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www.nature.com/scientificreports OPEN Tracking the early depleting transmission dynamics of COVID‑19 with a time‑varying SIR model Kian Boon Law1*, Kalaiarasu M. Peariasamy1, Balvinder Singh Gill2, Sarbhan Singh2, Bala Murali Sundram2, Kamesh Rajendran2, Sarat Chandra Dass3, Yi Lin Lee1, Pik Pin Goh1, Hishamshah Ibrahim4 & Noor Hisham Abdullah4 The susceptible‑infectious‑removed (SIR) model ofers the simplest framework to study transmission dynamics of COVID‑19, however, it does not factor in its early depleting trend observed during a lockdown. We modifed the SIR model to specifcally simulate the early depleting transmission dynamics of COVID‑19 to better predict its temporal trend in Malaysia. The classical SIR model was ftted to observed total (I total), active (I) and removed (R) cases of COVID‑19 before lockdown to estimate the basic reproduction number. Next, the model was modifed with a partial time‑varying force of infection, given by a proportionally depleting transmission coefcient, βt and a fractional term, z. The modifed SIR model was then ftted to observed data over 6 weeks during the lockdown. Model ftting and projection were validated using the mean absolute percent error (MAPE). The transmission dynamics of COVID‑19 was interrupted immediately by the lockdown. The modifed SIR model projected the depleting temporal trends with lowest MAPE for I total, followed by I, I daily and R. During lockdown, the dynamics of COVID‑19 depleted at a rate of 4.7% each day with a decreased capacity of 40%. For 7‑day and 14‑day projections, the modifed SIR model accurately predicted I total, I and R. The depleting transmission dynamics for COVID‑19 during lockdown can be accurately captured by time‑varying SIR model. Projection generated based on observed data is useful for future planning and control of COVID‑19. Compartmental mathematical models are critical for understanding the transmission dynamics of the Severe Acute Respiratory Syndrome Coronavirus 2 (SARS-CoV-2). Tese models are used to evaluate the impact of lockdown measures and various public health interventions during the COVID-19 pandemic1–7. Te susceptible- infectious-removed (SIR) model is the simplest compartmental model used to describe the epidemic pattern of an infectious disease. It functions on the principle that individuals can be classifed by their epidemiological status, based on their ability to host and transmit a pathogen. Most compartmental models assume that the number of cases increases exponentially until the epidemic can no longer be sustained due to the reduced number of susceptible individuals. Tis process continues until the number of infections drop, eventually leading to the extinction of an epidemic8. In COVID-19 pandemic, the inadequacy of efective pharmaceutical remedies forced many countries to impose various public health interventions to fatten the epidemic curve. Tese measures included public lock- down, physical distancing, prohibition of gathering and schools closure to reduce the contact rate between individuals9. Other interventions such as contact tracing and quarantine were implemented to prevent the occur- rence of transmission by isolating infected individuals before they could develop infectiousness10. However, the utility of any one intervention alone is likely to be limited, requiring multiple interventions to be combined to have a substantial impact on the dynamics of transmission11. Many countries also authorized legislative lockdown or movement control to optimize public response and compliance to those interventions. In China, the epidemic growth of COVID-19 was successfully fattened within three months by strictly enforced movement restrictions and lockdown. Te early extinction of COVID-19 was achieved with a high 1Institute for Clinical Research, Ministry of Health Malaysia, Setia Alam, Malaysia. 2Institute for Medical Research, Ministry of Health Malaysia, Kuala Lumpur, Malaysia. 3Heriot-Watt University Malaysia, Putrajaya, Malaysia. 4Ministry of Health Malaysia, Putrajaya, Malaysia. *email: [email protected] Scientifc Reports | (2020) 10:21721 | https://doi.org/10.1038/s41598-020-78739-8 1 Vol.:(0123456789) www.nature.com/scientificreports/ Figure 1. Te compartmental structure of the classical SIR model and modifed SIR model. degree of compliance to the public health interventions 12. In like manner, Malaysia frst implemented a 3-week nationwide lockdown or Movement Control Order (MCO) beginning 18 March 2020. Tereafer, in response to the continuous growth of COVID-19 in the country, the MCO was extended twice until 12 May 2020. Malaysia further enforced the MCO for the fourth time until 9 June 2020; a total MCO duration of 12 weeks. Te aim of this study was to develop and validate a modifed SIR compartmental model that factored in the early depleting transmission dynamics of COVID-19 and compare model predictions to observed COVID-19 cases during the lockdown period in Malaysia. Methods Model structure. In countries afected by the COVID-19 