Control Strategies to Curtail Transmission of COVID-19

Recently, the World Health Organization has declared the outbreak of a severe acute respiratory syndrome coronavirius as a pandemic, and declared it as Public Health Emergency of International Concern. More than 6,83,536 positive cases and 32,139 deaths caused by coronavirus 2019 affecting 199 countries and territories. This pandemic can transform into an extremely destructive form if we still do not take it seriously. In this present study, we propose a generalized SEIR model of COVID-2019 to study the behavior of its transmission under different control strategies.


Introduction
In December, 2019, the pandemic outbreak of a novel coronavirus disease named COVID-19 raised intense attention not only within China but internationally [11]. Doctors and scientists tested the previously developed drugs to treat the infected people, but that failed to succeed. Then, in February, 2020, the World Health Organization (WHO) declared the outbreak of this highly contagious COVID-19 as pandemic globally [13]. To control the human to human virus transmission, the central government of China as well as all local governments had tightened preventive measures. However, the virus had spread rapidly across most of the regions in China and in other countries and territories around the world. One major cause of the quick spread of COVID-19 is the lack of information and awareness about the virus during its early stages of infection. As on March 29, 2020, 13:29 GMT, has affected 199 countries and territories around the world with 683,536 confirmed cases, of which 32,139 have passed away, and 25,422 are serious or critical (worldometers.info). Still there is a possibility that spread of this virus could be more intense and cause high mortality. While the New Year was enjoying their spring vacations, the outbreak of COVID-19 has accelerated, as most of the people were on their way to hometown or else travelling to different places for relaxation.
Symptoms of COVID-19 takes at least 2 to 10 days, which makes it is tough to isolate infected individuals during initial stage of the infection. The major symptoms of COVID-19 include dry coughing, high fever with difficulties in breathing [16]. The virus may spread in the environment through respiratory droplets of infected individuals when they cough or sneeze. Further an unaffected population becomes infected when they are exposed by touching the infected surface or while breathing in an infected environment [10]. During the initial stages of COVID-19 outbreak, such human transmissions were taking place because, wide-range of public was unaware of these risk factors, and the infected individuals were also not isolated and were spreading the virus unknowingly to other individuals. Moreover, the risk factor of contamination is very high since the virus can remain viable in environment for several days in favourable conditions [3,12]. Several studies reveal that old-aged people, children and those with major diseases are having low immunity and tend to seriously affected once they become infected [8,17]. Still, we lack with any proper treatment or vaccines as a cure for this disease. Hence to control its transmission further, isolating the infected individuals in special quarantine cells has been implemented in most of the countries. Despite of these prevention strategies, we are in danger as the transmission is still ongoing and the mortality due to the virus maintains a high level.
To combat this situation, studies like mathematical modelling play a crucial role to understand the pandemic behaviour of the infectious disease. Several studies have already been undertaken to analyse the COVID-19 transmission dynamics. Based on these database studies of COVID-19 outbreak since 31 st December, 2019 to 28 th January, 2020, Wu et. al.
(2020) introduced a SEIR-model to estimate the spread of the disease nationally as well as globally [18].  proposed a compartmental model by dividing each group into two subpopulations, the quarantined and unquarantined. Moreover, they re-designed their previous model by using diagnose and time-dependent contact rates, and re-estimated the reproduction ratio to quantify the evolution in better way [14,15]. Peng et. al. (2020) planned a generalized SEIR model that suitably incorporates the intrinsic impact of hidden exposed and infectious cases of COVID-19 [9].  presented the incubation period behaviour of local outbreak of COVID-19 by constructing a dynamic system [2]. Khan & Atangana (2020) described brief details of interaction among the bats, unknown hosts, humans and the infections reservoir by formulating the mathematical results of the mathematical fractional model [7]. Chen et. al. In this work, a COVID-19 model is constructed to study human to human transmission of the virus in section 2. Optimal control theory is introduced and applied to the model to develop in section 3 and in section 4, the model is simulated numerically to observe effect of control strategies on the model.

