Charu Arora,Sudhakar Yadav
Optimal Control Strategies to Cope with Unemployment During the Covid-19 Pandemic
Modified unemployment model,Covid-19,Stability analysis,Control strategies,Numerical simulation
In the present paper, a mathematical model using the non-linear differential equations depicting the impact of Covid-19 on unemployment is discussed. The stability of the system is studied and model is reformulated as an optimal control problem. To assess the impact of unemployment on human population, two time-dependent controls are used. Providing education and training of job-oriented persons act as first control and campaigning about the awareness of coronavirus disease and self-employment business is the second control. Necessary conditions for optimal control are derived by Pontryagins maximum principle. Further, the results are illustrated by numerical simulation.
Optimal Control Strategies to Cope with Unemployment During the Covid-19 Pandemic

In the present paper, a mathematical model using the non-linear differential equations depicting the impact of Covid-19 on unemployment is discussed. The stability of the system is studied and model is reformulated as an optimal control problem. To assess the impact of unemployment on human population, two time-dependent controls are used. Providing education and training of job-oriented persons act as first control and campaigning about the awareness of coronavirus disease and self-employment business is the second control. Necessary conditions for optimal control are derived by Pontryagins maximum principle. Further, the results are illustrated by numerical simulation.

Unemployment is the state of a person who is willing to work but is not able to get the job. Unemployment is a big problem across the globe and it effects the socio-economic condition of a region. It is vividly a matter of concern for government authorities as it holds a major role in taxation and budgets. Hence, providing adequate number of jobs to meet the challenge of unemployment is a major agenda for every government body in elections. Apparently, supporting education from the primary classes itself is an essential and elementary step towards the solution of a bigger problem. Unfortunately, covid crisis has impacted the condition of unemployment in a more worse manner. Many people have lost their jobs due to the Covid-19 pandemic and their economic, mental and social health have suffered a lot. Advances in technology adds to the development economically but the advancement should be made in a planned and controlled way like proper training, developing skills etc. These factors often increases the employment. Also, entrepreneurship increases the employment in an efficient way (Yousefpour et al. 2020; Mallick and Biswas 2017). The rate of unemployment has increased dejectedly today than from the previous years due to the worldwide lockdown and curfews. According to research, India’s unemployment rate has increased to 7.1% in 2020 from 5.2% in 2019. In 2021–2022, unemployment rate is almost the same as 2019–2020 (https://www.businesstoday.in/jobs/story/india-unemployment-in-2020-at-worst-level-in-29-years-shows-study-297195-2021-05-28). One of the significant study depicts the cost effective measures for the control of unemployment which is to embed the control strategies on temporary employees and regular employees while creating some temporary and regular vacancies (Munoli et al. 2017). In Munoli et al. (2017) and Munoli and Gani (2016), models include the schemes of retirement, unavoidable circumstances like death, termination from job etc. and control analysis is executed by adopting the Pontryagin’s maximum principle. In Abu Arqub and Shawagfeh (2021), optimal control problem is solved by the method of reproducing kernel Hilbert space. Inverse problems are analyzed for diffusion equations with fractional Sturm–Liouville operator for space (Djennadi et al. 2021a) and fractional derivatives in time (Djennadi et al. 2021b). The class of fractional delay differential equations with Atangana-Baleanu fractional derivatives is discussed in Sweis et al. (2022). Effective measures like various policies to provide temporary or regular employment are discussed and numerical simulations are employed to validate the theoretical results. A mathematical model in Mallick and Biswas (2020) analyzes the stability conditions of a system which comprises of vacancies, employment and unemployment compartments. Optimal control techniques are adopted and model is simulated. The optimal control conditions for unemployment problem with effective use of policies in providing temporary and regular employment by creating new vacancies are derived. The study calculates the Unemployment Averted Ratio (UAR) and Incremental Cost-Effectiveness Ratio (ICER) to investigate the cost effectiveness of all possible combinations of four control measures. In this paper, we make an attempt to study and deal with this major problem of unemployment that has even increased more due to Covid-19 pandemic. Section 2 of the paper explains the formulation of the model and discusses the basic preliminaries of the proposed model. Global stability is examined in Sect. 3. Section 4 studies the model with control followed by the numerical simulation in Sect. 5. Section 6 concludes the paper with discussion and future scope.

