SEAMHCRD deterministic compartmental model based on clinical stages of infection for COVID-19 pandemic in Sultanate of Oman

The present novel coronavirus (COVID-19) infection has engendered a worldwide crisis on an enormous scale within a very short period. The effective solution for this pandemic is to recognize the nature and spread of the disease so that appropriate policies can be framed. Mathematical modelling is always at the forefront to understand and provide an adequate description of the transmission of any disease. In this research work, we have formulated a deterministic compartmental model (SEAMHCRD) including various stages of infection, such as Mild, Moderate, Severe and Critical to study the spreading of COVID-19 and estimated the model parameters by fitting the model with the reported data of ongoing pandemic in Oman. The steady-state, stability and final pandemic size of the model has been proved mathematically. The various transmission as well as transition parameters are estimated during the period from June 4th to July 30th, 2020. Based on the currently estimated parameters, the pandemic size is also predicted for another 100 days. Sensitivity analysis is performed to identify the key model parameters, and the parameter gamma due to contact with the symptomatic moderately infected is found to be more significant in spreading the disease. Accordingly, the corresponding basic reproduction number has also been computed using the Next Generation Matrix (NGM) method. As the value of the basic reproduction number (R0) is 0.9761 during the period from June 4th to July 30th, 2020, the disease-free equilibrium is stable. Isolation and tracing the contact of infected individuals are recommended to control the spread of disease.


Materials and methods
In this research, we formulate the transmission mechanism of COVID-19 using a compartmental model which is deterministic in nature. We have collected patient wise demographic data including symptoms from Royal hospital, Oman after taking formal ethical approval. The data highlighted the number of patients admitted in ICU, admitted in the hospital, and not required admission in the hospital. Their duration of stay in the hospital (1 to 30 days) is also given in the data depending on the severity of the disease. Also, we have studied various symptoms, along with the severity of the diseases which leads to arrive at different compartments. We have made a detailed study on the various stages from mild to severe that COVID-19 patient is undergoing while they are under treatment. These studies prompted us to make a compartment model that incorporates mild, moderate, critical, and severe cases separately instead of taking all together as an infected case.
To formulate the mathematical model, the total population is grouped into eight mutually exclusive compartments based on their disease status. The class of Susceptible individuals (S) are those who have never been infected with and thus have no immunity against COVID-19. Susceptible individuals become exposed once they are infected with the disease. The Exposed individuals (E) are those who are in the infected stage, but not yet infectious to others. The class (A) of infected individuals are those with no symptom or mild symptom developed. These individuals may have symptoms like fever and cough but with normal chest X-ray result. These individuals may either improve or progress to the Moderate stage of the disease. The Moderate class (M) of individuals suffers from a moderate infection symptom like fever and cough and may even have mild pneumonia but do not need hospitalization. In this stage, they are likely to show symptoms of fever and cough and with chest X-ray result that indicate major bilateral abnormality, pneumonia, or infiltrations/patchy shadowing. These individuals may either improve or progress to the severe stage of the disease.
The hospitalized individuals (H) suffer from a severe infection and have severe pneumonia and need hospitalization. These people show indications of serious pneumonia that prompts dyspnea, blood oxygen saturation that is less than or equal to 93%, respiratory frequency which is more than or equivalent to 30 breaths/min at rest, the ratio of the partial pressure of arterial oxygen to fraction of inspired oxygen being less than or equal to 300 mmHg, and/or lung infiltration that is more than 50% within 24 to 48 hours. Such cases require immediate hospitalization and supplemental oxygen. These individuals may either recover or progress to the critical stage of the disease. The class of critical (C) individuals are affected by a critical infection like a failure in the respiratory system, septic shock, and/or dysfunction in multiple organs and require treatment in an Intensive care unit (ICU), often with the requirement of mechanical ventilation. These individuals may either recover or die suffering from the disease during the time. The Recovered individuals (R) are those who have recovered and are expected to be immune to future infection with COVID- 19 Figure 1 depicts the various stages of the proposed compartment model. At any time, t, total population N (t) = S (t) + E (t) + A(t) + M (t) + H(t) + C (t) + R(t) + D(t). These 8 compartments are governed by 8 dynamic system of non-linear differential equations that vary with time, and nature of the solution and the stability of the system is investigated. Figure 2 depicts the diagram of the overview of the model. The corresponding equations and the description of the various parameters involved are given in Eq. (1). The compartment C is completely isolated to strict preventive measures and proper care in ICU. Hence the transmission from C is considered as negligibly small.   Analysis of the model. Nonnegativity of solutions. The model (1) represents a bilinear system consists of 8 differential equations. The system is considered positive when all the compartmental variables assume nonnegative values for t ≥ 0 . R(t) and D(t) are cumulative variables.

