Optimal control strategies to combat COVID-19 transmission: A mathematical model with incubation time delay

The coronavirus disease 2019, started spreading around December 2019, still persists in the population all across the globe. Though different countries have been able to cope with the disease to some extent and vaccination for the same has been developed, it cannot be ignored that the disease is still not on the verge of completely eradicating, which in turn creates a need for having deeper insights of the disease in order to understand it well and hence be able to work towards its eradication. Meanwhile, using mitigation strategies like non-pharmaceutical interventions can help in controlling the disease. In this work, our aim is to study the dynamics of COVID-19 using compartmental approach by applying various analytical methods. We obtain formula for important tools like R0 and establish the stability of disease-free equilibrium point for R0<1. Further, based on R0, we discuss the stability and existence of the endemic equilibrium point. We incorporate various control strategies possible and using optimal control theory, study their expected positive impacts on the spread of the disease. Later, using a biologically feasible set of parameters, we numerically analyse the model. We even study the trend of the outbreak in China, for over 120 days, where the active cases rise up to a peak and then the curve flattens.


Introduction
Over the years, humankind has seen many epidemics. In addition to the loss of priceless lives, the economic, social and psychological pressure on people (and in general, on the entire world) are some examples of immediate impacts of any epidemic. History is full of such eras, where an epidemic lead to an economy's downfall. For instance, the deadly Spanish Flu of 1918, which lasted for about two years, infected approximately 500 million people on the planet, resulted in around 20-50 million casualties [1] and led to a GDP loss of 11%, 15% and 17% in USA, Canada and UK, respectively [2]. The epidemic came to an end only because of the development of a natural herd immunity, however, no proper medication or vaccination could be developed to cope with it. Some diseases persist in the population for a very long time and cannot be eradicated for decades. For instance, the HIV/AIDS epidemic started in the year 1981 and the disease is still spreading among people with approximately 37.9 million cases as of 2018 [3]. There is no cure for the disease till date, however, with treatment an infected person can lead a long-healthy life, but there is no way to stop this disease from spreading other than taking some preventive measures. With the development of research and medicine, humankind has also successfully eradicated some epidemics in the past. The most iconic example is that of eradicating the Smallpox disease, which was said to have lasted for around 3000 years. The epidemic was brought under control by mass vaccination [4]. The dynamics of a delayed system is examined in Section 4 wherein the stability of the equilibrium points and the sensitivity of basic reproduction number have been discussed. In Section 5, we have studied the impact of control strategies. Section 6 includes numerical analysis of the COVID-19 model and comparing it to real time data of China. We have then concluded the paper in Section 7, summarizing the results and our findings along with a few strategies, highlighting the importance of non-pharmaceutical interventions.