Misra and Singh (2011) studied an employment model by assuming three classes such as U, T and R representing the unemployed class, temporarily employed and regularly employed persons at any time t which leads to the following model;We assume , , E and V as the unemployed class, unemployment occurred due to the pandemic, employed class and number of vacancies created at the time t. So, the proposed model comprises of the following variables; Unemployed persons due to the factors other than Covid-19 (). Unemployed persons due to Covid-19 (). Employed persons (E). Unfilled vacancies (V). It is depicted in the proposed system about the unemployed class getting employment to fill the vacancies which follows a linear functional pattern. On the other hand, unemployed persons due to the crisis of Covid-19, gets employed following a different functional response (Fig. 1). We differentiate the unemployed class into two subclasses i.e. unemployed persons due to Covid () and unemployed persons due to factors other than Covid (). Based on the above concepts, we reformulate the model given by Misra and Singh (2011) which is given as;The parameters used in the system are interpreted in Table 1.

Every solution of the system (2) with respect of initial conditions existing in remain positive for all .

The proposed model with initial conditions is written in the matrix equation form as,where such that (Khajanchi 2014; Gakkhar and Singh 2012).where and such that for By the classical theory proposed by Nagumo (1942), the solution is positive for all finite and positive time t.

Every solution of system (2) with respect to initial conditions are bounded for all .

Let . Assuming and . Then,which implies,If, Hence, we obtain, Sup This proves the boundedness property as discussed in Birkhoff and Rota (1982).

The formulated system 2 possesses the nonzero equilibrium point , where,and is the root of quadratic equation defined by,where , , . By using Descartes rule of signs which depends upon the number of changes in sign of coefficients (Wang 2004), the quadratic equation 4 has a unique positive real root if any one of the following condition holds true,     Thus, , and can easily be obtained using . Now, the jacobian matrix corresponding to the system 2 is given by, where , , , , . In addition, the characteristic equation at nonzero equilibrium point is given by,where , , , . Using Routh Hurwitz criteria, the stability conditions become, (i) , (ii) , (iii) .

Global stability signifies the high resistance property pertaining to the changes in the system. Section 3 analyzes the global stability conditions of nonzero equilibrium point using the positive definite function based on the work in Santoprete and Xu (2018). Considering a positive definite function as follows,Differentiating with respect to ‘t’, we get,After the simplification process, we get the following as,Since, , , .If and , , . So, , with,whereAll the principal minors of the matrix are positive when the following holds, (i) , (ii) The matrix H is invertible provided the determinant is nonzero i.e.,

The various ways in optimal control develop optimal strategies to control the factors that affect the system. In this section, we develop a model using control variables and verify how effective controls in the model work. Pontryagin’s maximum technique provides the necessary conditions for the principle of optimal control system (Schättler and Ledzewicz 2012; Lenhart and Workman 2007). An optimal control problem is an optimization technique and defined as the process of designing control and state trajectories over time in order to maximize (or minimize) a performance index. The collection of variables used to represent the mathematical state of the system is referred as the state variables. The Cost functional is formulated using state and control variables in an optimal control problem. Let u(t) be the control variable that is used on the state variable P(t), and the state variable constitutes the differential equation that is dependent on the control variable i.e. The goal is to find a control u(t) and its corresponding state variable P(t). The objective functional (maximize/minimize) is given as,Necessary Conditions Let u(t) be a control affecting the differential equation (DE) and state function P(t) that satisfies the differential equations(DE). Defining Hamiltonin H asLet is an optimal control and be the corresponding state. Then the following conditions hold true, at (optimality condition), (adjoint equation), (transversality condition). Sufficient Conditions