Model and equations
Proposition 1 Consider a system F(t) = (S, E, A, M, H, C, R, D) with the initial conditions F(0) ≥ 0 , then the solutions of (1) are non-negative for all t ≥ 0.
Proof The integrating factor of first equation in (1) is Similarly, we can establish other variables also. Thus F(t) ≥ 0 for all t ≥ 0.
The compartmental system satisfies mass conservative property, and it can be verified easily from Hence the total population is a constant. When we consider each variable as the fraction of population, we can assume Assuming an initial condition S(0) , E(0) , A(0),M(0) , H(0) , C(0) , R(0) , D(0) summing to 1, it is possible to prove that the variables converge to an equilibrium S ≥ 0 , So only the susceptible, recovered and dead populations are eventually present. This means that the epidemic phenomenon has come to an end. All the possible equilibria are given by (S, 0, 0, 0, 0, 0, R, D) with S + R + D = 1 . In our model (1) the rate of the transmission parameter values β, γ is to be reduced by strict social measures. Also, the rate of the main transition parameters σ,µ depends on the strictness of the home isolation and quarantine process. Even if the strict procedure is adopted in hospital admission, there is always a chance for the disease to spread from the hospital. This is because long-time exposure to huge numbers of infected patients directly increments the risk of infection among healthcare workers. In addition to this, inadequate personal protection of healthcare workers, shortage of personal protective equipment (PPE), as well as lack of organized training, practice, professional administration, and guidance also may aid the spread the disease. The increase  Transmission rate due to contact with asymptomatic exposed individuals www.nature.com/scientificreports/ in awareness of personal protection, availability of sufficient personal protection equipment (PPE), and proper awareness would play a significant role in reducing the risk of infection among healthcare workers 23 . Hence by considering all these factors, we incorporated compartment class H in our model in determining the spread of the disease, specifically in calculating the basic reproduction number. The model has mainly three subsystems: The first subsystem is the susceptible individuals (S), the second subsystem is the infected and exposed individuals ( E , A , M , H , C ) which are non-zero in the transmission state , the third subsystem includes the recovered and removed individuals ( R,D ). The second subsystem is more important in the epidemic study. When this subsystem is zero the remaining variables are at equilibrium. The variables R , D , are monotonically increasing and converge to the values R , D , respectively and S is monotonically decreasing and converges to S if and only if the second subsystem converges to zero.
Basic reproduction number (R 0 ). It is not possible to predict the endemicity of the disease based on the increase or decrease in the number of cases alone. We need to find equilibria and linearization about each equilibrium to determine whether the disease is pandemic or not. If an equilibrium is unstable with all susceptible population, then it is an epidemic, and if it is asymptotically stable, then it is the end of the pandemic. To get a meaningful realization of the situation, the basic reproduction number (R 0 ) is calculated, and this determines whether there is an epidemic or not. The basic reproduction number (R 0 ) is debatably the most significant quantity in infectious disease epidemiology. It is among the quantities most critically evaluated for evolving infectious diseases in epidemic states, and its value delivers insight when scheduling control interventions for conventional infections. It can be found easily whether the disease-free equilibrium of the system exists and is calculated from this system of the equation after converting to next-generation matrix (NGM).
If R 0 < 1, the infection stops, and if R 0 > 1, there is an epidemic. The basic reproduction number (R 0 ) is characterized as the number of secondary diseases brought about by a single infected individual from a susceptible population and is an indicator of the severity of the disease. Diekmann et al. 1 and Van den Driessche and Watmough 2 proposed a generalized approach to determine the basic reproduction number using the next generation matrix approach. The linearization of (1) the disease-free equilibrium is of the form (S, 0, 0, 0, 0, 0, R, D) . The infectious sub-system from (1) can be written as The Jacobian matrix from (2) is Now the Jacobian matrix is decomposed into two matrices (transmission matrix and Transition matrix).
The NGM with classical domain is denoted by D C . Here, we assume an auxiliary matrix E of K is of the form: Now.