Model formulation
Now, we begin the formulation of our model. Our aim is to come up with a mathematical model that can capture the real aspect of the COVID-19 disease, as much as possible. We have based our model on the compartmental modelling approach, where change in each compartment is denoted by an ordinary differential equation. To begin, we assume that the entire population ( ) at any time can be split into seven compartments as can be seen in Fig. 1. This also means that at any time, the population of all the seven compartments add up to the total population at that time. Further, the population may decrease when individuals in any compartment die naturally (at a rate ) or due to COVID-19 (at a rate ) and may increase due to the recruitment of susceptible individuals (at a rate ). Next, we also assume that initially the entire population is at a risk of getting infected, i.e., everyone is susceptible to the disease. These susceptible individuals will move to the exposed compartment after they come in contact with an infected individual.
While modelling transmissible diseases, it is very important to use an appropriate rate of incidence in order to make accurate predictions. Authors frequently use a bilinear incidence rate , which is based on the law of mass action. suggests that the number of infectives increase linearly without a bound, which seems odd when the symptoms of the disease are well identifiable (because when symptoms are well identifiable the behaviour of susceptible class changes). In [67], Capasso and Serio used a nonlinear saturated incidence rate 1+ , where is the force of infection and 1 1+ measures the inhibition effect due to the change in behaviour of the susceptible individuals when is large. Such an incidence rate ensures that 1+ tends to a saturation level of , i.e., the increase in the number of infected individuals is not unbounded. In view of the above discussion, we take the incidence rate as + 1+ , where is the contact rate and is the probability of transmission per contact. This is justified because the asymptomatic individuals do not show any symptoms, therefore, there is no inhibition towards disease transmission. On the other hand, symptomatic individuals are identifiable, which leads to a behavioural change of susceptible class leading to the inhibition effect.
After the mean incubation period ( −1 ), the individuals go to infection classes , and in proportions 1 , 2 and 3 , respectively. Individuals having severe symptoms after −1 days go in the class, individuals showing no symptoms after −1 days go to class and individuals having very mild symptoms after −1 days home isolate themselves and move to class. In general, quarantined individuals are those people that home isolate themselves as a precautionary measure because they doubt that they might be infected (probably because they met an infected person recently and are now showing very mild symptoms). Similarly, we have a movement from 'Asymptomatic' compartment to the 'Quarantined' compartment in order to acknowledge those asymptomatic individuals who home isolate themselves as a precautionary measure (because they might have come in contact someone who was later diagnosed as COVID-positive). Since and individuals have only mild symptoms or no symptoms at all, we assume that they naturally recover under home isolation and move directly to the 'Recovered' compartment. And lastly, since symptomatic individuals are adversely affected they move to the 'Hospitalized' compartment (i.e., seek treatment), and post recovery move to 'Recovered' compartment. We assume that any individual moving to the 'Hospitalized' compartment is already diagnosed as positive for COVID-19.
Since we are still new to the COVID-19 disease we cannot be completely sure if recovery from the disease provides permanent immunity. Motivated by this, we incorporate parameter ▵ in the model that denotes the rate at which the recovered individuals become susceptible again. Consistent with the above description the model has been described with the help of a flow diagram in Fig. 1. Although, in reality the movement from one stage to another is a much more complex process, we have aimed to keep the model as realistic as possible, while ensuring it still can be mathematically solved and interpreted. Combining everything that we have discussed above, we formulate the following system of seven ordinary differential equations that represent our model mathematically: (2.1) As discussed earlier, it is known that sometimes it can take a few days more than the incubation period of 2-14 days [68][69][70][71][72][73] for the COVID-19 symptoms to be visible. Therefore, individuals that were identified as asymptomatic because they did not show any symptoms after −1 days can actually turn out to be symptomatic after a delay and hence there is a movement from 'Asymptomatic' compartment to 'Symptomatic' compartment. Thus, we incorporate a delay parameter which takes into account this delay in the development of symptoms in infected individuals. Thus, the time delayed model is as follows:  Table 1 briefly describes all the parameters of the model.

Positivity and boundedness of solutions
For system (2.1) to be biologically meaningful, it is required that the solutions with positive initial data are positive and bounded for all ≥ 0, as the state variables in system (2.1) represent populations. Now, as done by Naresh et al. in [74], using a theorem on differential inequalities [75], it can be easily shown that,  Simplifying and solving the above differential equation we get, = ( + + + + + + ) ≤ + − .

Basic reproduction number ( 0 )
In this section, we discuss about important analytical tools required for stability analysis: like the basic reproduction number ( 0 ) and disease-free equilibrium points. Disease-free equilibrium point can be thought of as that equilibrium position where there is no trace of infection in the population, i.e., the point where the disease no longer persists in the population. Proceeding as in Section 3 of [76], the DFE ( 0 ) of system (2.1) can be easily computed by setting RHS of each equation in (2.1) equal to zero. Basic reproduction number is an epidemiological term used for the mean number of individuals turning into infectives in a susceptible population due to one infected individual existing in it. 0 has an important role to play in epidemiology, because it determines important factors, like whether the disease will remain in the population or will it be eliminated. In general, we can say if 0 < 1 then the disease will be eradicated eventually otherwise it will persist. Although there are a variety of methods that can be employed to calculate 0 , in this paper we make use of the next generation matrix method [6,51] to come to a formula (3.2).
Step 1: To begin, we use the notation for system (2.1) as follows: Which can be rewritten as, 7 1 is such that 11 = ( + 1+ ) and every other element in this column vector is zero. Further, each element of this vector corresponds to a group of terms resulting in new infectious individuals in each of the seven classes; and , is a collection of left over terms.
Step 2: Next step is to find the Jacobian matrices of ( ) and ( ) at DFE, which are as mentioned below: Step 3: Now, we compute next generation matrix which is defined as follows: Finally, 0 is computed by finding the spectral radius of −1 and is expressed mathematically as follows: . (3.2)

