key: cord-0801368-1tq4z3uw authors: Alshammari, F. S. title: A mathematical model to investigate the transmission of COVID-19 in the Kingdom of Saudi Arabia date: 2020-05-06 journal: nan DOI: 10.1101/2020.05.02.20088617 sha: ccfe1455524bb1da264c8d868ecb5247ba869508 doc_id: 801368 cord_uid: 1tq4z3uw Since the first confirmed case of SARS-CoV-2 coronavirus (COVID-19) in the 2nd day of March, Saudi Arabia has not report a quite rapid COVD-19 spread compared to America and many European countries. Possible causes include the spread of asymptomatic cases. To characterize the transmission of COVID-19 in Saudi Arabia, this paper applies a susceptible, exposed, symptomatic, asymptomatic, hospitalized, and recovered dynamical model, along with the official COVID-19 reported data by the Ministry of Health in Saudi Arabia. The basic reproduction number R0 is estimated to range from 2.87 to 4.9. As of April 22, 2020, more than 12772 cases and 114 deaths of coronavirus disease 2019 caused by the SARS-CoV-2 virus had been confirmed in Saudi Arabia. Since the 4 th of March [17] , control measures have been implemented within Saudi Arabia to try to control the spread of the disease. Isolation of confirmed cases and contact tracing are crucial part of these measures, which are common interventions for controlling infectious disease outbreaks [26] [27] [28] . For example, the severe acute respiratory syndrome (SARS) outbreak SARS and Middle East respiratory syndrome (MERS), were controlled through tracing suspected cases and isolating confirmed cases because the majority of transmission occurred concurrent or after symptom onset [27] [28] [29] . However, it is unknown if transmission of COVID-19 can occur before symptom onset, which could decrease the effectiveness of isolation and contact tracing [26, 27, 29] . 1 In this paper, the impact of asymptomatic COVID-19 cases on the spread of the disease will be considered using a modified version of the susceptibleexposed-infected-recovered (SEIR) dynamical model, along with the official COVID-19 data reported by the ministry of health in Saudi Arabia. Other main objectives of this paper include: estimating the basic reproduction number (R 0 ) of COVID-19 in Saudi Arabia and how interacting with infected individual (symptomatic and asymptomatic) affect the estimated number, estimating the maximum required number of hospital beds and intensive care units (ICU). The population will be divided into six categories: susceptible (S), exposed (E), symptomatic (Y ), asymptomatic (N ), hospitalized (H), and recovered (R) individuals (SEYNHR). Individuals moves from the susceptible compartment S to the exposed compartment E after interacting with infected individuals with transmission rates β 1 , β 2 , and β 3 as shown in Figure 1 . COVID-19 is Figure 1 : Schematic diagram of SEIHR compartment model. The arrows, except the black ones, represent progression from one compartment to the next. known to have an incubation period, from 2 to 14 days, between exposure and development of symptoms [6, 30] . After this period, exposed individual transits from the compartment E to either compartment Y at a rate α, or to compartment N at a rate α(1−γ). An individual could move from compartment N to Y at a rate K if they show symptoms. Once an individual becomes infected with the coronavirus that causes COVID-19, that individual develops immunity against the virus with a rate Φ Y or the individual will be hospitalized with a rate of or die because of the disease with a rate of µ 1 . When individual becomes hospitalized, that individual receives treatment and develops immunity against the virus with a rate r or die because of the disease with a rate of µ 2 . As shown in Figure 1 , the SEYNHR model has six compartments, and there-2 . CC-BY-ND 4.0 International license It is made available under a is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity. (which was not certified by peer review) The copyright holder for this preprint this version posted May 6, 2020. . https://doi.org/10.1101/2020.05.02.20088617 doi: medRxiv preprint fore a discrete dynamical system consisting six non-linear differential equations will be formed as the following: where . The next-generation matrix will be used to derive an analytical expression for the basic reproduction number (R 0 ), for the compartmental model above. Calculating R 0 is a useful metric for assessing the transmission potential of an emerging COVID19 in Saudi Arabia. 