Optimal control of a fractional order model for the COVID - 19 pandemic.

In this paper a fractional optimal control problem was formulated for the outbreak of COVID-19 using a mathematical model with fractional order derivative in the Caputo sense. The state and co-state equations were given and the best strategy to significantly reduce the spread of COVID-19 infections was found by introducing two time-dependent control measures, u 1 ( t ) (which represents the awareness campaign, lockdown, and all other measures that reduce the possibility of contacting the disease in susceptible human population) and u 2 ( t ) (which represents quarantine, monitoring and treatment of infected humans). Numerical simulations were carried out using RK-4 to show the significance of the control functions. The exposed population in susceptible population is reduced by the factor ( 1 - u 1 ( t ) ) due to the awareness and all other measures taken. Likewise, the infected population is reduced by a factor of ( 1 - u 2 ( t ) ) due to the monitoring and treatment by health professionals.

Introduction

In December 2019, a novel corona virus COVID-19 was identified from three patients with pneumonia in connection to cluster acute respiratory illness cases in Wuhan, China [1]. The virus was genetically related to SARS-COV. Severe acute respiratory syndrome (SARS-COV) and middle-east respiratory syndrome (MERS-COV) are the two corona viruses infecting animals evolved and caused outbreak in humans since 2002 [1].The SARS-COV and MERS-(COV) were both from zoonotic Coronaviruses (whose source were likely from bats) that were first found in the mid-1960.

COVID-19 outbreak spread and become global pandemic, as at this time the outbreak spread across 150 countries, with over 700,000 infected people and 30,000 death. The disease can be transmitted from an infected to healthy individual through nose, eye and mouth, by means of droplets produced by coughing or sneezing, contact with contaminated surfaces, objects, or items of personal use [2]. The most common signs of infection include respiratory symptoms, cough, fever, and shortness or difficulties in breathing. In more severe cases, the infection can cause SARS, pneumonia, kidney failure and even death [3].

Over the years mathematical modeling has been used in analyzing different dynamics of diseases, like Malaria, Tuberculosis and HIV/AIDS, it also plays a very important role in understanding the epidemiological patterns of the diseases control, because it can provides short and long term prediction of the incidence of the diseases [4], [5], [6], [7], [8], [9], [10], [11], [12], [13], [14], [15].

Mathematical model (for MERS) using nonlinear system of differential equations was given in [16], in which camel was assumed to be the origin of the infection that spread the virus to infective human population, then transmission from human to human, then to clinic center then to care center.

An optimal control theory is yet another effective tool used in diseases modeling, it comes in to existence after the World War II by the formulation of the most popular Pontryagin maximum principle [17], and it gives more insight into the dynamics of the diseases and also provides more appropriate control and preventive strategies [18]. Nowadays, the use of optimal control theory in modeling diseases is significant [18], [19], [20], [21], [22], [23], [24], [25], [26], [27], [28].

Optimal control problems that involves fractional calculus are called fractional optimal control problems (FOCPs),they are considered to be the generalized form of classic optimal control problems (OCPs), in which the differential equations are in form of fractional differential equations (FDEs), and the performance index can be given by a fractional integration operator [29]. There are several research papers in the literature that provide the theoretical basis and fundamentals of FOCPs, most of these papers extensively investigate how to formulate the FOCPs and derived the optimality conditions for several states and control variables using analytical and numerical methods [30], [31], [32], [33], [34], [35], [36], [37], [38], [39], [40]. Nowadays, the FOCPs have been applied to epidemiological models for faster and more accurate behavior of controlling the diseases, because the fractional-order depends on the memory. Hence, the FOCPs can becomes potentially the most flexible tools in modeling epidemiological and biological systems related to memory.

In [41] fractional OC was used on the HIV-Immune system model and forward backward algorithm was used to solve the problem. Fractional OC for an enzyme kinetic model was developed in [42] and also its numerical solution was provided [43]. Formulated and discussed HIV/AIDS with treatment fractional model and included three control efforts (effective use of condoms, ART treatment, and behavioral change control) into the model in order to control the spread of HIV/AIDS epidemic [44]. Present fractional OC of a novel West Nile virus model and solve the problem using two simple numerical methods. Formulation and investigation of the transmission dynamics of pine wilt disease using FOCP with three suggested controls measures to control the spread of the disease was provided in [45].

For this outbreak of COVID-19 (whose source was assumed to be from bat), the model in [16] was modified in which susceptible human and bat population was incorporated. Fractional derivative in the Caputo sense was also used due to the fact that fractional order derivative gives better result than the integer order and also the fractional optimal control problem was formulated, in which the state and co-state equations were given.

Optimal control theory is a powerful mathematical tool that is used extensively to control the spread of infectious diseases. It is often used in the control of the spread of most infectious diseases for which either vaccine or treatment is available. For COVID – 19, some researchers considered only optimal control to their models [52], [53], [54] and some of them used only fractional order model [55], [56], [57], [58]. In this paper we considered fractional order model and we incorporated optimal control.