Local stability of disease free equilibrium point
Here, we discuss the conditions required for local stability of the disease-free equilibrium point based on the basic reproduction number obtained in the previous subsection and state a theorem for the same.
The characteristic equation corresponding to Disease-Free Equilibrium point is as follows: Further, 0 + 0 can be simplified as follows: It can be observed from Eqs. (3.4) and (3.5) that, (0) = 0 + 0 < 0 for 0 > 1 and lim →∞ ( ) = ∞. This means that Eq. (3.4) has a positive real root and hence disease-free equilibrium is unstable for 0 > 1. Using Routh-Hurwitz criteria Eq. (3.4) will have roots with negative real part if the following conditions are satisfied: Using Eq. (3.5), the first condition in (C1) is satisfied if 0 < 1. Similarly, it can be shown that second, third and fourth conditions in (C1) are satisfied if 0 < 1. Hence, we have the following theorem.
Theorem 1. The disease-free equilibrium point of system (2.1) is locally asymptotically stable if 0 < 1.
Looking at Eq. (3.15), we can observe that 0 is always greater than 0 whereas 0 is greater than 0 when 0 is less than 1 and 0 is less than 0 when 0 is greater than 1. With the help of Descartes' rule of signs, the below theorem has been obtained for the existence of 1 .
Theorem 2. The system (2.1) has: Remark: 0 < 1 is not a sufficiency condition to eradicate any disease. Extra efforts are required.
Theorem 3. Let 1 be an endemic equilibrium point of system (2.1). Then, 1 is locally asymptotically stable iff for each of the seven Hurwitz matrices defined as in Eq. (3.17), | ′ | > 0.

Dynamics of delayed system (2.2)
The positivity of system (2.2) can be proved on similar line as done in [77] and the boundedness of system (2.2) can be proved in a similar manner as in Section 3.1.

Equilibrium points and its stability
As mentioned by Tipsri and Chinviriyasit [78], the equilibrium solutions are same for the system with and without time delay. Therefore, to obtain the equilibrium points, we use = 0. Hence, the Disease-Free and Endemic Equilibrium points of the system (2.2) are the same as obtained in Sections 3.1 and 3.4 respectively.

Local stability of disease free equilibrium point
In this subsection we will discuss stability of the system (2.2) around disease free equilibrium point.
The characteristic equation corresponding to the disease free equilibrium point 0 is : Clearly, the four eigen values corresponding to the first four roots of the above equation are negative. The remaining three eigen values can be obtained from the following characteristic equation: where the coefficient ′ and ′ are same as obtained in Section 3.3. For > 0, Eq. (4.1) is a transcendental characteristic equation and the roots will be of the form, = ( ) + ( ), where > 0. As explained by Mukandavire [79], the roots of a transcendental equation will have positive real parts if and only if it has purely imaginary roots. We will aim to obtain the conditions for which no such purely imaginary root exists for Eq. (4.1). These conditions will be then sufficient to conclude that all the roots of Eq. (4.1) for > 0 have negative real parts.
Theorem 5. Let 1 be an endemic equilibrium point of system (2.2). Then for ≥ 0, 1 is locally asymptotically stable if for each of the seven Hurwitz matrices defined as in (4.9), | | > 0 and for each of the seven Hurwitz matrices defined as in (3.17), | ′ | > 0.

Hopf bifurcation of endemic equilibrium point
In the previous subsection, we listed conditions for local stability of 1 for > 0. However, if these conditions are not satisfied, then 1 looses its stability. In this subsection, we will work to obtain the conditions for local stability of 1 based on the delay parameter and will determine the critical value of (i.e., 0 ), post which 1 ceases to be locally stable.
Remark: Choosing as the hopf bifurcation parameter helps us understand the dependence of epidemic transmission on the delay (over and above the mean incubation period) in development of symptoms in infected individuals. Theorem 6 suggests that an infectious disease can be easily controlled and the system is asymptotically stable if is under a certain critical level, but the level of infection in the system undergoes fluctuations once the delay reaches the critical level. From a biological point of view these fluctuations can be viewed as the frequent ups and downs in the cases during an epidemic. For instance, during current COVID-19 crisis the number of infected individuals kept on fluctuating. There were times when the disease seemed to be under control but it was followed by a sudden increase of infected individuals. This is an example of hopf bifurcation. Although controlling the development of symptoms is practically out of our hands, but being aware of the critical value of the delay can help us proactively deal with the fluctuating stability, for instance, large scale testings can be prioritized in order to identify infected individuals and provide necessary treatment or isolate them so as to prevent further infection.
In the next subsection, we have discussed sensitivity of 0 to various parameters.

