Title: Estimating the reproduction number and forecasting the impact of COVID-19 in Kuwait using a modified compartmental epidemiological model

A modified compartmental epidemic model was developed to simulate the state of Kuwait protocol in fighting COVID-19 pandemic. The next generation matrix method was used to drive an expression for the basic reproduction number, R 0 . Basic and effective reproduction numbers were calculated using data from the intrinsic growth rate of the confirmed COVID-19 cases. R 0 was found to be 2.18. Three scenarios that varied by effective reproduction number were used to estimate the future course of the disease: a high value of R = 1.98, a middle value of R = 1.62, and a low value of R = 1.2. The maximum number of beds required in general hospitals in each scenario were estimated at 141 184, 85 341, and 16 412, respectively. For intensive care units, the estimated numbers of beds required were 16 461, 9 645, and 1788. Maximum deaths also varied and were estimated to be 29 202, 23 973, and 11 565. For the maximum value of R, it is estimated to peak on August 27, 2020. For the middle value of R, it is estimated to peak on September 20, 2020. For the minimum value of R , it is estimated to peak on December 21, 2020.


Introduction
The compartmental model in epidemiology helps health authorities make intelligent decisions and has been proven to be effective in predicting the spread of pathogens since the time it was developed by W. O. Kermack and A. G. McKendrick [1][2][3]. Since then, other models have been developed to address specific needs for comprehending the spread of emerging or established pathogens [4][5][6]. The recent emergence of the SARS-CoV-2 virus and COVID-19 disease has paralyzed the world. Research has focused on the virus since it surfaced in Wuhan, China, in December 2019 [7]. The characterization of virus characteristics has been rapid as a result of the intense focus. Understanding the dynamics of disease spread is critical to help the policymakers prepare for what is to come and plan accordingly. In each country, health authorities establish procedures to delay the emergence of disease and to curb its spread. In this study, a compartmental epidemiological model was developed to estimate the spread of SARS-CoV-2 and COVID-19 in Kuwait. Kuwait acted swiftly to control the spread of the disease; all arrivals from countries with reported COVID-19 cases were institutionally quarantined beginning February 24, 2020; schools and universities were closed on February 27; non-essential government employees were ordered to stay home on March 12; passengers flights were suspended, prayer services in Mosques were banned, and public parks were closed on March 13; stricter measures, such as closing non-essential retail shops, were instituted on March 14. Finally, on March 22, 2020, a partial lockdown was imposed. Despite these precautions, Kuwait reported its first untraced domestic verified COVID-19 case on March 18. Since then, the daily reported cases have followed an increasing trajectory [8]. When the government imposed a total lockdown on May 11, the daily number of confirmed COVID-19 cases reached 920 cases with a cumulative total of 9701 cases. Although the total lockdown was partially lifted on June 1, the disease continues to spread, with daily new cases in the hundreds ( Figure 1).