Some important definitions and theorems

[46]: a gamma function of is defined as

[46]: The Rieman-Liouville fractional derivative of order of is defined as

[46]:The Caputo fractional derivative of order of is defined as

[47]: (Linearity of fractional derivative)

Let be continuous and be scalars, then

[48]:(Contraction)

An operator that maps a metric space onto itself is said to be contractive if for

[48]: (Picard-Banach fixed point or Banach contraction mapping principle)

Any contractive operator that maps a metric space onto itself has a unique fixed point. Furthermore, if is a contractive operator that maps a metric space onto itself and is its fixed point: then for any iterative sequence

Converges to .

In other words is a solution or an equilibrium for continuous dynamical system and fixed point for discrete dynamical system.

[49]: The equilibriums solutions of the system is locally asymptotically stable if all the eigenvalues of the Jacobian matrix evaluated at the equilibrium points satisfy

[50]: Let be an equilibrium of system, let be a domain containing

Let be continuously differentiable function such that

and

Where are continuous positive definite function on and is a Lyapunov candidate function, then is globally asymptotically stable.

[51]: Let be continuous and derivable function. Then, for any time instant and

Model formulation

The model consists of eight compartments; Susceptible bat population , Infected bat population , Susceptible human population , Infected human population , Human to human transmission population , Infected individual to family members transmission population , Patient to clinic center transmission population and Patient to care center transmission population With bats as the origin of the novel Covid-19 virus, it is assumed that the new born of bats are born into susceptible class , at the rate .Which joined the infectious class at rate .It is also assumed that the new born of humans are born into susceptible class which later became infectious as a result of contact with an infected bats at the rate. Then the virus spreads from an infected human to human , to a family member , then to clinic center and care center at the rates , , and respectively. Table 1 gives the descriptions of the parameters as used in the model.

Table 1

model parameters and their descriptions.

Model parameters	Description
μ_i,i=1,2,…,8	Natural death rates in S_b,I_b,S_h,I_h,H_h,F_m,P_c,andC_c compartments
δ_1	Disease induced death in I_b
δ_i,i=2,3,…,6	Disease induced death in I_h,H_h,F_m,P_c,andC_c compartments
λ_b	Birth rates of bats
λ_h	Birth rates of human
β_i,i=1,2,…,6	Transmission rates

The transmission dynamics can be described by the nonlinear system of fractional order differential equations (FODE) in the Caputo sense as given in Fig. 1 .and

Fig. 1

Schematic diagram describing the transmission dynamics of COVID-19.

Stability analysis and derivation of basic reproduction number

In this chapter, we find the equilibrium solutions and carry out local stability analysis of the solutions. We use the condition for the local stability to derive the basic reproduction number.

Equilibriums solutions

To find the equilibrium solutions we equate system (1) to zero and solve simultaneously.

We obtain two important equilibrium solutions; Disease free equilibrium and endemic equilibrium. Where;andwith

In the disease free equilibrium all the populations are zero except that of susceptible human and susceptible bird. This equilibrium point always exists. For the endemic equilibrium, it is an equilibrium where none of the populations is zero. Mathematically this is the most important equilibrium point.

Stability analysis

We construct the following Jacobian matrix from (1) by letting

The disease free equilibrium is locally asymptotically stable.

we have;

: Consider the Jacobian matrix at, then letting

Evaluating we get the following eigenvalues as

Since the then clearly

Then by theorem [2] , is locally asymptotically stable.

Note: For the DFE to be stable, we need the eigen value , that is

Simplifying, we get;

Define basic reproduction number

The endemic equilibrium is stable if

: Since all the equilibrium points in the endemic equilibrium depend on then it suffices to investigate its stability,

Since, then

Either

This will lead to a chance of to be complex.

Or

Optimal control

In this section, we extend our model by introducing two time-dependent control measures, (which represents the awareness campaign, lockdown, and all other measures that reduce the possibility of contacting the disease in susceptible human population) and (which represents quarantine, monitoring and treatment of infected humans). It is assumed that the exposed population in susceptible population is reduced by the factor () due to the awareness and all other measures taken. Likewise, the infected population is reduced by a factor of () due to the monitoring and treatment by health professionals.

Hence, the model system of equations (1) becomes:

With the objective function given as:Whereis the susceptible human population and is the infected human population. is the final time andthe coefficients a, b, c, d are positive weights. Our aim is to minimize the susceptible and infected human populations while minimizing the cost of control. Thus, we search for an optimal control , such thatWhere the control set is

The termsand represent the cost of reducing the exposed susceptible and infected population respectively, while is the cost of awareness, lockdown and other measures and also, is the cost of quarantine, monitoring and treatment. The necessary conditions that an optimal control must satisfy come from the Pontryagin's Minimum Principle. This principle converts Eqs. (3) and (1) into a problem of point-wise minimizing a Hamiltonian M with respect to stated asfollows:Where, are the co-state variables or adjoint variables

The transversality conditions are

On the interior of the control set, where 0 < <1, for = 1, 2 we have:

From where;

The control parameters ( ) that minimizes over are given by:

Where are the adjoin variables satisfying (1-8) and the following transversality conditions: = = = = 0 and

The existence of an optimal solution with the corresponding optimal control result can be achieved from the convexity of integrand of J with respect to control and,a priori boundedness of the state solutions, and the Lipschitz property of the state system with respect to the state variables. Applying Pontryagins Maximum Principle, we obtain:

With:= = = = 0

The optimality conditions is obtained by differentiating Hamiltonian with respect to thecontrol variables and:

The adjoint system (6) and (7) is obtained by solving (8), while the optimal control pair (9) come from the optimality condition (10). The optimality system constitutes of the controlled system (2) with its initial conditions, adjoint system (4) and transversality conditions (5).