3 Basic reproduction number R 0 An important concept in epidemiology is the basic reproduction number, defined as "the expected number of secondary cases produced, in a completely susceptible population, by a typical infective individual" [10] . The next generation method will be used to calculate R 0 [11] . The system in Equation 1 can be rewritten as follows CC-BY-ND 4.0 International license It is made available under a is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity. The copyright holder for this preprint this version posted May 6, 2020. . https://doi.org/10.1101/2020.05.02.20088617 doi: medRxiv preprint The Jacobian matrices of F and V evaluated at the disease-free equilibrium (DFE) of the system in Equation 1, M = ( A µ , 0, 0, 0, 0, 0), are given by Direct calculations show that Denoting the 4x4 identity matrix by I, the characteristic polynomial Γ(λ) of the matrix F V −1 is given by where . CC-BY-ND 4.0 International license It is made available under a is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity. The copyright holder for this preprint this version posted May 6, 2020. The solutions λ 1,2,3,4 are given by Therefore, the reproduction number for the SEY N HR model in Equation 1 is given by At Table 1 . I assumed that the mean asymptomatic infectious period is the same as the mean symptomatic infectious period because there is no estimation available in the literature [9, 29] . Based on those estimated, assumed and measured values, the basic reproduction number R 0 is estimated to range from 2.87 to 4.9. The variation of the basic reproduction number R 0 for different values of β 1 , β 2 , Φ Y , Φ N and K are shown in the heat maps in Figure 3 . The upper heat map of Figure 3 shows that practicing physical distancing could significantly reduce the value of R 0 and hence control the spread of the disease. The center panel of Figure 2 shows that about 18% of the entire Saudi population will be asymptomatic in the last week of May 2020 and about 17% will be exposed in the third week of May. The percentage of the entire population being symptomatic at anytime will not exceed 1%, which is estimated to occur in the third week of May. Moreover, about 60000 hospital beds and 18000 ICU beds are required (30% of the hospitalized cases [6] ) immediately after the second week of May. Currently, the Ministry of Health designated 25 hospitals for COVID-19 infected patients with up to 80,000 beds and 8000 intensive care units (ICU) beds [25] and therefore extra 10000 ICU beds could be required. . CC-BY-ND 4.0 International license It is made available under a is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity. The copyright holder for this preprint this version posted May 6, 2020. . The parameter with the hight degree of uncertainty are the effective contact rates from symptomatic to susceptible β 1 , from asymptomatic to susceptible β 2 and from hospitalized to susceptible β 3 (expected to be a fraction of β 1 because of the protective measures in hospitals), as well as the mean infectious periods for symptomatic Φ −1 Y and asymptomatic Φ −1 N individuals. I estimated a maximum value of β 1 to be 0.5 which is one half of the value reported by Li et al. [18] . This could be a reasonable estimation as we have not seen similar scenario in Saudi Arabia after 5 weeks since reaching 100 confirmed cases on the 14th of March (week 7 since the first case) as we have seen in many other countries like China, America and different European countries in the same timescale. This could be a result of the precautionary measures taken by the Saudi authorities, including closure of schools and universities that started as early as the 8 th of March (six days after the first confirmed COVID-19 case in Saudi Arabia). Based on the above estimation for R 0 , the center panel of Figure 3 suggests that the infectious period for symptomatic patient could be in the range from 6.6 to 12.5 days (i.e., Φ Y ∈ [0.08, 0.15]) and the infectious period for asymptomatic patient could be in the range from 6.6 to 25 days (i.e., Φ N ∈ [0.04, 0.15]). The infectious period for symptomatic cases are consistent with what is being observed clinically [30] . In reality, R 0 is not a biological constant; it could fluctuate daily depending on environmental and social factors such as percentage of entire susceptible 6 . CC-BY-ND 4.0 International license It is made available under a is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity. The copyright holder for this preprint this version posted May 6, 2020. . population wearing suitable medical mask and practicing physical distancing. In the literature, estimates of R 0 vary greatly: from 1 to 6 [18] [19] [20] [21] [22] [23] [24] up to 26.5 [9] . This variation is because of different assumptions and factors they had considered in the calculations. In general, considering asymptomatic infection sub-population will increase the estimated values of R 0 . The contribution of undocumented COVID-19 infections (asymptomatic cases) on the transmission of the disease deserves further studies and investigations. This paper shows that asymptomatic cases of COVID-19 will drive the growth of the pandemic in Saudi Arabia. Therefore, more testing is needed to identify COVID-19 patients (symptomatic and asymptomatic) and to contain the spread of the disease. . CC-BY-ND 4.0 International license It is made available under a is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity. The copyright holder for this preprint this version posted May 6, 2020. . . CC-BY-ND 4.0 International license It is made available under a is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity. (which was not certified by peer review) The copyright holder for this preprint this version posted May 6, 2020. . CC-BY-ND 4.0 International license It is made available under a is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity. 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