Sensitivity of basic reproduction number( 0 )
0 is an important tools in epidemiological modelling. In Section 3.2, we have already derived a formula for 0 using the next generation matrix method. In this section, we calculate the sensitivity index of 0 in response to various parameters of the model. In plain words, sensitivity index is a measure of how much 0 changes with respect to a changing parameter. In order to compute the sensitivity index of 0 in response to a parameter , we use the following formula [80]: The sensitivity index of 0 corresponding to different parameters has been listed in Table 2 and shown graphically in Fig. 2. It is to be noted that Table 2 lists an index only for those parameters that appear in the formula for 0 (see Eq. (3.2)), for all other parameters there is no direct dependence of 0 on them.
The sign of the indices refers to the nature of change (increase/decrease) in 0 in response to the changing parameters while the value of the indices refers to the magnitude of this change. For instance, in the bar graph in Fig. 2, for every parameter having a bar pointing in the right direction, there will be an increase in 0 when the parameter increases while for all those with bars lying towards the left, 0 decreases as these increase. Further, it can be seen from Table 2 that 0 = +1.000), meaning that 0 will increase by 1% when increases by 1%. Similarly, 0 = −0.7689, means that 0 will decrease by 0.7689% when increases by 1%.
It can also be observed that 0 has the strongest negative relation with while it has a strongest positive relation with , and . Further, Figs. 3(a) and 3(b) show contour plots of 0 as a function of two parameters , and , , respectively. From Fig. 3(a) it is clear that 0 increases with the increasing and decreasing . Similarly, Fig. 3(b) shows how 0 increases with increase in and . Since, 0 has a direct impact on the spread of the disease, it is important to be aware of its dependence on different parameters, to be able to take appropriate steps to decrease it. For instance, since the COVID-19 pandemic have started, countries all over the  world have been focusing on social distancing and imposing lockdown, which now makes sense because this way they have been able to bring down the parameters like and which correspond to the probability of transmission per contact and contact rate, respectively. However, it is important to note that controlling certain parameters is out of our hands and no meddling can be done with these to decrease 0 , but a mere knowledge about the dependence can help us take proactive decisions.
In the next section, we introduce various control strategies and try to look for optimal control, so that suitable policies can be implemented for controlling the disease.