pandemic, the SIR model provided the simplest framework that matched the reporting structure with the least underlying assumptions. Figure 1A shows the compartmental structure of a classical SIR model, with three state variables: S for susceptible, I for infectious and R for removed, and two transition rates: (1) Force of infection, βI/N that controls the transition of individuals from S to I and (2) Removed rate, δ that controls the transition of individuals from I to R, respectively. Te δ is the reciprocal of the infectious period and N is the sum of all three state variables. Te force of infection is the rate at which individuals acquire an infection, which relies on the transmission coefcient, β and the fraction of infectious individuals, I/N . Te model assumes that the entire population remains equally susceptible during infection. As infected individuals were being isolated immediately once detected, we attributed the transition of individuals from compartment S to I to the transmission period and the transition of individuals from compart- ment I to R to isolation period or admission period. Te reproduction number, Rˆ is defned as the average number of secondary cases generated by an index case in a large entirely susceptible population. Te basic reproduction number, Rˆ 0 is estimated at the beginning of an outbreak when there is no population immunity or any deliberate intervention in disease transmission 13. Te Rˆ 0 is a valuable concept to determine if an emerging infectious disease can spread in a population. When Rˆ 0 > 1, an outbreak is expected to continue in the population. In this study, using the classical SIR model, the Rˆ 0 was calculated by β × transmission period whilst the efective reproduction number, Rˆ t was estimated at the current state of the population. Studies show that the Rˆ t of COVID-19 gradually decreases over time, as a result of enforced lockdown and public health interventions4,13,14. To overcome the limitation of the classical SIR model in capturing the early reducing trend of COVID-19, a partial time-varying force of infection was incorporated into the SIR model. Figure 1B presents the modifed SIR model with a partial time-varying force of infection, given by zβt I/N , where zβt is the partial transmission coefcient at time t. Te fractional term, z allows the transmission dynamics to decrease with the number of infected individuals who can spread the coronavirus. A power decay log function representing gradually depleting βt over time t is given by, t βt = βt=0(1 − p) , (1) where p is the proportion of depletion between 0 and 1. By incorporating the derivative of function (1) into ordinary diferential equations (ODEs) as shown in Fig. 1B, the early depleting transmission dynamics of COVID-19 can be simulated. Te modifed SIR model enables the outcome of lockdowns and public health interventions to be explicitly quantifed by p and z. For instance, a larger value of p signifes a more efective intervention in reducing contact rates and βt over time, whilst the value of z signifes the efectiveness of an intervention in preventing infected individuals from spreading the coronavirus. Figure 2 illustrates the observed βt at p = 0.2 (20%), which is deplet- ing faster than p = 0.1 (10%). Te Rˆ t can be estimated by zβt × transmission period during the lockdown period. Data sources. Te frst wave of COVID-19 in Malaysia occurred between 25 January and 26 February 2020, involving only 22 individuals with 20 of them being oversea travelers. Te second wave of COVID-19 emerged exponentially following a large religious gathering held in Sri Petaling, Kuala Lumpur between 27 February and 1 March 2020. Te massive gathering involved more than 16,000 attendees, including many foreign nation- als from countries with COVID-19 outbreak15. As of 26 April 2020, 2130 (37%) among 5780 confrmed cases Scientifc Reports | (2020) 10:21721 | https://doi.org/10.1038/s41598-020-78739-8 2 Vol:.(1234567890) www.nature.com/scientificreports/ Figure 2. Graph of βt vs t, for two diferent values of p, illustrating the rate of depletion. Te βt=0 is set at 0.5 (arbitrary). Figure 3. Te phases of infection period for COVID-19. were related to the Sri Petaling cluster16,17. Te Ministry of Health (MOH) Malaysia reports on a daily basis the number of cumulative total cases, cumulative active cases, newly confrmed cases, recovered and death cases for COVID-1918. For this modeling, we denoted daily confrmed cases as I daily, cumulative total cases as I total, cumulative active cases as I and cumulative removed cases as R. Removed cases comprised of both recovered and death cases. Te frst day of Sri Petaling gathering (27 February 2020) was denoted as the start date of the second wave outbreak. The reporting structure of COVID‑19 in Malaysia. All cases of COVID-19 were confrmed by real- time reverse transcriptase-polymerase chain reaction (RT-PCR) assays. Once confrmed, an infected individual was isolated immediately for treatment until recovery or death.

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