key: cord-0917190-17hgx6s7
authors: Aldila, Dipo; Shahzad, Muhammad; Khoshnaw, Sarbaz H.A.; Ali, Mehboob; Sultan, Faisal; Islamilova, Arthana; Anwar, Yusril Rais; Samiadji, Brenda M.
title: Optimal control problem arising from COVID-19 transmission model with rapid-test
date: 2022-04-20
journal: Results Phys
DOI: 10.1016/j.rinp.2022.105501
sha: 271aade741f5fe215d417942e3d5e350a018e4ab
doc_id: 917190
cord_uid: 17hgx6s7

The world health organization (WHO) has declared the Coronavirus (COVID-19) a pandemic. In light of this ongoing global issue, different health and safety measure has been recommended by the WHO to ensure the proactive, comprehensive, and coordinated steps to bring back the whole world into a normal situation. This is an infectious disease and can be modeled as a system of non-linear differential equations with reaction rates which consider the rapid-test as the intervention program. Therefore, we have developed the biologically feasible region, i.e., positively invariant for the model and boundedness solution of the system. Our system becomes well-posed mathematically and epidemiologically for sensitive analysis and our analytical result shows an occurrence of a forward bifurcation when the basic reproduction number is equal to unity. Further, the local sensitivities for each model state concerning the model parameters are computed using three different techniques: non-normalizations, half-normalizations, and full normalizations. The numerical approximations have been measured by using System Biology Toolbox (SBedit) with MATLAB, and the model is analyzed graphically. Our result on the sensitivity analysis shows a potential of rapid-test for the eradication program of COVID-19. Therefore, we continue our result by reconstructing our model as an optimal control problem. Our numerical simulation shows a time-dependent rapid test intervention succeeded in suppressing the spread of COVID-19 effectively with a low cost of the intervention. Finally, we forecast three COVID-19 incidence data from China, Italy, and Pakistan. Our result suggests that Italy already shows a decreasing trend of cases, while Pakistan is getting closer to the peak of COVID-19.

It's likely that this infectious disease (COVID-19) originated in an animal species, and then spread to humans. Person to person spread of the novel coronavirus reported daily throughout the world. This virus involves serious respiratory tract infections [1, 2] . Therefore, all the countries are making all-out e↵orts to deal with a rapidly evolving situation which is a challenge for the whole world. An emergency 5 has been declared in infected areas of the world and a serious public health concern has been paid at a global level. Now, it's important to understand how to get control by monitoring the spreading of this disease. Within this urgency, doctors and paramedical sta↵ are on the front line for treating the COVID-19 patients. While to stop the impact of this infection and to avoid further spreading some mathematical estimations are also being performed at each level. Some method has been proposed by the researches interventions are very risky for a country's economic stability, Pakistan, India, Iran. Therefore, as a step to prevent the increasing number of infections, social distancing interventions to minimize the successful contact of infections and rapid-test to map the spread of infection into options in various countries [5, 7] , instead of implementing lockdown in their countries. Computational results give an essential way to identify the key critical elements based on the modern decomposition techniques [8, 9, 10] in di↵erent 20 available reaction routes [11, 12] that allow us to discuss the dynamical properties of the suggested models of the COVID-19. Recently, some models of the COVID-19 have suggested, they provide a good step forward to understand the dynamics of this disease [13, 14, 15, 16] . Accordingly, some suggested mathematical models were reviewed and some computational simulations investigated for the confirmed cases in China [17] . More recently, we developed an updated model of the COVID-19, we have also 25 identified some key critical parameters with sensitivity analysis [18] .

Although some mathematical models have been projected so far for new coronavirus disease prediction, a lot can still be improved. Defining such models based on mass action law with reaction rate constants and calculating the sensitivities for each model state with respect to model parameters could improve the outcomes. In a complicated modeling case like new coronavirus dynamics, it is necessary to pay more 30 attention to the optimal control problem and sensitivity analysis more accurately and widely.