Optimal control problem
In this section, we aim to reduce infection in system using various controllers. Our aim is to see fewer people become sick and more people recover from infection. In our proposed model we introduce four control variables 1 , 2 , 3 4 . The first control variable 1 ( ) is applied on the recruitment of susceptible individuals. It is assumed that the 'Susceptible' class has a constant recruitment rate of (1 − 1 ( )) and individuals self-isolate themselves at a rate of 1 ( ) and move directly to the 'Recovered/Removed' class. The second control variable 2 ( ), is applied on the contact rate and refers to the preventive measures (like social distancing, using mask, sanitizing, etc.) that can be taken by the susceptible class to avoid getting exposed/infected. As assumed in the proposed model, the symptomatic individuals move to the hospitalized compartment and receive treatment. However, due to several reasons (like, financial bounds or lack of information or misinformation) a lot of symptomatic individuals resist from getting hospitalized. Therefore, 3 ( ) is applied on the 'Symptomatic' class and refers to the government initiative of tracking and hospitalizing more and more individuals showing symptoms. The model proposes that 'Quarantined' individuals show mild symptoms and can recover naturally while under home isolation. However, certain individuals may exhibit severe symptoms and may require medical help. Therefore, the fourth control variable 4 ( ), is applied on the 'Quarantined' class and refers to the frequent monitoring of quarantined individuals by government in order to hospitalize the one's that need medical care. A real-life application of 3 ( ) and 4 ( ) is how government kept a data base of all infected individuals and also reached them on regular basis via calls, in order to keep a check on their condition. Let, [0, ] be the time interval over which the control strategies are applied in the system. Then relative to the seven state variables ( ( ), ( ), ( ), ( ), ( ), ( ), ( )), the admissible set of control variables is defined as: = {( 1 ( ), 2 ( ), 3 ( ), 4 ( )) ∶ 0 ≤ ( ) ≤ 1 is Lebesgue integrable; for = 1, 2, 3, 4 and ∈ [0, ]}.
Introducing the control variables 1 ( ), 2 ( ), 3 ( ), 4 ( ) in system (2.2), we obtain the control system for the optimal control problem as follows: Before formulating the optimal control problem, it is important to show the existence of solution of control system (5.2).
Our aim is to minimize the cost functional (5.6), which involves minimizing the populations, Exposed ( ), Symptomatic ( ) and Asymptomatic ( ) along with minimizing the socio-economic costs associated with resources required for self isolation given by 2 1 , social distancing measures, sanitizing methods, using masks, and etc given by 2 2 , tracking and testing of symptomatic individuals given by 2 3 and tracing Quarantined individuals requiring medical help given by 2 4 . Here, (for = 1, 2, 3) and (for = 1, 2, 3, 4) are the weight constants and denotes the relative cost of interventions over [0, ]. Therefore, we want to find an optimal control pair ( * 1 , * 2 , * 3 , * 4 ) such that the objective functional in (5.6) is minimized. In the two subsections that follow, we show the existence of the optimal control pair followed by finding the Lagrangian and Hamiltonian of the control problem, and then using the Pontryagin's Maximum Principle to obtain the optimal control pair.
As done by Abta et al. [83], the existence of the optimal control pair can now be proved using the result by Lukes [82]. □
In the control problem given by (5.2), (5.3) and (5.6) the final state is free as there is no terminal cost. Therefore, as in [84] we can say that the transversality condition is satisfied and ( ) = 0. Next, using the second condition (Eq. Using the properties of the admissible set U (defined in (5.1)), 0 ≤ ( ) ≤ 1 for = 1, 2, 3, 4. This gives the optimal control pair as required in the theorem (in (5.12)). □ The formula provided by Eq. (5.12) for * ( ), = 1, 2, 3, 4 is known as the characterization of the optimal control pair. We can find the optimal control and the state variables by solving the optimal control problem which consists of the control system (5.2), the adjoint system (5.10), the boundary conditions (5.3) and (5.11), and the characterization of the optimal control pair (5.12). Also, it is observed that second derivative of the Lagrangian with respect to all the control variables ( ), = 1, 2, 3, 4 is positive showing that the optimal control problem is minimum at optimal control * ( ), = 1, 2, 3, 4.

Numerical analysis
In this section, we focus on the numerical analysis of the model (2.1). It is important for a proposed epidemiological model to be consistent with the real world, otherwise all the obtained analytical results turn out to be futile. It is needed to be ensured that if required, after the calibration of the model, the model can be used to forecast the future trends of the disease, so that various mitigation strategies can be adopted beforehand.
In the next few sub-sections, we have investigated the sensitivity of different compartments to various model parameters using the One-way Sensitivity Approach [52,91], followed by studying the behaviour of the model in the presence of time delay and control strategies.