From this, the basic reproduction number by considering the classical domain matrix is the spectral radius of the NGM.

Steady state analysis. Proposition 2
The disease-free equilibrium is unstable for R 0 > 1 , and stable for R 0 < 1. Proof The dynamical matrix of the linearized system around the equilibrium (S, 0, 0, 0, 0, 0, R, D) is. www.nature.com/scientificreports/ The matrix (4) has three null eigenvalues and five eigenvalues which are the roots of the equation. Then Now consider the following cases Since P(s) is a continuous function of s , hence by Bolzano theorem on continuous function implies that P(s i ) = 0 for some s i > 0 , at least one eigenvalue must be positive. So, in this case, the equilibrium is unstable.
Assume that P(s) = 0 has a root of the form x + iy where x, y ∈ R and x ≥ 0 . Then P(x + iy) = 0. Now This shows 1 < 1 , clearly a contradiction. Hence all roots of the equation P(s) = 0 have the form x + iy where x, y ∈ R and x < 0 . Thus, all eigenvalues of the coefficient matrix have negative real part and this case equilibrium is stable, the solution of (1) die out exponentially 9 .
Proof Define x = [E, A, M, H, C] T , we can write the subsystem as.
. and the other variables in the differential Eq. (1) can be written as The number of infectives always approaches zero and the number of susceptible always approaches a positive limit as t → ∞.

Extension of the model by including exposed period
The study on COVID-19 reveals the fact that transmission of disease happens in various stages, namely symptomatic transmission, pre-symptomatic transmission, and asymptomatic transmission. In the case of a symptomatic transmission, the infected persons develop signs and symptoms compatible with COVID-19 virus infection and transfers to the contacted person(s). Specifically, symptomatic transmission denotes the transmission from a person while they are suffering from the symptoms and this transmission takes place through respiratory droplets or by contact with the contaminated objects and surfaces 5,[30][31][32][33][34] . Research studies on virus infection reveal that flaking of COVID-19 virus is peak in the upper respiratory tract (nose and throat) early during the disease, which is, within the first three days from the start of symptoms [24][25][26] . Preliminary data proposes that people may be more infectious around the time of the beginning of the symptoms as compared to the later stages of the disease [35][36][37] .
In the case of pre-symptomatic transmission, there is an incubation period (on average 5-6 days and can be extended up to 14 days) which is the duration between the first day of infection and symptom onset, is, however, can be up to 14 days. During this period, some infected persons can be contagious. Therefore, transmission from a pre-symptomatic case can occur up to a certain extent before symptom onset. Various studies [24][25][26][27][28][29] reveal that people can be tested positive for COVID-19 from 1 to 3 days before the development of symptoms. Thus, it is probable that infected people could transmit the disease before the onset of any visible symptoms. It is very significant to understand that pre-symptomatic transmission still needs the virus to be spread through infectious droplets or through touching contaminated surfaces.
While in asymptomatic transmission, a person is infected with COVID-19, but no visible symptoms. Asymptomatic transmission refers to the transmission of the infection from an infected individual who does not show any symptoms. Even though there is no solid report on asymptomatic transmission, we cannot avoid the possibility that it may occur. In some countries, asymptomatic cases have been reported through an effective contact tracing mechanism. WHO frequently monitors all evolving evidence about this and is delivering updates whenever more information becomes available. So in our model (1), the exposed classes are divided into two separate classes namely E N defines no symptom or transmission and E T with no symptom but can transmit the virus ("pre-symptomatic transmission"), based on the various studies 24-29 about COVID-19 pre-symptomatic infection. The rate of exit from E N is ε and from E T is υ . Accordingly, the modified diagram by expanding exposed case is given in Fig. 3. The modified equation of (1) is also given in (18). Figure 3. Extension of the model by including exposed period. www.nature.com/scientificreports/ We choose θ 3 = 1 − (θ 1 + θ 2 ). Accordingly, the basic reproduction number using next generation matrix can be written as The model including exposed period was implemented in MATLAB and findings are depicted in "Results" section.

Results
The simulation of the extended model is implemented in MATLAB. The data for this study has been taken from the official Tarassud Plus application, Ministry of Health, Oman 38 and the sources 38,39 . The data of Oman during the period June 4th, 2020 to July 30th, 2020 (about 57 days) has been used for this study. During this period, the country has undergone lockdown, partial lockdown and relaxation. The given data is categorized into 9 compartments (S, E N , E T ,, A, M, H, C, R, D) as mentioned in the methodology Section 2.1. The data for the study is given in Table 2.