Here in this article, we further developed our previous model, some transmission paths and parameters are added. We focused on analyzing the e↵ect of COVID-19 rapid-test as an alternative to suppress the spread of COVID-19. Another novelty of the paper is the identification of the critical model parameters, which makes it easy for the biologists to be used with less knowledge of mathematical modeling and also 35 facilitates the improvement of the model for future development. Consequently, here we measure the e↵ect of rapid-test infection identification on the COVID-19 free equilibrium point and the reproduction number for local stability. Interestingly, the optimal control problem applied to the established model shows that the time-dependent interventions which adapt to the number of infections are able to reduce the number of COVID-19 infections well and at a much lower cost. Finally, we give some short time 40 forecasting of three countries (China, Italy and Pakistan) using our proposed model.

Let assume the human population can be separated depending on their health status respected to infection status on COVID-19 disease, both visually (symptoms) or through a medical test. Next, let us consider that there is a random test to check whether someone is infected with COVID-19 or not. Then, 45 we split the human class into 5 di↵erent classes. The explanation about model construction is as follows. Susceptible individual increased caused by natural birth rate ⇤, and infection from A, U and I with e↵ective contact rate 1 , 2 and 3 , respectively. Note that 2 > 1 > 3 since undetected infected individuals still have full access to perform a contact social with randomly. On the other hand, I has the smallest infection rate caused by an infected individual already detected by the government, which isolated in the hospital or monitored by the government to conduct self-isolation in their home. Individuals in A increased cause by infection from S, and decreased caused by recovery to R with a rate of ⌘ 1 , progression in symptomatic to U with a rate of and the infectious detected individual with the rate of 1 . 1 present both human awareness to report their 70 health status to the government about their symptoms, so they can get treatment by the government, or detected by rapid test intervention by the government. Please note that 2 > 1 since we assume that the government has more concerned to bring the symptomatic individual into the hospital. Undetected symptomatic individual U increased by progression from A and decreased by recovery with constant rate ⌘ 2 , rapid test 2 and death rate induced by COVID-19 ⇠. Detected symptomatic infectious individual 75 I increased by progression from A, rapid test from U , and decreased by recovery rate ⌘ 3 and death rate induced by COVID-19. Last, recovered compartment R increased by recovery rate from all infected individuals. Each compartment decreased by natural death rate µ.

Based on the transmission diagram in Fig. 1 and aforementioned explanation, the model equation to J o u r n a l P r e -p r o o f Journal Pre-proof describe the e↵ect of rapid test in the spread of COVID-19 is as follows :

(1)

This model supplemented with the non-negative initial condition, and note that all parameters are positive. (1) is one of the paramount analyses in this section since it describes the human population. We perform the following theorem about the positiveness solution of the system (1).

Theorem 1. Let the initial conditions :

, then the solutions S(t), A(t), U(t), I(t) and R(t) of system (1) are positive for all t 0. Proof. From system (1), we obtain :

The above rates are all non-negative over their boundary planes of the non-negative cone R 5 + . Therefore, we have the direction of vector fields intended inward from their boundaries. Consequently, we are starting 90 from the non-negative initial conditions so that all the solutions of the system (1) remains positive for all the time t > 0. Hence, the following theorem implies the boundedness solution of the system (1). 

Basically, N (t)  ⇤ µ with respect to the condition N (0)  ⇤ µ . Therefore, we have that ⌦ to be positively invariant and attracting which su ces system (1) can be 95 considered in ⌦. Hence, system (1) considered being well-posed mathematically and epidemiologically.

The COVID-19 free equilibrium point of system (1) is given by :

To analyze the local stability of E 1 , first, we construct the valued basic reproduction number of system (1) using the next-generation matrix approach (Please see [19] for further detail, and more example in [20, 21, 22, 23, 24, 25] ). The basic reproduction number of system (1) is given by :

where

.

Note that R 0 is the summation of three types of "local" basic reproduction numbers depending on the origin of the infection, whether it from asymptomatic (R asymptomatic ), undetected symptomatic (R undetected ) 100 or detected symptomatic (R detected ) individuals.

Having the basic reproduction number in hand, and using [26] , we have the following theorem regarding the local stability of E 1 .

Theorem 3. The COVID-19 free equilibrium E 1 of the system (1) is locally asymptotically stable if R 0 < 1, and unstable otherwise.

R 0 presents the expected number of secondary cases of COVID-19 which generated by a single infection introduced into a community of totally susceptible individuals. The results in Theorem 3 shows that COVID-19 can be eliminated in the community when the basic reproduction number is less than unity.