COVID-19 Model Formulation
To analyse the human to human transmission dynamics of COVID-2019, a compartmental model is constructed. The model consists all possible human to human transmission of the virus. The COVID-2019 is highly contagious in nature and infected cases are seen in most of the countries around the world, hence in the model the susceptible population class is ignored and whole population is divided in five compartments, class of exposed individuals (t) E (individuals surrounded by infection by not yet infected), class of infected individuals by COVID-19 I(t) , class of critically infected individuals by  (t) C , class of hospitalised individuals (t) H and class of dead individuals due to COVID-19 . CC-BY-NC 4.0 International license It is made available under a author/funder, who has granted medRxiv a license to display the preprint in perpetuity.
is the (which was not peer-reviewed) The copyright holder for this preprint   Rate at which hospitalised individuals get recovered and become exposed again 0.35 Assumed 10 β Rate at which infected individuals recovered themselves due to strong immunity and again become exposed 0.32 Assumed Table 1 Parameters used in the model Using the above depiction, dynamical system of set of nonlinear differential for the model is formulated as follows: . CC-BY-NC 4.0 International license It is made available under a author/funder, who has granted medRxiv a license to display the preprint in perpetuity.
is the (which was not peer-reviewed) The copyright holder for this preprint . https://doi.org/10.1101/2020.04.04.20053173 doi: medRxiv preprint  B  EI  ED  H  EI  E  dt  dI  EI  I  I  I  EI  I  dt  dC  I Note that, all the parameters used in this COVID-19 model are non-negative. Consider the feasible region, 5 ( , , , , ) : The region Λ is positively invariant, all the solutions of the system (1) are remain in the feasible region (2).

Equilibrium points
By solving the above system (1), we get two equilibrium points: II. Endemic equilibrium point: * 2 6 8

Basic reproduction number
Basic reproduction number ( ) 0 R for the model can be established using the next generation matrix method [4,5]. The basic reproduction number ( ) 0 R is obtained as the spectral radius of matrix 1 ( ) FV − at disease free equilibrium point. Where F and V are as below: 7 1 0 1 0 7 Defined effective basic reproduction number is:

Optimal Control Theory
Control measures have made significant role to control the epidemic of COVID-19 at certain level. In this control theory, five control variables are used as five possible control strategies.
Since the virus is highly contagious, it quickly infect any people come in contact with an infected individual. To avoid this situation we have taken 1 u control variable to selfquarantine exposed individuals and 2 u control variable as an isolation of infected individuals.
Moreover, to minimise mortality rate of COVID-19, 3 u control variable is taken which helps to reduce critically infected cases by taking extra medical care of infected individuals. 4 u and 5 u control variables are taken to improve hospitalisation facility for infected and critically infected individuals respectively. Purpose of this study of control theory is to protect people from the outbreak by applying control or treatment in each stage. Objective function for the required scenario is,    where, φ is a smooth function on the interval [0,1] . The optimal effect is found by using results of Fleming and Rishel (2012) [6]. Associated Langrangian function with adjoint variable 1 2 3 4 5 , , , , λ λ λ λ λ is given by, The partial derivatives of the Lagrangian function with respect to each variable of the compartment gives the adjoint equation variables . CC-BY-NC 4.0 International license It is made available under a author/funder, who has granted medRxiv a license to display the preprint in perpetuity.
is the (which was not peer-reviewed) The copyright holder for this preprint Hence, the optimal controls are given by And optimal conditions given as, ( ) This calculation gives analytical behaviour of optimal control on the system. Numerical interpretation of optimal control theory is simulated in the next section.
. CC-BY-NC 4.0 International license It is made available under a author/funder, who has granted medRxiv a license to display the preprint in perpetuity.
is the (which was not peer-reviewed) The copyright holder for this preprint . https://doi.org/10.1101/2020.04.04.20053173 doi: medRxiv preprint Figure 3 variation in all compartments Figure 3 represents the variations in all the compartments of COVID-19 model with respect to time. Also, the pandemic behaviour of the COVID-19 outbreak can be clearly seen here. We can say that, a large population of exposed individuals become infected before a week. Moreover, the critically infected cases and hospitalisation cases also increase with matter of time. Further, it clearly shows that after one week the mortality rate is also increased.
. CC-BY-NC 4.0 International license It is made available under a author/funder, who has granted medRxiv a license to display the preprint in perpetuity.    figure 5. Moreover, the figure also demonstrates which control can be applied at how much intensity to control COVID-19 outbreak in seven weeks. The high fluctuation in 3 u control variable at an initial stage suggests that it is very important to control infected individuals to move at critical stage to reduce mortality due to COVID-19. And, this can be achieved easily if an infected individual gets proper vaccination for this disease. Since effective vaccination is not available for coronavirus, one should take proper care of infected individuals to improve their immunity so that their body becomes capable to fight against the virus and not reaching to a . CC-BY-NC 4.0 International license It is made available under a author/funder, who has granted medRxiv a license to display the preprint in perpetuity.
is the (which was not peer-reviewed) The copyright holder for this preprint . https://doi.org/10.1101/2020.04.04.20053173 doi: medRxiv preprint critical stage. Moreover fluctuation in 2 u control variable suggests that it is very important to isolate or quarantine infected individuals to control this pandemic outbreak. Figure 6 Variation in each compartment under individual effect of control variables Separate effect of control variables on each compartment can be observed in figure 6. From the figure, we can interpret that 2 u control variable is highly effective to stabilise this epidemic situation. Figure 6(b) depicts that population class of infected individuals is lowest under the influence of 5 u control variable which suggests that rapid hospitalisation of . CC-BY-NC 4.0 International license It is made available under a author/funder, who has granted medRxiv a license to display the preprint in perpetuity.
is the (which was not peer-reviewed) The copyright holder for this preprint . https://doi.org/10.1101/2020.04.04.20053173 doi: medRxiv preprint infected individuals is an effective step to reduce infected cases of COVID-19. Figure 6(c) suggests that to reduce critical cases of COVID-19, first we should make control on infected individuals to become critically infected and we should improve hospitalisation and medical facility for critically infected individuals to save their lives. Figure 6(e) shows that mortality rate due to COVID-19 can be reduced effectively within three weeks of outbreak by applying 1 u , 2 u and 3 u control strategies. That means self-quarantine for an exposed individual, isolation of an infected individual and reducing critical cases by taking extra care of infected individuals are effective strategies to control further transmission of COVID-19.  . CC-BY-NC 4.0 International license It is made available under a author/funder, who has granted medRxiv a license to display the preprint in perpetuity.
is the (which was not peer-reviewed) The copyright holder for this preprint . https://doi.org/10.1101/2020.04.04.20053173 doi: medRxiv preprint (c) Figure 8 Scatter diagram for COVID-19 outbreak Figure 8 demonstrates the scatter diagram representing chaotic situation created during COVID-19 outbreak. Figure 8(a), 8(b) and 8(c) shows periodic mortality from classes of exposed individuals, infected individuals and critically infected individuals respectively, when under hospitalisation. On comparison of figures 8 (a) and (b), it can be interpreted that mortality ratio in class of infected individuals is higher and much quicker than in the class of exposed individuals. The chaotic figure 8(c) represents a very high mortality rate of critically infected individuals. Hence, in the absence of vaccination for COVID-19, it becomes a challenging situation to cure critically infected individuals.

Conclusion
In this study, a compartmental model is constructed to examine transmission of COVID-19 in human population class. Moreover, basic reproduction number is formulated to calculate threshold value of the disease. In order to develop strategies to prevent the epidemic of COVID-19, optimal control theory is applied to the model. Further to advance control theory, five control variables are introduced in the model in the form of control strategies. These strategies include self-quarantine of exposed individuals, isolation of infected individuals, taking extra care of infected individuals to reduce critical case of COVID-19, increase hospitalisation facility for infected and critically infected individuals. Distinctive and combined effects of these control variables on all the compartments are observed and examined graphically by simulating the COVID-19 model. Numerical simulation of the model reflects that quarantine and better medical treatment of infected individuals reduce the critically infected cases, which will further reduce the transmission risk and demises.

Data Availability
https://www.who.int/emergencies/diseases/novel-coronavirus-2019/situation-reports . CC-BY-NC 4.0 International license It is made available under a author/funder, who has granted medRxiv a license to display the preprint in perpetuity.