Method
The health authority in Kuwait has established a protocol for dealing with newly discovered cases of COVID-19, which was outlined during a press conference [8]. When a case is discovered, the patient is sent into institutionalized quarantine and will remain there until recovered. If the case requires medical attention, then the patient will be transferred to a hospital, and if required, transferred to an intensive care unit. Some infected individuals are asymptomatic and would remain undetected. We assume that no deaths will arise from these cases. A compartmental epidemiological model was developed to simulate this protocol ( Figure 2). The total population (N) was divided into susceptible (S), exposed (E), hidden (H), infected (I), first-level quarantine (QA), second-level quarantine (QB ), first-level hospitalization (HOA), second-level hospitalization (HOB), intensive care (IC), recovery (HOC), death (D), and recovered (R) compartments (Table 1). The S population includes individuals who can get infected but have not yet contracted the virus. The E compartment includes individuals who are infected but not yet infectious. It was established that the E individuals would be infectious two days before the onset of the symptoms [9]. Therefore, an adjustment term was applied to individuals in the E state to account for the latent period of infection with a mean incubation period of 1/ days [5]. The H compartment includes infected individuals who are not showing symptoms; therefore, these individuals are undetected but will contribute to the spread of the virus during their infectious period. This compartment accounts for = 17.9%-30.8% of the infected pool [10,11]. Individuals in the H compartment have a mean infectious period of 1/ 1 , after which it is assumed that they recover [9,12]. The I compartment includes infected individuals who are showing symptoms. It is expected that individuals in this state will be detected after a mean period of 1/ 2 days from the onset of the symptoms. This mean rate depends on how fast the health authorities identify and isolate the infected individuals. The mean period was defined after the model was compared to the daily counts of new cases reported in Kuwait. After identifying infected individuals, health authorities institutionally quarantine the patients; therefore, they cease being infectious to the general population. The three compartments E, H, and I are infectious and could transmit the virus from being in contact with the S compartment at a daily contact rate of . It was estimated by the WHO [13] that 80% of infected individuals with symptoms do not need medical attention. To model this, individuals in quarantine were classified into two levels.
Initially, a patient is classified as QA, immediately after being detected and removed from the I compartment. While in the QA compartment, the condition of a patient is monitored for a mean period of 1/ 1 . A QA patient will be either classified as QB if mild symptoms appear or HOA if medical attention is required. The fraction of cases that become classified as QB is 1 . Patients will stay in the QB compartment for a mean period of 1/ 2 until they cease to be infectious. When a patient ceases to be a threat to the susceptible population, that individual will be moved into the R compartment. The health condition of patients in the HOA compartment will either deteriorate or become stable after a mean time of 1/ 3 . A fraction 2 of the patients will move into the IC compartment because of their need for ventilators. The remainder will be moved to the HOB compartment. Patients in the HOB compartment will recover at a mean period of 1/ 4 . Patients in the IC compartment will remain for a mean period of 1/ 5 , at which time a fraction 3 of IC patients will die and the rest will be moved to the HOC compartment. Subsequently, these patients will be moved at a rate of 1/ 6 to the R compartment. Vital dynamics were ignored in our model. The value of the mean periods for each compartment and percentages are presented in Table 2.
The equations for the model explained above are listed below.
At the initial state the total population N was assumed to be constant and equal to 4.7 million [14], which is the total population of the country. It is also the sum of all the compartments.

The reproduction number R and the contact rate calculated from the infection data
The basic reproduction number, 0 , is critical for determining whether a disease will die out or spread. It can be calculated from the infection data and the initial rate at which the number of infected individuals increases. In the model Equations (1)(2)(3)(4)(5)(6)(7)(8)(9)(10)(11)(12)(13)(14), the next generation method,N.G.M, [15][16][17] (appindex) was used to calculate an expression for 0 as follows: R0 is important for identifying the equilibrium state of the disease [16]. A disease will die out when 0 < 1, or it will persist and become endemic when 0 = 1, or it will increase epidemically when 0 > 1. There are two ways to control such a spread, regardless of location. The first approach is to decrease the daily contact rate , and this can be achieved by social distancing measures and lockdowns. The second approach is to implement an aggressive locate and isolate campaign, where individuals in the infectious E, H, and I compartments are located and isolated, thereby minimizing their exposure to the susceptible S compartment. In other words, eliminate , 1/ 1 , and 1/ 2 , each of which reduces 0 .
We relied on the recorded data of the confirmed number of infected individuals, which was provided by the ministry of health [8], to calculate the effective reproduction number in Kuwait. After linearizing the model for the infectious compartments in Equations (2), (3), and (4), the eigenvalue equation for the disease free linear Jacobean matrix of the model (appendix) was found to be In the equation above 1 , 2 , and values are tabulated in Table 2. and the leading eigenvalue, , are unknowns to be determined. of Equation (16) is the intrinsic growth rate of the epidemic [18]. The growth rate was calculated using the following expression [19]: For 0 , 1 is the total number of cases on March 18 and 2 is the total number of confirmed cases on June 10. 2 − 1 is the number od days from March 18 th until June 10. For the effective reproduction number, R, 1 and 2 are the seven-day rolling averages of the reported cases. 2 − 1 was taking to be seven days. COVID-19 cases due to travel were excluded from the calculation because we assumed that they were located and isolated at the airport and did not contribute to the local spread of the disease. The daily contact rate was then calculated using Equations (17) and (16). Once was found, then 0 was calculated from equation (15). The seven-day rolling average of the daily effective reproduction number, which was used in the calculation until its last recorded date, is shown in Figure 3. It ranges from a maximum value of 5.79 on March 31, 2020, to a minimum value of 0.59 on June 10, 2020. The 0 from the beginning of the epidemic on March 18 until June 10 is 2.18. Three scenarios were used for predicting the epidemiology of cases: a high value equal to the average of all the effective reproduction numbers (R = 1.98), a middle value that was calculated 7 days after the partial lockdown on April 7 (R = 1.62), and a low value that was calculated 7 days after the total lockdown on May 11 (R =1. 20). The values were calculated until June 10, 2020, which was the last recorded data available to include in this study. The high value represents a return to normal life with some restrictions, the middle value represents a partial lockdown in which some businesses return to normal operation, and the low value represents a partial lockdown in which only essential businesses are open. The 7day period was considered to be a good estimate for the measures taken by the authorities to be reflected in the recorded data.