The COVID-19 endemic equilibrium point of system (1) is given by

where 110 

I 2 is taken from the positive root of linear equation given by :

where

It can be seen from (8) that I 2 will be positive if R 0 > 1. The existence of the endemic equilibrium is given in the following theorem. 

On the system (1), we assumed

Therefore, system (1) can be re-written as

Next, let 1 as the bifurcation parameter. To do this, we solve R 0 = 1 respect to 1 to yield ⇤ 1 which is given by

Next, substitute E 1 and ⇤ 1 to the Jacobian matrix of system (9) which will gave us :

. This matrix has one simple zero eigenvalues, while the other four are negative. Therefore, we can use a center-manifold approach to analyze the stability of the endemic equilibrium when R 0 close to one. Firstly, we look for the right eigenvector and left eigenvector.Let vectorw = (w 1 , w 2 , w 3 , w 4 , w 5 ) as the right eigenvector of simple zero eigenvalue of J E1, 1 = ⇤ . The right eigenvectorw is given by

Similarly, let the left eigenvector of

. Therefore, the left eigenvectorṽ is obtained as follows:

Since the eigenvector v1 = 0 and v 5 = 0, so there is no need to look for a partial derivative of g 1 and g 5 . Therefore, we find the derivatives of g 2 , g 3 , and g 4 to get the values A and B in the Castillo-Song bifurcation theorem [27] . From non-zero g 2 , g 3 and g 4 , the derivatives are as follows:

So that A and B are obtained as follows:

J o u r n a l P r e -p r o o f Journal Pre-proof Figure 2 : Transcritical bifurcation diagram of system 1 using 1 as the bifurcation parameter

As A < 0 and B > 0, the E 1 become unstable when R 0 > 1, but close to one. At the same time, it appears an endemic equilibrium, which is locally asymptotically stable. The following results are stated in the form of the following theorem.

To illustrate the result of Theorem 3, 4, and 5, we give a bifurcation diagram of system 1 presented 125 by detecting symptomatic variables with R 0 in Figure 2 . To derive Figure 2 , we use parameter value as shown in table 1 except 1 which used as the bifurcation parameter. When 1 < 0.019, then we have that R 0 < 1, which gives a stable COVID-19 free equilibrium. When 1 = 0.019, zero eigenvalues appear, and change of stability appears when we can see that the COVID-19 free equilibrium becomes unstable, while the endemic equilibrium arises and locally stable. basic reproduction number, various ways can be adopted, i.e., such as reducing to avoid infection with the asymptomatic individual. Unfortunately, the asymptomatic individual does not show any symptoms; therefore, avoid contact with this type of infected individual is di cult to be estimated. Therefore, the only way is to reduce random contact between each individual, whether it's a healthy or infected individual. In several countries [28] , the government is campaigning for the use of medical masks to all people, 140 regardless of whether they are infected or healthy individuals. The other way to reduce R asymptomatic is to enlarging the rate of rapid-test 1 . As mentioned before, the purpose of this rapid-test intervention is to map the infected individuals (asymptomatic and symptomatic), so several actions can be done to these individuals to prevent further infection. Enlarging 1 will help the government to be able to focus all health measures on this individual group, such as monitoring self-quarantine, isolation in hospitals, 145 and so forth. The long-term e↵ect is that the government can reduce the number of infections in the field from this group of individuals. Another way to reduce R asymptomatic is by increasing the recovery rate ⌘ 1 . Since these individuals do not show any symptoms, no medical intervention can not be given to asymptomatic individuals. Therefore, encourage a healthy lifestyle to enhance self immunity through a media campaign is a reasonable option.

The second component in R 0 is R undetected , which describes new infection from the unreported symptomatic cases during its infection period. Since @R undetected

reducing the progression rate will reduce R undetected . In COVID-19, the incubation period estimated between 2-14 days [29] . Smaller means that the infection needs a longer time to show the symptoms. The other way except increasing rapid-test rate 2 , to reduce R undetected can be done by improving the quality 155 and quantity of health services in hospitals,which our model represents, i.e., by reducing the death race induced by COVID-19 (⇠) and increasing recovery rate (⌘ 3 ). By increasing the capacity of the hospital, more infected individuals can get proper medical treatment, which is expected to reduce the rate of death due to disease, and shorten the duration of infection.