One-way Sensitivity analysis in the absence of control and delay
In this section we have used the One-Way Sensitivity analysis approach to analyse the behaviour of various classes when only a single parameter changes, and the rest of the parameters are still at their base value.
Such an analysis on an epidemiological model helps in predicting various steps that can be taken instantly to cope with the spreading disease at the initial stage itself, while proper treatment and medication for the disease are being figured out. We have worked with factors like , , , and , because these are some of the parameters that can be controlled in real sense by social distancing, imposing lockdown, precautionary measures and regular testing being conducted by the government. Fig. 5 depicts how the seven populations of the model change in response to the change in probability of transmission per contact . Fig. 5(c) depicts the trajectories at the base level of . It can be observed, that with a decrease in the trajectories shift towards the right (see Fig. 5(b)), indicating that in the presence of lower levels of , the spread of the infection will be delayed.
While with an increase in , the trajectories shift towards left (see Fig. 5(d)), indicating that the greater the level of , the earlier the spread begins. Further, it can be observed that the height of the curve for Exposed compartment change with the changing . The peak is lower when is decreased while it is higher when is increased, indicating that the higher the probability of transmission per contact, more the number of Exposed individuals and hence greater the infection. It can also be noted from Fig. 5 that as increases the rate at which the susceptible population decrease in the system keeps on increasing. For instance, when = 5.62 * 10 −10 (in Fig. 5(c)) the susceptible population start falling around the 50th day, whereas, when is increased to 6 * 10 −10 (in Fig. 5(d)) the susceptible population start falling around the 40th day itself. Similarly, Fig. 5(a) depicts that at zero probability of transmission of disease upon contact within the population, the susceptible population thrives at positive levels and there is no infection in the system due to which the trajectories of all other compartments rest at zero. While it is difficult to achieve = 0, it can be brought to lower levels by taking proper precautionary measures. By the above discussion, it can be understood why wearing masks has been employed as a mitigation strategy against COVID-19 all across the globe. Fig. 6 shows how various compartments behave in the presence of changing parameters. Fig. 6(c) shows that with increase in , exposed individuals are decreased. This is because is the rate with which asymptomatic people start showing symptoms and shift to the class, and hence due to the reduction of asymptomatic individuals there are less accidental cases of a susceptible individual coming into contact of an infective individual without knowing about it. Further, it can be seen from Fig. 6(b) that higher the rate with which exposed population become infected, more the level of symptomatic individuals in the system. And Fig. 6(d) depicts how the number of recovered individuals increase with the increase in , which is the rate of hospitalization of individuals. Therefore, there should be more focus on medical facilities. Also, temporary lockdowns can significantly help in reducing the infection as it leads to a reduced contact rate. It can be observed from Fig. 6(a) that when is equal to 0, the infected population is at zero levels. This mean that if the people are not coming in contact with each other at all, the disease will stop spreading due to the lack of new host bodies for the virus. But, it is next to impossible to achieve = 0 permanently, because measures like lockdown drastically affect the economy and this is only a short term solution.