ConsiderAssuming f(t, P, u) and g(t, P, u) as both continuously differentiable functions in their arguments and concave in P and u. If is a control with associated states and are the piecewise differentiable functions. Moreover, and fulfil on and,Then, for all controls u, we have When the problem is minimization rather than maximizing, a similar approach gives the same required conditions. We minimise the Hamiltonian pointwise in a minimization problem, and the inequality in Pontryagin’s Maximum Principle is reversed. However, f and g are convex in u in minimization problem. Optimal control assessment is performed to study the effectiveness of all possible combinations of control measures in the system. The Covid-19 problem has a significant impact on employment across the globe. The set of measures to improve employment, includes the monetary and fiscal policies, education and training to decrease structural unemployment, geographic incentives to encourage businesses to invest in struggling regions, a lower national salary to minimize the real pay unemployment, more flexible labor markets and easy measures to hire employees. For economic growth and sustainable development, a transformational strategy is adopted. The strategy is supported by seven engines, including logistical infrastructure, mass transit, rivers, airports, and trains. This strategy provides numerous career and business options. The comprehensive effect of all these measures are anticipated to increase output and create jobs jointly. The creation of job possibilities is another goal of infrastructure development and industrial corridors. The aforementioned strategies can all be used as a control to enhance employment. Firstly, providing education and training of job-oriented and secondly, campaigning about awareness of the corona and self-employment business are the two taken control measures. The self-employment business means drop shipping, blogging, chocolate making, toy making, brew making, interior designer/home decorator, event planning, fashion designing, dog walking, makeup artist, personal trainer/gym instructor, landscaper architecture, tutoring, food cart franchise, logo designing, E-Bay merchant, computer repairing, picture framing, local tour guide, recycling services, soap/detergent making, repair services, food business, home-based restaurant, laundry services, social media expert, resume writing, agriculture works etc. Hence, a model is considered on unemployment by taking two time-dependent controls. Initially, the formulated problem is studied with its existence of control and time dependent control parameters are considered. The total impact of all measures in controlling unemployment due to corona is examined. The optimal level needed to control unemployment is minimizing the number of unemployment population. The model is discussed by inculcating the following two controls: (i) -providing education and training of job-oriented and (ii) -campaigning about the awareness of corona and self-employment business especially during the pandemic times. The control functions, and are bounded and Lebesgue integrable functions. So, the model with two control strategies is given below,where , , and are weights. corresponds to the unemployed class and associates with the unemployed class due to Covid-19. and are the parameters attached with and respectively.where U is control set defined as,Based on the relative significance, the weights of state variables are allocated. Using Pontryagin’s maximum principle (Pontryagin et al. 1962), the evaluation criteria is obtained for the control value such that J is optimized. Occurrence of this feasible control is recognized as optimal control (Clark 1990; Lenhart and Workman 2007).

Given the objective functional , with initial data . u(t) is measurable with , then there are optimal controls and to minimize the objective functional such that

To prove the existence of optimal control, the following conditions are to be fulfilled, (i) The non empty state variables and controls. (ii) The control set U is closed and convex. (iii) Objective functional is convex in U. (iv) The system of the state must be bounded by a linear function having state and control variables with coefficients as dependent on state and time. (v) The integrand bounded below by for and Using (Fleming and Rishel 2012; Lenhart and Workman 2007), the above stated conditions are true. Clearly, the state and control variables attain non-negative values and convexity of the objective functional in u(t) exist in the minimization problem. The control along with the admissible Lebesgue measurable control variables set u(t) are convex and closed. Further, is convex in and . For verifying (iv), RHS of equation (9) is written as,where , , as vector valued function of and ,where . depends on the coefficients of the system and is maximum norm value. For the last conditions,Thus, the integrand bounded below by for and

If and be an optimal control which minimize . Let , and be the optimal state solutions of control system (9), such that there exist adjoint variables , , and which leads to the following:with the transversality conditions as,Further, the optimal control variable and that minimize are given by,

The transversality and adjoint conditions as given in Lenhart and Workman (2007), Kamien and Schwartz (2012) and Schättler and Ledzewicz (2012) and using the Hamiltonian function H, we get,where is integrand of the functional and are state equations.where , , and are the adjoint functions. Using the Pontryagin’s maximum principle (Schättler and Ledzewicz 2012) provides adjoint system, which can be followed as,The transversality conditions are,The optimality condition provides,Using bound on control (Lenhart and Workman 2007), we obtain optimal control,Further, can be represented in a compact form (by combining all the above three cases) as,andUsing bound on control (Lenhart and Workman 2007), we obtain,Further, is represented in compact form by combining all the cases (above).