The third component in R 0 is R detected which describes a new infection caused by symptomatic 160 reported cases during its infection period. Here, note that this class is monitored by the government, whether it in the hospital, or monitored by the government tracking system to conduct self-quarantine in their home. This intervention will reduce 3 . Discipline implementation of this contact reduction will reduce 3 significantly, and more easy to be monitored compared with minimizing 1 and 2 .

To finalize our discussion on R 0 , we perform a sensitivity analysis of R 0 respect to 1 and 2 . Using 165 parameters value in Table 1 except 1 and 2 , we plot a condition of R 0 = 1 on 1 2 plane in Figure 3 . Since @R0 @ 1 and @R0 @ 2 are negative, the 2 nd and 3 rd (a) area represents a combination of 1 and 2 and gives R 0 < 1, while 1 st and 3 rd (b) area representing a condition when R 0 > 1. The first information that we can take from Figure 3 is that 1 is more sensitive in determining R 0 rather than 2 . It means that rapidtest intervention into asymptomatic individuals is more advisable to encourage for the implementation 170 in the field. The reason is this, by bringing/identify the infected individuals in the field will give the government flexibility and more focus in the intervention. Another information that can be taken from Figure 3 is that the existence of "useless" intervention, that is the 1 st area. The 1 st area ( 1 < 0.0173) representing a condition of R 0 is always larger than one, no matter the value of 2 . On the other hand, the 2 nd area ( 1 > 0.228), represents an area when R 0 is always less than one even though the 175 government does not take a rapid-test 2 in unreported symptomatic individual. If the government only implements 1 in the area between 0.0173 and 0.228, then implementation of 2 must be taken carefully, since it can end up in an unsuccessful intervention (3 rd (b) area) or the successful intervention (3 rd (a) area). Therefore, implementing rapid-test need a very careful justification for the implementation, since a large rapid-test intervention leads to a very costly intervention, but small rapid-test could end up in an 

Consider a system of di↵erential equations

where x 2 R m and ↵ 2 R n . The functions h j , j = 1, 2, ..., m are often non-linear therefore model di↵erential equations may not solve analytically. An important technique to analyze system 13 is the idea of sensitivity analysis. According to this approach, the sensitivity of each variable concerning parameters can be calculated. The main equation of sensitivity is given below

The first order derivatives given in equation 14 represent the time-dependent sensitivities of all variables {x j , j = 1, 2, ..., m} with respect to each parameter value {↵ p , p = 1, 2, ..., n}. Furthermore, the di↵erential equations can be solved for sensitivity coe cients as below

Using the chain rule of di↵erentiation, equation (15) can be further driven and the sensitivity equations take the Jacobian matrix as followsṠ = H ↵p + J .S, p = 1, 2, ..., n, 

For more details and applications of sensitivity analysis in the field of systems biology, the readers are refereed to [30, 31, 32, 33, 34, 35, 36, 37] . The local sensitivity values are given in equation (16) can be computed using SimBiology Toolbox in MATLAB with three di↵erent techniques: non-normalizations, 185 half normalizations, and full normalizations. Accordingly, in a complicated modeling case like new coronavirus dynamics, it is necessary to pay attention to sensitivity analysis more accurately and widely. This helps us to identify the key critical model parameters and to improve model dynamics.

Mathematical models and computational simulations help and provide a good environment to analyze To perform our numerical experiments in this section, due to a short time interval of simulation, we ignore new-born and natural death rate in our model. Therefore, we have that A = 0 and µ = 0. With this assumption, our model now read as :

The model dynamics of susceptible, asymptomatic infected, reported symptomatic infected, unreported symptomatic infected and recovered individuals are shown in Figure 4 . The number of susceptible individuals decreases dramatically and becomes stable after four days while the dynamics of recovered people increase gradually and get flat after 15 days. Interestingly, the number of asymptomatic infected 205 individuals reaches a high level after 5 days while the number of infected people in both reported and unreported symptomatic are dramatically changed between 3 days to 15 days. Furthermore, Figure 5 explains the relationship between asymptomatic infected people with the other groups in the COVID-19.

