some numerical simulations in order to illustrate our theoretical results. We consider system (13) with the initial conditions, and and the following parameter values in the first age groups () and second age groups () (see Tables 1 and 2 ).

Table 1

The description of the parameters used for the definition of systems with the first age group.

S01	E01	Ih,01	Il,01	H01	R1,01	R2,01	N^1	b_1
1000	400	200	200	100	60	200	4320	200
μ_1	α11	α21	α31	β11	β21	β31	β431	β531
0.045	0.03	0.03	0.01	0.4	0.4	0.4	0.6	0.75
δ11	δ21	γ11	γ21	γ31	β631	η11	η21	η31
0.0001	0.001	0.03	0.03	0.001	0.7	0.25	0.15	0.001

Table 2

The description of the parameters used for the definition of systems with the second age group.

S02	E02	Ih,02	Il,02	H02	R1,02	R2,02	N^2	b_2
1000	400	200	200	100	60	200	4320	200
μ_2	α2	α22	α32	β12	β22	β32	β422	β522
0.045	0.06	0.03	0.01	0.6	0.5	0.5	0.6	0.75
δ12	δ22	γ12	γ22	γ32	β622	η12	η22	η32
0.0001	0.001	0.03	0.03	0.001	0.7	0.25	0.15	0.001

We begin by presenting the evolution of the solution of our model (12) in second age groups () with contact and without controls with age groups () at as shown in Fig. 3 .

Fig. 3

Number of infected and recovered individuals in the second age group when they are in contact or not with adults people.

Fig. 3 (a)–(c) shows that the number of the symptomatic infected individuals with high risk, symptomatic infected individuals with low risk and recovered individuals with psychological complications in the second age group has increased due to their contact with individuals from the age group compared to the absence of contact with the first age groups. There are other cases where contact occurs between different age groups, which are children age group and the adults age group, as well as individuals from the children age group and the elderly age group.

We aim here to highlight the specificities of each age group when formulating control strategies which seek to achieve some strategies.

Strategy 1: Protecting susceptible individuals of the second age groups from contacting infected persons of the same or other age groups.

To realize this objective, we apply only the control i.e. the implementation of healthy protocol programs and awareness campaigns to protect susceptible individuals from contacting infected individuals of different age groups, reducing contact with vulnerable individuals, reducing family visits to elderly people.

Fig. 4 (a) shows that the number of the symptomatic infected individuals with high risk decreases more when approaches 1(with controls) at the end of the implementation of the proposed strategy. Fig. 4(b) demonstrates that the number of the symptomatic infected individuals with low risk decreases more when approaches 1(with controls) at the end of the implementation of the proposed strategy. Fig. 4(c) demonstrates that the number of the recovered individuals with psychological complications decreases more when approaches 1(with controls) at the end of the implementation of the proposed strategy. We take different values of to present the effect of the fractional derivative on the number of the infected individuals and and the recovered individuals with psychological complications . Also, the effects of fractional (different values of ) have been investigated in this case. We observe that there was a low increase in order 1, followed by order 0.9, till it reaches the last order.

Fig. 4

The number of infected and recovered individuals for different values of with and without control .

On the other hand, the fractional derivatives play an important role in describing the effects of memory in dynamic systems. As limits to the memory effects are reduced. Also, the fractional derivative order plays the role of time delay in ordinary differential models. We see from Fig. 4(a) and (b) that when the derivative order is reduced from 1, the memory effect of the system increases, and therefore the number of infected individuals and increases for a long time.

Strategy 2: Protecting susceptible individuals of the second age group from contacting infected individuals of the same or other age groups and encouraging diagnosis for the symptomatic infected individuals with high risk to go into hospitals.

To realize this objective, we apply only the controls and i.e. the implementation of awareness campaigns for the second age group which aim at protecting them from being infected by the virus, encouraging diagnosis and going into hospitals.

Fig. 5 (a) shows that the number of the symptomatic infected individuals with high risk decreases more when approaches 1(with controls) at the end of the implementation of the proposed strategy. Figure 5(b) demonstrates that the number of the symptomatic infected individuals with low risk decreases more when approaches 1(with controls) at the end of the implementation of the proposed strategy. We take different values of to present the effect of the fractional derivative on the number of the infected individuals and We observe that, there was a low increase in order 1, followed by order 0.9, till it reaches the last order.

Fig. 5

The number of infected and recovered individuals for different values of and when controls and are applied together and without controls.

Strategy 3: Protecting susceptible individuals of the second age groups from contacting infected persons of the same or other age groups, encouraging the symptomatic infected individuals with high risk to go into hospitals and get treatment, psychological follow-up for individuals with psychological complications.

To realize this objective, we apply only the control and i.e. the implementation of awareness campaigns that aim at protecting individuals from being infected by the virus, encourage the infected individuals with symptoms to go into hospitals. In addition to that, we reduce the number of the recovered individuals with psychological complications using treatment and psychological follow-up which represents virtual social interactions, maintaining a healthy lifestyle and maintaining physical activity.