We investigated the model parameters and corresponding effective reproduction number R 0 during the period from June 4th to July 30th when the country has undergone various intervened measures to control the disease.
Parameter estimation. Mathematical models are useful in capturing patterns in data, thereby predicting the nature of a pandemic to take the right policy to control it. Hence it is very essential to find out the best-fit parameters that give the closest correspondence between model predictions and data. The parameter estimation is done based on parameterized nonlinear functions which solve the min F(x i − y i 2 , where F(x i ) is a nonlinear function and y i is data. The parameters for each period are chosen in such a way that it gives the minimum value for R 0 . Figure 4 shows the R 0 over a different period, and active cases versus R 0 . Figure 5 shows the various transmission parameters over a period from June to August based on the fitted data' .
Sensitivity analysis of the model with respect to varying parameters. We analyzed the sensitivity of the model on infected and, recovered cases and cases of death cases concerning transmission parameters, which would help the policymakers to plan appropriate strategies to mitigate the spread of the disease. Figure 8 depicts 'how infected case changes with respect to transmitting parameters. Figure 8a-d illustrates how the nature of pandemic changes along with R 0 based on the transmitting parameters, α, β, γ, δ. The parameters are scaled using scaling factors 0.5, 0.8, 1.4, 1.8 and 2,2 on their current value during period 5. Figure 9 shows the sensitivity of transmission parameters with respect to the recovered cases. Figure 10 shows the sensitivity of transmission parameters with respect to the death cases.

Discussion
Based on the public data 38,39 , the situation in Oman at the end of May was not very severe. Muscat governorate reported more cases compared to other governorates. From the beginning itself, the Oman government has imposed strict measurements to control the disease. Oman Supreme committee has taken a good decision to close the main containment zone, Muscat governorate by April 10 which resulted in the reduction of the transmission rate. By April 29, 2020, the government lifted the restriction on travel inside the governorate except in and out of the Muscat governorate. More number of random tests were conducted in the main containment zones of Muscat governorate such as Muttrah, Seeb, Muscat Province and Al Amerat. It was made compulsory to wear masks in public places and thermal imaging was also done while entering majority of shops in Muscat. The entry to shops was restricted to people with a temperature of more than 37.5 °C, and children of age less than 12 years of old were also not permitted. These decisions were a part of several measures broadcasted by the Municipality as per decision number 199/2020, which lists out 15 safety measures to be practiced by the commercial establishments. Fines were also imposed on those who violates the rule. Lockdown in Muscat province, including the capital Muscat, was lifted entirely on Friday, May 29, as the government began easing the containment measures in place, nationwide. Additionally, at the end of May 2020, the government announced that at least 50 percent of public sector employees could resume working from their offices from Sunday, May 31. In this research, the significant 15 parameters mentioned in the proposed model is calculated during the said period (Period 1-Period 5) mentioned in Sect. 3, where significant change is observed in the data. The corresponding basic reproduction number R 0 is calculated and this depicts the nature of the pandemic with respect to the the spread of the disease.
During Period 1, a partial lifting of the restrictions was observed after a 2-month lockdown period. The value of R 0 is found to be 1.5802. During Period 2, the ministry ensured strict social measures to stop the spreading of the disease. Timely instruction is given through official social media account 39 , such as the importance of washing hands, social distancing etc. At the same time, authorities took necessary care to convey the people not to be panic so that they can reduce the overstress in this situation. Otherwise, ultimately it will affect the living conditions www.nature.com/scientificreports/ and the business and working environments. More tests were conducted to avoid panic in people. As a result, a slight change resulted in social parameters like α , β and the new reproduction number becomes R 0 = 1.4811. During Period 3, isolation and quarantine policies were made stricter during the period from 28th June-10th July 2020. Anyone who enters the Sultanate was subjected to quarantine for 14 days. The Omani residents were not permitted to go outside the Sultanate. The land and air borders were closed. Because of this strict isolation and quarantine process, and basic social distancing measures among people (such as washing hands often, not touching one's face, avoiding handshakes etc.) the value of R 0 became 1.3071. During the period 11th July-20th July 2020, due to the lack of strict lockdown and an increase in the number of random tests in the containment zones, more infected cases were reported. The Supreme Committee in Oman advised the public to wear face masks in public places, and for business establishments to ensure that the visitors also followed the required safety measures. The spread of the disease was little more during this period, which resulted in R 0 value of 1.4392. From 21st July-30th July, a nationwide lockdown came into effect. Travel between the governorates was not allowed. The emergency services such as ambulances, service vehicles that provide electricity and water maintenance, and vehicles belonging to the police and Armed Forces, were exempted from this ban. People remained indoors between 7 P.M. and 6 A.M. It was not permitted at all to go outside after 7 P.M., either on foot or in a vehicle. Such incidents were imposed with proper fines. Royal Oman Police set several checkpoints on the roads that connect diverse areas of the Sultanate, to ensure domestic travel restrictions. Working was no longer a valid reason for going out of the house; progressively non-indispensable activities were also stopped. As a result, at the end of July, R 0 value became 0.9761. It shows that the restriction of the contact with each other in various form significantly brings down the value of R 0 , thereby decreasing the spread of the disease.