There are almost the same model dynamics for reported and unreported symptomatic states whereas there are slightly di↵erent model dynamics for susceptible and recovered groups. Figure 6 shows that the impact of the transition rate 1 on all model variables. The e↵ect of this parameter can easily occur on in the dynamics of the model states. For example, if the value of 1 is increased then the number of asymptomatic, unreported symptomatic and reported symptomatic infected people are also increased, see Figure 6 (b,c,d). Also, the dynamics of susceptible and recovered people become stable when the value of this parameter becomes larger and larger, see Figures 6(a) and 6(e). Figure 7 shows that the impact of transition rate 1 on asymptomatic infected people, reported symptomatic infected, unreported symptomatic infected, and recovered people. The e↵ect of this parameter can easily occur on the variables A, I, U and R. It can be seen that the model dynamics for such states become flattered when the value of 1 is increased, see Figure 7 (a,b,d) . On the other hand, the number of reported symptomatic people becomes larger when the value of this parameter is increased, see Figure   220 7 (c). This is an important key element for controlling this disease. The impact of transition rate ⌘ 1 on asymptomatic infected, reported symptomatic infected, unreported symptomatic infected, and recovered people is shown in Figure 8 . Intestinally, the number of infected people in A, U, I groups is dramatically increased when the value of ⌘ 1 gets smaller, this is illustrated in Figure 8(a,b,c) . Furthermore, the dynamic of recovered individuals reaches stable very quickly when this 225 parameter becomes large, see Figure 8 (d). The idea of sensitivity analysis has an important role in identifying the model critical element. The main equation of local sensitivity is presented in equation (16). We use SimBiology Toolbox for MATLAB to calculate the local sensitivity of each model state concerning model parameters for the model equations (1) . We compute the model sensitivities using three di↵erent techniques: non-normalizations, half 235 normalizations and full normalizations; see Figures 12. Interestingly, results provide us further understanding of the model and helps us to identify the key critical model parameters. For example, it generally seems that the susceptible, asymptomatic infected, recovered individuals are more sensitive to the set of model parameters compared to the reported and unreported symptomatic individuals, this result is based on non-normalization approach, see Figure 12 (a). Another interesting result is that the group of 240 asymptomatic, reported symptomatic, unreported symptomatic people are very sensitive to almost all model parameters, see Figure 12 (b). This is an e↵ective step to identify the model critical parameters for controlling the spread of COVID-19. Accordingly, the parameters { 1 , 2 , ⌘ 1 , ⌘ 2 , ⌘ 3 } are the key critical elements for understanding and to prevent this disease because they are very sensitive according to full normalization method presented in Figure 12 

In this section, we analyze the optimal control problem related to the model (1). This optimal control approach aims to minimize the number of an infected individual (A, U, R) using rapid-test intervention. As explained in section 2 that with the rapid-test, policymakers could map and detect the infected individual; hence, the controlled isolation could be implemented to these individuals to avoid contact with a susceptible individual. Therefore, we re-define 1 and 2 in system (1) as a time-dependent parameter u 1 (t), u 2 (t), respectively. Due to a short time of interval simulation and short term of COVID-19 pandemic, we neglect natural new-born and the natural death rate from our model, similarly with section 5. Also, since R does not appear in another part of the equation in (17) except in dR/dt, we may ignore R from our optimal control problem. Therefore, system (17) now read as :

Mathematically, our goal is to find an optimal rate of rapid test which minimize the following cost functional :

The first three-component in the integrand is a cost related to a high number of COVID-19 infections in the community, while the last to component related to rapid-test intervention. Note that a 1 , a 2 , a 3 , b 1 , and b 2 are the weight constant that will balance each component in this cost function.

Here we seek an optimal solution (u ⇤ 1 , u ⇤ 2 ) such that

where U = u i |u i is Lebesgue measurable and 0  u min i  u i  u max 2  1 be the set of admissible control. To investigate the existence of optimal control, we use Pontryagin's Maximum Principle to govern 255 the necessary condition of the optimal control problem. The Lagrangian for the optimal system (18) can be defined as:

where j for j = 1, 2, 3, 4 are the costate variable related to S, A, U, I, respectively. The costate variable J o u r n a l P r e -p r o o f Journal Pre-proof j satisfies the following system of ordinary di↵erential equations :

with the transversality condition j (t f ) = 0 for j = 1, 2, 3, 4. Using the optimality condition @L @ui = 0, we 260 get :

Taking into account the lower and upper bound for u 1 and u 2 , we get the characterization :