Numerical simulation

Numerical simulations using the variables and parameter values as given in [16] are carried out in this chapter, Table 2 gives the values of variables while Table 3 gives the values of the parameters used for the simulation.

Table 2

Values of the variables used.

Notation	Variable	Value	Source
Sb	Susceptible bat population	00-600	[16]
I_b	Infected bat population	200-500	[16]
S_h	Susceptible human population	10,000,000	[16]
I_h	Infected human population	240-440	[16]
H_h	Human to human transmission population	100−400	[16]
F_m	Infected individual to family members transmission population	40−200	[16]
P_c	Patient to clinic center transmission population	00−300	[16]
C_c	Patient to care center transmission population	00−300	[16]

Table 3

Values of the parameters used.

Notation	Parameter	Value	Source
β_1	Transmission rate from susceptible bats population to infected bats population	1.2300	[16]
β_2	Transmission rate of infected bats population to infect human population	0.1000	[16]
β_3	Transmission rate of infected individual to healthy individual(human to human)	0.0060	[16]
β_4	Transmission rate of infected individual to own family member	1.0090	[16]
β_5	Transmission rate of patient to clinic center	0.0040	[16]
β_6	Transmission rate of patient to care center	0.0900	[16]
λ_b	Birth rates of bats	1.5000	[16]
λ_h	Birth rates of humans	1.2500	[16]
μ_1	Natural death rate in susceptible bats population	1.7000	[16]
μ_2	Natural death rate in infected bats population	0.1340	[16]
μ_3	Natural death rate in susceptible human population	0.5000	[16]
μ_4	Natural death rate in infected human population	0.1343	[16]
μ_5	Natural death rate in infected human to healthy human population	0.0024	[16]
μ_6	Natural death rate in infected human to own family population	0.0074	[16]
μ_7	Natural death rate in patient to clinic center population	0.3440	[16]
μ_8	Natural death rate in patient to care center population	0.5410	[16]
δ_1	Disease induced death in infected bats population	0.0143	[16]
δ_2	Disease induced death in infected human population	0.3002	[16]
δ_3	Disease induced death in infected human to healthy human population	0.0054	[16]
δ_4	Disease induced death in infected human to own family population	0.0019	[16]
δ_5	Disease induced death in patient to clinic center population	0.0640	[16]
δ_6	Disease induced death in patient to care center population	0.4400	[16]

The simulation results were given in Fig. 2, Fig. 3, Fig. 4, Fig. 5, Fig. 6 , Fig. 2 gives the dynamics of all the populations with respect to time. Susceptible population and the total infected human population are presented in Fig. 3. Fig. 4 presented the dynamics of the total infected human population with respect to time for various values of lastly, Figs. 5 and 6 show the significance of the control programs in curtailing the pandemic.

Fig. 2

Dynamics of the populations over time.

Fig. 3

Susceptible human population versus Total infected human population.

Fig. 4

Total infected population for various values of .

Fig. 5

Total infected population with control and without control.

Fig. 6

Susceptible population with control and without control.

Results discussion

From Fig. 3 it can be seen that many people will be infected with time, and we can see from Fig. 4 that FODEs have rich dynamics and are better descriptors of biological systems than traditional integer – order models. We also note that the solution of the model, with various values of continuously depends on the time – fractional derivative, but arrives to the equilibrium points. Fig. 5 shows the infected population with and without the control measures, as we can see from the graph by using the control measure the total infected population reduced significantly due to the impact of the control measure. Fig. 6 shows the susceptible population with and without the control measures, it can also be seen that when using the control measures, the susceptible population increases, this is due to the fact that the exposed population in susceptible population is reduced because of the control impact.

Summary and conclusion

We have presented a general formulation for an FOCP, in which the state and co-state equations are given. Fractional order model for transmission of COVID-19 was considered in which two time dependents controls measures, (which represents the awareness campaign, lockdown, and all other measures that reduce the possibility of contacting the disease in susceptible human population) and (which represents quarantine, monitoring and treatment of infected humans) were incorporated in the model. Two equilibrium solutions were found and their local stability analyses were carried out. Numerical simulations were also carried out to show the significance of the control programs. It is clear that when the control measures are applied optimally the number of infectives reduces and the number of susceptibles increases.

Declaration of Competing Interest

None.