Based on the parameters obtained during period 5 (21st July-30th July 2020) and with the assumption that the policy will remain the same, we predicted the infected size, recovery size and death cases up to 150 days. It is expected to have the number of cumulative infected cases as 236,436, the number of recovery cases as 207,839 and no of deaths as 9234 at the end of 150 days starting from June 04, 2020, provided no change in the policy decisions as on during period 5.
The sensitivity of the model is also analyzed on infected, recovered and death cases concerning transmission parameters as given in "Results" section. The sensitivity analysis on infected case shows that the transmission parameters alpha and gamma are very much sensitive to the spread of the disease. The reason for the high sensitivity of gamma is due to the fact that Moderate category people (M) have a higher probability rate of transmitting the disease as they are in the infectious stage and they are only isolated to their homes and, not admitted in the hospital. The people in the exposed (E) category have a higher rate of a chance of contact with other people as www.nature.com/scientificreports/ at this stage disease is not diagnosed, and it is infectious to a certain extent which results in higher sensitivity of alpha parameter. The lockdown and social distancing measures such as wearing a mask, avoiding gathering, closing schools and mosques etc. will help significantly reducing the value of these parameters. The sensitivity analysis on recovered case shows that as the value of transmission parameters increases the number of recovered cases also increases. This is because the infected cases increase with the rise in parameters, and accordingly the number of recovered cases also rise due to the dedicated service of our health centers. In sensitivity analysis of death cases, as parameter value increases, the number of death case may also increase. But due to our medical service, the rise in the value may not be alarming. So, in addition to fineness in our medical field, it is a vital role to frame an appropriate policy to reduce the value of the parameter to control the disease like COVID-19 due to the absence of a specific vaccine for the disease.
It is observed that the transmission parameters are very much influenced by various remedial measures such as lockdown, social distancing and quarantine. This would reduce/increase the basic reproduction number which is the probability of disease transmission in a single contact multiplied by the average number of contacts per person. Since the transmission rate α is due to the contacts between susceptible and exposed (Et), its rate can be reduced through contact tracing and isolate them from the contact of others. The mass random test will also help to identify this category of exposed population (Et) who transmits the disease without showing any symptoms.

Conclusion and future
In this work, we have developed a mathematical model to analyze the nature of the pandemic COVID-19 meaningfully on the data of the Sultanate of Oman. The model is an extension of SEIR where we expanded the infected compartments into mild (A), moderate (M), severe (H) and critical (C), based on the clinical stages of infection. The exposed category is also extended to E N and E T where E T is an infectious stage. The parameters are estimated by fitting the data of Oman on the differential equations, and R 0 is computed. The proof for the computation of R 0 , steady-state analysis and final pandemic size equations are also shown to validate the model. The parameter value is justified with the effective actions taken by the government to lessen the spread of the disease. The sensitivity of the transmitting parameters is also identified which is a direct indicator of the impact of the strict or weak lockdown measures on the spread of the disease. It is observed that the transmitting parameters alpha and gamma are sensitive to the number of infected cases, and their value can be reduced by proper contact tracing mechanism and effective social distancing measures. In conclusion in Oman, there is no need for any complete lockdown as per the present situation. This research can be extended by analyzing the data of Oman by taking Wilayat wise or governorate wise data in Oman. This can also be done for any country by considering the www.nature.com/scientificreports/ different factors such as population, culture, age distribution, regulations etc. Also, the model can be extended by incorporating the impact of suitable vaccination once it is discovered.