The optimal system (includes the adjoint system and state system) along with the initial and transversality conditions is,and subject to the following conditions,

As the optimal control problem is arduous to solve, so forward-backward sweep method with the help of fourth order Runge–Kutta method is used and the successive steps are given in Lenhart and Workman (2007) and Kar and Ghosh (2012). To solve the optimality system, we use the Runge–Kutta forward-backward sweep procedure, which is a more accurate and effective technique. Firstly, we start with the initial guess of the control variable and then the system of state equations are approximately solved with the given initial condition forward in time. After that, all adjoint is simultaneously solved backward in time. Thus, the state equation is solved in forward time using the Runge–Kutta method with initial conditions as and . Using initial guessed control value and state values, the adjoint equations are solved backward in time along with transversality conditions. Subsequently, the value of control is updated using the state values, adjoint values together with the average value of previous control and new control. Thus, the process is repeated for updating control until convergence occurs. The choices of the bound for the control are selected between 0 and 1 because the control would not be 100% effective. In this problem, we have considered hypothetical data for the purpose of verifying the results. In Fig. 2, the computational analysis on the control of unemployment populations is addressed. The results represent the unemployment and employment populations due to Covid-19 in the presence and absence of control. In Fig. 2a, population is 10.1 unit in the presence of control, while 20 units in the absence of control. In Fig. 2b, is approximately 0.5 unit populations with control and 5.9 without control. Consequently, the unemployment declines in the presence of control as compared to the absence of control. Further, Fig. 2c, d show that the population of E and V increase in the presence of control. Figure 2e, f show the variation and convergence of the control. Figure 3 represents the variation of the adjoint variables , and with respect to time. In Fig. 4, the numerical output of control problem shows the variation in unemployed and employed population with the optimal strategy compared to prey populations not having a control technique. Here, we take some other value of the parameters compared to Fig. 2, i.e., which are shown in Fig. 4. In Fig. 4, population has an approximate value of 12 individuals and 28 individuals with control and without control respectively at 2 units time. Similarly, population has a value of approximately 1 individual with control and 5.8 value without control at 2 units time. Further, the population E and V have more value with control compared to without control. In Fig. 4, the cost-effectiveness is less compared to Fig. 2. Figure 4e, f show the variation and convergence of the control variables. The Figs. 3 and 5 represent the four adjoint variables and in the population, and we solve these adjoint by a backward fourth-order Runge–Kutta procedure. It is seen in Fig. 3 that declination in the value of leads to decline in control . Similarly, when the value of declines, the control plunges too. Initially, the values of and are extremely low, but after sometime, the values of adjoint variables grow with respect to time t. Figure 5 depicts the variation of and when different parameter values are varied in comparison to Fig. 3.

In this study, we have presented mathematical model with control and without control using a system of ordinary differential equations. Various parameters used in the model influences the system and the proposed approach. Like and illustrates the conversion rate of unemployed class to employed and employed class to unemployed respectively. High or low value of these parameters will lead to increase or decrease the class of population (Unemployed and employed). Similarly, other parameters are highly impactful for optimizing the process of employment. Local and global stability results are derived. Two controls have been introduced in the model where providing education and training of job-oriented is one control parameter, and the other one is campaigning about awareness of the corona and self–employment business. Further, b1 and b2 are the coefficients of control variables which are used to reduce the number of jobless people and raise the number of employed people. In examining the effect of optimal control strategies, we have analysed the existence conditions for optimal control. Further, the control model is solved by numerical techniques by taking assumed suitable parameters and various numerical results which are obtained with different values of parameters. Figures 2 and 4 shows that the unemployment is reduced in the presence of control variables. More focus can be drawn on taking some other control in place of the present examined control in the future or remodel the system by introducing more parameters to provide an alternative way of controlling the predator, which also restores the balance of the ecosystem.