Effect of control parameters and delay parameter
This sub-section, discusses the numerical simulations on the controlled dynamics of all state variables based on the set of parameters provided in the Table 2. We use Euler method to study and compare the controlled and uncontrolled model presented above graphically. Using MATLAB we have obtained the graphical results with varying conditions with the combination of both delay parameter and control parameters 1 , 2 , 3 and 4 . We have assigned values to all the parameters, initialized the values of state variables and the weight constants provided in the objective functional. All the state equations have been solved with the help of forward Euler method and then the adjoint equations have been solved by backward Euler method. Next we have control updates for 1 , 2 , 3 and 4 using weighted convex combinations. We have done our graphical interpretation for 120 days and analysed the behaviour of all the state variables.
In Fig. 7 we have discussed the behaviour of different compartments under various combinations of delay and controllers. As our analytical results suggest implementation of control strategies and a reduction in the delay factor can significantly reduce the infection in the population, therefore, discussing numerical simulations of model in presence of controllers and delay is very important. If we compare Fig. 7(a) with Fig. 7(b), and Fig. 7(c) with Fig. 7(d) we can observe the impact of control strategies very clearly. In the absence of any control strategies (in Figs. 7(a) and 7(c)), we can see that the peaks of infective classes are considerably high. For instance, in Fig. 7(a) (where we have no control strategies in the system) the number of exposed, symptomatic, asymptomatic, quarantined and hospitalized individuals have peaked to approximately 4 * 10 8 , 1.5 * 10 7 , 0.5 * 10 7 , 1.8 * 10 7 and 1.2 * 10 8 , respectively while in Fig. 7(b) (where control strategies have been implemented) the numbers are only 2500, 100, 30, 100 and 700, respectively. A similar positive effect of controllers can be observed in the presence of delay as well, if we do a similar comparison of Figs. 7(c) and 7(d). These observations suggest that, control strategies can significantly help to control the spread of the disease and in their absence this spread can go out of hands. A delay in the development of symptoms in some exposed individuals (who are currently thought of as asymptomatic) can also speed up the spread of infection. The longer an individual remains asymptomatic (and hence unrecognizable), the lesser would be the inhibition from susceptible class and more individuals will keep on getting infected. Comparing Fig. 7(a) with Fig. 7(c) and Fig. 7(b) with Fig. 7(d), we can clearly see how the peaks in the absence of delay are significantly low and the number of recovered individuals are comparatively higher. Fig. 7 also suggests that the best strategy is when there are non-zero controllers and an absence of delay (see Fig. 7(b)). Delay can be controlled to some extent by doing mass testings but it is impossible to achieve a condition where delay is exactly zero. Therefore, the next best situation, i.e. a combination of non-zero controllers and non-zero delay within certain limits (see Fig. 7(d)), turns out to be the most sensible strategy to cope with an epidemic. Also, it is worthwhile to note the worst case scenario in Fig. 7(c) where we have zero controllers and a non-zero delay, which is the case during the initial stage of any epidemic.
Talking about the best and the worst combinations, Fig. 8(a) depicts the behaviour of the susceptible class in the two scenarios. We can see how in the presence of delay and absence of controllers (green curve in Fig. 8(a)) the susceptible class undergo a steep fall but in the absence of delay and presence of controllers (yellow curve in Fig. 8(a)) the susceptible population is stable. It is to be noted that, it may seem as if the susceptible population (the yellow curve in Fig. 8(a)) is constant and do not change at all in the presence of controllers and absence of delay, but it is misleading as Fig. 8(b) clearly shows a decrease in the susceptible population. However, this decrease is very small as in this case the infection is very low (refer to Fig. 7(b)) and as a result the susceptible population is thriving and is not affected much. However, as discussed before this is only an ideal situation and not a realistic one. In reality, we have seen that it takes a few more days (than the ideal incubation period) to develop symptoms after exposure with a COVID-positive person. China is the only country that could bring the situation under control in a very short period, when the country suffered at the hands of COVID-19. Therefore, it is worthwhile to study the case of China and talk about the possible reasons that helped in controlling the disease rapidly. Fig. 9 discusses the possible control strategies that could be implemented to deal with this pandemic. The figure is basically a numerical solution of the optimal control pair obtained in Section 5.4 (see Eq. (5.12)) and is a pictorial representation of the time dependent solution. It tells the levels of control strategies implemented in order to cope with the COVID-19 disease. Fig. 9 depicts the variation in the effectiveness of control strategies over time. Furthermore, the graph shows which controls may be used and the amount of intensity that can be applied to the controllers in order to prevent the spread of COVID-19 infection in 120 days. The fluctuations in the control strategy 1 imply that it is of utmost importance to keep track of the new recruited susceptible individuals, then they will not come in contact with others and will be removed from the stages of infection for the entire period. Similarly, the fluctuations in 2 imply that it is mandatory to adopt preventive measures (such as social distancing, using mask, sanitizing, etc.) which can be implemented by raising awareness of the issue through advertising that stresses the value of preventative actions in battling the disease. Therefore, 3 ( ) is applied on the 'Symptomatic' class and refers to the government initiative of tracking and hospitalizing more and more individuals showing symptoms. Additionally, due to a lack of hospital beds, even those who desired to be hospitalized were unable to do so. This can also be done by putting up helplines, apps, and other means of helping people locate hospitals with open beds. In real life government set up helplines, created apps to aid people in getting vacant hospital bed. This is how 3 was realized in real life. Therefore, the fourth control variable 4 ( ), is applied on the 'Quarantined' class and refers to the frequent monitoring of quarantined individuals by government, so that they remain in quarantine till they recover and in case their symptoms deteriorate, provide necessary hospitalization facilities. A real-life application of 4 ( ) is how government kept a data base of all infected individuals and also reached them on a regular basis via calls, in order to keep a check on their condition. Now, we compare the results of our model with the actual data from China. We use the data of active infected cases from [92] to numerically simulate our model. Model parameters are same as in Table 2. We have fitted our model with real time data of active infected count from China's population and compared the predictions of our model with actual numbers. This analysis has been done to give a basic yet viable and informative model for the future predictions, and depict the viability of regulatory and precautionary measures.
In Fig. 10(b) the actual data of active infected individuals in China is represented by the green curve. The red, yellow, blue and purple curves represent the trajectory of active infected individuals predicted by our model in the presence of controllers and a delay of = 0, = 1, = 2 and = 3, respectively. It can be observed that the numbers predicted by the model are close to the actual numbers, if we assume that there was a delay of 2 days in the development of symptoms (or identifying asymptomatic individuals) when the infection spread in China. Fig. 10(a) depicts that in the absence of strict control strategies China could have witnessed active infected cases as high as 130 million. But as can be seen in Fig. 10(b), the maximum number of active infected cases were only around 60,000 which implies China had a very thoughtful combination of control strategies and was very quick in imposing restrictions and doing aggressive mass testing. This mass testing really helped to reduce the delay in identifying asymptomatic persons and consequently reducing infection. Thus, we have verified the accuracy and effectiveness of our model, equipped with control strategies and time delay, which fits best with real data. Therefore, when both time delay and optimal control parameters are introduced into the model there is a significant reduction in the spread of the infection.