The optimal control system which involves the system of state variables in (18) , costate variables in (21) , and optimal characterization in (22) is analyzed using the Runge-Kutta forward-backward iterative numerical approximation method [23, 21, 41] . The idea of this method as follows. First, give an initial guess for control variables u 1 and u 2 for all time t 2 [0, t f ]. Using this value, and the initial condition for state variables, solve system (18) forward in time to find values of state variables in all-time t 2 [0, t f ], and calculate the cost function (19) . Next, solve the costate system (21) backward in time using the transversality condition of costate variables, an initial guess of u i and solution of state variables from the previous step. Next, update the control variables using eq. 22. Repeat this scheme until the convergence criteria achieved. In this article, the terminate condition is until the error of the optimal solution ⇤ = {S ⇤ , A ⇤ , U ⇤ , I ⇤ , u ⇤ 1 , u ⇤ 2 } in iteration-k is less than small constant , or in this case :

For the base-case, we choose the weight cost:

(a 1 , a 2 , a 3 , b 1 , b 2 ) = (10, 10, 1, 10 6 , 10 5 ).

Note that a 1 and a 2 are larger than a 3 because it is quite di cult for the policymaker to handle the undetected crises. Furthermore, we have that c 1 > c 2 since the rapid test for the asymptomatic individual 265 is easier to implement if the candidate had already shown the symptoms. We run our simulation for t 2 [0, 120].

J o u r n a l P r e -p r o o f

For the base case, we use parameter value as shown in Table 1 , and the numerical results are given in Figure 13 , and the final number of infected individuals at t = 30, total cost function (19) , and a number of an averted infected individual given in Table 2 . Please note that we use two di↵erent y-axes to identify 270 the number of the infected individual, with and without control, since the scale of an infected individual without control is almost ten times larger than with controls. It can be seen that the time-dependent control succeeds in suppressing the number of infected individuals in all classes. The Control profile for u 1 and u 2 is relatively di↵erent. It can be seen that rapid test for the symptomatic individual (u 2 ) should be given at a maximum rate since the early time of simulation, and start to decrease to it minimum 275 value after day 70, where the total of infected individuals are already decreasing. On the other hand, rapid-test for asymptomatic individual start to be implemented at day 35 when a total of individual infected start to increase, and it starts to decrease when the number of asymptomatic individuals also shows a decreasing trend. It is interesting to see that the dynamic of the symptomatic individual has three outbreaks. The first outbreak occurs as a consequence of u 1 start to increase in day 35, while the 280 second outbreak occurs when u 1 starts to decrease to its minimum value in day 62. When u 2 starts to decrease in day 70, the infected population will start to increase, and the symptomatic class will reach the third outbreak on day 90. Using the control profile, which depends on time, as shown in Fig. 13 , we can avoid new cases as much as 138 thousand cases with the cost of intervention more than 50% cheaper than without the intervention. Our second simulation conducted to see the impact of a cheaper rapid-test cost on the dynamic of the infected population. To do this, we redefine the weight parameter of the base case as (a 1 , a 2 , a 3 , b 1 , b 2 ) = (10, 10, 1, 10 5 , 10 4 ), while the other parameters remain the same. It can be seen that b 1 and b 2 for this scenario are ten times smaller than in the base-case. The dynamic of the infected population and the control profile can be seen in Fig. 14 , and the numerical results are shown in Tab. 3. Similar to the 290 base-case scenario, the time-dependent control could avoid more than one hundred thousand new cases. As a consequence of a cheaper rapid-test cost, it can be seen that u 1 and u 2 can remain at the maximum rate for a longer period than in the base-case. Furthermore, we can see that instead of having three outbreak as in the base-case, the symptomatic undetected case only have two outbreak in this scenario. Therefore, we can conclude that a massive implementation of rapid-test as a consequence of a cheaper 295 cost for the implementation could prevent a future outbreak of COVID-19. In this section, we present some examples of how our model in (17) can fit COVID-19 incidence data. The incidence data used in this article can be accessed in [42] . We fit the daily infected data to compartment I in the proposed model (17) for the early outbreak period. We use parameter value 300 as shown in Tab. 1, except 1 , 2 , and 3 which we estimated from the real data. We use COVID-19 incidence data from China (Date of 22 January 2020 -27 May 2020), Italy (Date of 6 March 2020 -27 May 2020) and Pakistan (6 March 2020 -27 May 2020). To fit these real incidence data, we use the software MATLAB. The results are shown in Fig. 15 , and the parameters that been estimated can be seen in Tab. 4. It can be seen that our model could fit the Incidence data in China. For data of 305 Italy, our model suggests that the disease In Italy shows a decreasing trend and will tend to zero cases approximately after day 150 of simulation (Late of July 2020). On the other hand, our numerical results for Pakistan data show that our proposed model can capture the dynamics of COVID-19 in the early period of the outbreak. Let us summarized the discussion started with the mathematical modeling and concluded with the prime results, while all the data for studies are obtained from the WHO situation reports NHCRC.