Fig. 6 (a) that the number of the symptomatic infected individuals with high risk decreases more when approaches 1(with controls) at the end of the implementation of the proposed strategy. Fig. 6(b) demonstrates that the number of the symptomatic infected individuals with low risk decreases more when approaches 1(with controls) at the end of the implementation of the proposed strategy. Fig. 6(c) demonstrates that the number of the recovered individuals with psychological complications from second age groups decreases from 512.79 (without control and ) to 282.92 (with controls and ), from 458.16 (without control and ) to (with controls and ) and from 402.86 (without control and ) to 226.08 (with controls and ) respectively at the end of the implementation of the proposed strategy. Fig. 6(d) demonstrates that the number of the recovered individuals without psychological complications from second age groups increases from 160.285 (without control and ) to 394.2929 (with controls and ), from 155.1585 (without control and ) to 348.5148 (with controls and ) and from 151.5806 (without control and ) to 308.048 (with controls and ) respectively at the end of the implementation of the proposed strategy. Through the use of those controls, the above mentioned objective has been achieved.

Fig. 6

Number of infected and recovered individuals for different values of and when all controls are implemented and without control.

In the rest of this section, we analyse the cost effectiveness of the previous three strategies to show the most cost-effective strategy. To do this, we use the well known method called Incremented Cost Effectiveness Ratio (ICER), which allows comparing the differences between the cost and health outcomes of two competing intervention strategies (see for instance [48], [49], [50]). It is defined by the difference in the cost between two possible interventions, divided by the difference in their effect. It represents the average incremental cost associated with 1 additional units of the measure of the effect. Given two competing strategies and where strategy has higher effectiveness than strategy the ICER values are calculated as follows where (TC) is the total cost and where (TA) is the total cases averted, defined during a given period for each strategy. In our case, for strategy (where ) where and corresponds to the person unit cost of the four possible interventions, while ( )is the optimal solution associated to the optimal control ( ).

Using the simulation results (for we ranked, in the Table 3 our control strategies in order of increased numbers of averted infections.

Table 3

Total Costs and Total Averted Infections for Strategies 1, 2 and 3 for age groups =2 and .

Strategy	Total averted infections (TA)	Total cost(TC)
1	5,6632.10^4	6,4086.10^4
2	6,4077.10^4	6,6575.10^4
3	6,4769.10^4	6,7717.10^4

Strategy 1 is compared with Strategy 2 with respect to increased effectiveness, in reference to Table 3. So:

Since then strategy 1 is less effective than strategy 2. Therefore, strategy 1 is excluded from the set of alternatives.

Next, strategy 2 is compared with strategy 3. The ICER values for strategy 2 and strategy 3 are calculated below:

Since then strategy 3 is less effective than strategy 2. Therefore, strategy 3 is excluded from the set of alternatives. Thus, the conclusion is that strategy 2 (Protecting susceptible elderly individuals from contacting infected persons of the same or other age groups and encouraging diagnosis and the symptomatic infected with high risk to go into hospitals) is the most effective strategy.

Conclusion

In this work, we contribute to the global efforts aiming to understand and control the transmission of the novel coronavirus COVID-19. Our contribution consists of formulating and solving an optimal control problem based on a more accurate SEIR model, which includes an age structure, fractional order derivative equation and three controls.

Nowadays, it is well known that the reaction of the immune system to the infection with COVID-19 depends on the age factor. The risk of dying from this disease is higher for elderly people than young people or children. This is why, it is important to include age group classification in our model. Also, clinical studies have shown that there are many categories of infected people depending on symptoms apparition, risk level and diagnosis. In this context, taking into consideration this diversity, we have divided the infected class into three compartments: infected people with high risk, those with low risk and patients under treatment in the hospital. Additionally, as many recovered individuals have suffered from psychological complications, we proposed to consider two sub classes of the recovered people, with and without psychological problems.

Many scholars have investigated the transmission and control of COVID-19 disease through formulating mathematical models with classical integer order differential equations. However, this type of differential equations do not reflect an important characteristic of the dynamic behaviour of biological systems, which is the memory effect. So, to have a more accurate model, we used fractional derivative equations instead of the local derivative. This promising tool could simulate many phenomena that integer order cannot, and seems to be more appropriate for modelling the transmission of a complex disease like COVID-19.

To control the spread of COVID-19 pandemic, we propose three different control strategies, namely, awareness programs, diagnosis of infected people and follow-up of people with psychological complications. By considering these control interventions, we aim to protect susceptible individuals who are more vulnerable due to their age, reduce the number of infected individuals (with high and low risk) and minimize the number of recovered individuals with psychological complications. Based on some results from the optimal control theory, we derived a characterization of each control.

To investigate the effectiveness of our control measures, we proposed three strategies: the first one consists of applying the awareness program alone, for the second one we combined awareness programs with diagnosis and as the last strategy, we implemented the three controls together. Moreover, in the numerical part, we were interested by the effect of the order of the fractional derivative on the efficiency of the control strategies. Our numerical results show that the three strategies we proposed are effective and can reduce the number of infected patients, while the cost-effectiveness analysis show that the best choice is combining awareness with diagnosis which is the most effective solution.

For future work, it would be interesting to propose an extension of our model by adding a multi-region structure, which takes into account the important role of the human mobility in the spread of COVID-19. Furthermore, with the development of new vaccines, it will be important to investigate the impact of a vaccination campaign on the optimal behaviour of our model.

CRediT authorship contribution statement

Bouchaib Khajji: Writing - original draft, Writing - review & editing. Abdelfatah Kouidere: Writing - original draft, Writing - review & editing. Mohamed Elhia: Writing - original draft, Writing - review & editing. Omar Balatif: Writing - original draft, Writing - review & editing. Mostafa Rachik: Writing - original draft, Writing - review & editing.

Declaration of Competing Interest

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