Results
The recorded data for the number of confirmed COVID-19 cases and deaths count and rate were compared with the model calculations until June 10 to determine the accuracy of the model. The recorded data were almost identical to the model, as shown in Figures 4 and 5.
The number of confirmed infected cases reported by the health authority followed the pattern of the model with a slight deviation that began on May 14 and reached its maximum on June 3. There was a small peak on May 19 for the model results, but the peak from the actual cases appeared two days later. The number of confirmed cases and the model results began to decrease after that. The number of confirmed cases peaked because of the total lockdown that was imposed on May 11. However, a perfect opportunity to control the epidemic was missed by not extending the lockdown for another three weeks. According to the model, the number of cases will start to rise again and reach a peak value on August 11 for the high value of R, on September 6 for the middle value of R, and on December 5 for the low value of R. The number of cases per day estimated for these three peaks are 60 000, 32 589, and 5652, respectively. The number of cases per day is estimated to decrease after these dates until they reach single-digit values on November 18, 2020, January 2, 2021, and July 18, 2021, respectively ( Figure 5). The hospitalization rate is predicted to peak on August 27, 2020, September 20, and  Table 3.

Discussion
Knowing the reproduction number is crucial to understanding how the epidemic will develop.
To halt or slow the spread of disease, R must be reduced. In Kuwait, the health authority is waging a campaign to stop the spread of the disease, but to save more lives, they needed to impose a total lockdown for at least six weeks combined with an engaged trace and locate strategy. When the number of the new cases per day decreases to double digits, and the locations of outbreaks are pinpointed, opening up in stages can begin with partial lockdowns in the areas that are still experiencing outbreaks. Such a strategy should reduce the number of new cases, but it will not eradicate the disease. Therefore, any return to normal activities will revive the spread of the virus. The course of the disease will likely go through waves. Whenever an easing of restrictions occurs, the number of cases will start to increase. Authorities would likely re-introduce restrictive measures to help lower the cases again. This cycle could be repeated until herd immunity is achieved. Achieving herd immunity when R = 2.18 means that 54.1% of the population of Kuwait needs to have been infected [20]. This percentage needs to be achieved in the absence of any infected individuals (i.e., I = 0). This slow process is estimated to take two years to reach, but in the absence of a vaccine or effective treatments, it would be the best approach to contain the virus and reduce the number of deaths. If no such measures are implemented, then the health authority and government of Kuwait must prepare for the worst-case scenario and increase the capacity of hospital beds to 142 000 and intensive care unit beds to 16 461.