• Here we have modeled the dynamics of all possible cases of human to human transmission, i.e., susceptible, asymptomatic infected, reported symptomatic infected, unreported symptomatic infected, and recovered individuals to analyze accurate transmission dynamics of the COVID-19 outbreak.

• The solutions of the model equations for di↵erent parameters and initial populations have been numerically approximated using System Biology Toolbox (SBedit) for MATLAB.

• The modeling and simulation based on the suggested sensitivity analysis indicate that almost all model parameters may have a role in spreading this virus among susceptible, infected, recorded symptomatic, unrecorded symptomatic, and recovered individuals.

• The e↵ect of the control strategies on the model is analyzed graphically and analytically. From the analysis of the basic reproduction number, we found that rapid-test intervention, which aims to detect infected individuals among the human population, is promising to suppress the spread of the COVID-19. The success level of rapid-test also depending on how the government follows up the cases that have been detected with the help of rapid-test, for instance, with self-quarantine 325 monitored for the infected individual if the case is not yet serious. Increasing the quality of the hospital by upgrading the capacity of the hospital or increase the number of medical sta↵ can also increase the chance of COVID-19 eradication program.

• The prospect of rapid test intervention to help the eradication program of COVID-19 analyzed using the optimal control theory. We find that a time-dependent rapid test intervention could reduce the 330 number of new COVID-19 infection at a lower cost. We also find that rapid tests could reduce the size of future outbreak, delay the time of it appearance , furthermore, it also could eliminate the possibility of outbreak occurrence if the implementation of rapid-test is set to be adapted to the increasing number of infections.

• The coronavirus, which is the cause of COVID-19, is very easy to mutate. Based on [], until 2022, the 335 virus variants for COVID-19 are divided into three, namely variants of concern (4 serotypes), Variants of Interest (3 serotypes), Variants under monitoring (7 serotypes), and De-escalated variants ( 27 serotypes). These serotype di↵erences are shown in the virus's physiology and its consequences on people infected with COVID-19, such as the speed it spreads, the response to the environment, and how dangerous the serotype is to humans. Therefore, further analysis is needed regarding these 340 serotype di↵erences. Di↵erences in vaccine e cacy against di↵erent serotypes make it di cult to predict the dynamics of COVID-19 with simple models. Multiple variants model can be considered to understand this issue better.

• Many types of vaccines have been introduced in various parts of the world. Not all communities in various countries accept this vaccination policy. In addition, several countries have not yet achieved the high vaccination coverage recommended by WHO. Based on this, the mathematical model in this paper needs to be developed by considering the complexity of this vaccination problem, such as di↵erences in vaccine e cacy, multiple phases and vaccine booster, and others.

the reader could consider the existence of the maximum capacity of the hospital since the low capacity of the hospital will prolong the infection period. An application of optimal control problems to model the rate of rapid-test intervention as the time-dependent variable could be considered to handle the budget limitation on the COVID-19 eradication program.

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.

Results in this study provide a good step forward in predicting the model dynamics in the future for development programs, interventions, and health care strategies. For further development of the model

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A mathematical modelling approach for childhood vaccination with some computational simulations

Mathematical Model for the Ebola Virus Disease

An updated estimation of the 455 risk of transmission of the novel coronavirus (2019-nCov)

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Understanding Unreported Cases in 460 the COVID-19 Epidemic Outbreak in Wuhan, China, and the Importance of Major Public Health Interventions

Co-dynamics of measles and dysentery diarrhea diseases with optimal control and cost-e↵ectiveness analysis

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First author financially supported by Universitas Indonesia, with PUTI Research grant scheme, 2022.