Discussion and conclusion
In this study, we looked into the epidemiological model, with the help of an − − − − − compartmental model. Although the dynamics of many communicable diseases, including influenza, Ebola virus disease, measles, tuberculosis, etc., might be studied using this model. However, since the entire world is now battling COVID-19, we in our study take this particular scenario into consideration. We obtained some very important and useful analytical results, for instance the basic reproduction number ( 0 ), the disease-free ( 0 ) and endemic ( 1 ) equilibria. Then we derived the conditions for which 0 and 1 are stable based on 0 . We established that 0 < 1 would imply local asymptotic stability of 0 , as was seen in Fig. 4(a), where the system could be seen converging to 0 when 0 < 1. For system (2.1), it was seen in Fig. 4(b) that the system converged to 1 for 0 > 1. We also derived conditions for Hopf bifurcates at 1 for the bifurcation parameter = 0 . We discussed the importance of as bifurcation parameter and dependence of the epidemic transmission on length of the delayed period, . It was proved that if the delay is beyond a certain critical level, 0 , the endemic equilibrium point looses its stability. The effect of parameters on 0 was studied using sensitivity analysis and it was seen in Fig. 2, how 0 is highly sensitive to certain parameters like and . For instance, Fig. 5 suggested, with a decrease in infection is reduced in the system. Similarly, Fig. 6(a) suggested in case of zero contact rate there is no infection in the system. Hence, measures like social distancing, lockdowns and using face masks can be employed to bring down the values of these parameters, hence reducing the spread of the infection. Similarly, Fig. 6(c) suggested, that if testing is being regularly conducted, more and more asymptomatic individuals can be identified, which can help in reducing cases of new exposed individuals.
Also, we discussed various control measures to cope with the disease. We investigated the following four non-pharmaceutical precautionary and preventive control strategies for coping with novel coronavirus: (1) Home-isolation of the susceptible individuals; (2) Taking preventive measures; (3) Government intervention to track and hospitalize symptomatic individuals; and (4) Government intervention to monitor and hospitalize quarantine individuals, if necessary. Our main focus was to relatively set up an optimal control problem and find an optimal solution to significantly reduce infection and increase the count of recovered individuals. We first proved the existence of optimal control pair * , = 1, 2, 3, 4 and then to achieve our goal, we used the Pontryagin's maximum principle to obtain the optimal solution. In addition, the significant numerical findings of time delayed model were mathematically verified using MATLAB. We compared combination of with and without controls (see Fig. 7) and with and without time lag (see Fig. 7). It was analysed that a combination of all the controllers (see Fig. 9) can slow down the growth of infected individuals and prevent any outbreak. We showed through the graphical results that control strategies help in increasing the susceptible individuals and decreasing the infection. Also, with increasing time delay, infection kept on increasing (see Fig. 10(b)), which was due to the increased delay in the development of symptoms in some asymptomatic individuals and hence increased chances of contact with asymptomatic individuals.
Next, we compared the predictions of our model with the real-time data from China. Our estimations fitted well with the real data (see Fig. 10(b)). It was deduced that with a combination of control strategies for around 120 days in China, the count of infected individuals decreased. Thus, we can conclude that in order to reduce the spread of infection, imposing strict non-pharmaceutical measures (like home isolation, social distancing, increased hospitalization facilities and isolation) as control strategies can prove to be viable. Since, our model fits well with the COVID-19 data from China, hence our model is realistic. Applying control policies on epidemiological model provides a great help to the researchers in making necessary future predictions. Therefore, until people are properly vaccinated all over the globe, control measures will play an important role in dealing with the disease. Although, our findings suggest that non-pharmaceutical interventions like self-isolation of susceptible individuals, reduced contact with infected individuals, and government monitoring can help in reducing the rate of transmission and bringing down the disease-induced mortality rate, but the success of these strategies will only depend upon their proper implementation.

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.