key: cord-1049167-sxz39swi
authors: Rabbani, H.; Osei-Bonsu, K.; Osei-Bonsu, P. K.; Seers, T. D.
title: Modelling COVID-19 using the Fundamentals of Fluid Dynamics
date: 2020-06-25
journal: nan
DOI: 10.1101/2020.06.24.20139071
sha: e3df2e8ce117c705c491f00fb7ede5d87de5ea03
doc_id: 1049167
cord_uid: sxz39swi

As of 21st May 2020, there have been 4.89M confirmed cases worldwide and over 323,000 deaths of people who have tested positive for SARS-CoV-2. The outbreak of COVID-19, has not only caused widespread morbidity and mortality, but has also led to a catastrophic breakdown in the global economy and unprecedented social disruption. To lessen the global health consequences of COVID-19, sweeping COVID-19 lockdown and quarantine measures have been imposed within many nations. These measures have significantly impacted the world's economy and in many cases has led to the loss of livelihood. Mathematical modeling of pandemics is of critical importance to understand the unfolding of transmission events and to formulate control measures. In this research letter, we have introduced a novel approach to forecasting epidemics like COVID-19. The proposed mathematical model stems from the fundamental principles of fluid dynamics, and can be utilized to make projections of the number of infected people. This unique mathematical model can be beneficial for predicting and designing potential strategies to mitigate the spread and impact of pandemics.

In December 2019, a cluster of cases of pneumonia, subsequently associated with a novel coronavirus (Severe Acute Respiratory Syndrome -Coronavirus-2, SARS-CoV-2), named Coronavirus Disease 2019 (COVID-19) by the World Health Organization (WHO), emerged in Wuhan, China. It was rapidly declared a pandemic on March 11, 2020 , in view of its exponential spread worldwide 1 Clinically, infected subjects exhibit a wide range of non-specific features, from mild-to-moderate symptoms such as cough, fever and fatigue to severe, life-threatening respiratory and systemic complications. On the other hand, it has been well documented that infected persons may exhibit no symptoms at all (asymptomatic) or may be yet to manifest symptoms (pre-symptomatic), but are potentially infectious 3, 4 . In such cases, infected individuals may be likely to maintain normal social interactions, without realizing the need for self-isolation due to the obscurity of their symptoms.

Currently, our understanding of the transmission risk is incomplete. Epidemiologic examination in Wuhan at the beginning of the outbreak identified an initial connection with a live animal seafood market, where patients had worked or visited 5 . As time progressed, person-to-person spread became the main mode of transmission 6 . Although SARS-COV-2 has been detected in nonrespiratory samples such as stool and blood, transmission is primarily thought to occur through close contact, via respiratory droplets and aerosols 7 . The virus released in these secretions when . 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 June 25, 2020. . https://doi.org/10.1101/2020.06.24.20139071 doi: medRxiv preprint an infected subject coughs, sneezes, or talks can then infect another person if it makes direct contact with the mucous membranes 8,9 . Under ideal circumstances, an effective vaccine might be administered to mitigate the dire effects of the virus. Regrettably, the development of an acceptable vaccine to this end appears unlikely in the short-term. Consequently, government and public health responses have focused mainly upon non-pharmacological interventions 10 . These measures include physical/social distancing to minimize the rate of person-to-person contact, frequent hand washing, the utilization of masks, gloves and other forms of personal protection equipment (PPE), mass testing, contact tracing and isolation/quarantine of persons with suspected and confirmed cases of COVID-19 infection.

Although these interventions have contributed significantly to the gradual decline of the transmission rate and by extension deaths worldwide, there are increasing concerns that the easing of these measures may result in the surge of new cases 10 .

In pandemic situations where data could be sparse, mathematical modelling can be a powerful tool to understand and predict the course of the outbreak in order to inform the development of potential control strategies 11, 12 . The most frequently used framework in the case of human transmissions is the so-called SIR model 13 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 June 25, 2020. . https://doi.org/10.1101/2020.06.24.20139071 doi: medRxiv preprint al. 17 utilized available data to model the case fatality risks of symptomatic persons in Wuhan, China. Giordano et al. 18 further extended the model to distinguish between detected and undetected cases and the level of severity of the manifested symptoms during the course of the outbreak.

In this article, we introduce a novel approach to forecasting disease outbreaks, specifically COVID-19, using the fundamental principles of fluid dynamics. In this model, we consider a carrier of the virus as a fluid containing a dissolved ionic species (Figure 1 ). With this intuition, we attempt to derive a simplified theoretical model using the well-known Fluid Transport Equation showing an excellent match between the data set and model predictions. We further discuss the mitigating strategies for controlling the COVID-19 pandemic optimally, based on our model parameters. . 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 June 25, 2020. . https://doi.org/10.1101/2020.06.24.20139071 doi: medRxiv preprint

Using the fluid transport equation, we derived the following mathematical model to predict the infected cases of COVID-19. The details concerning the derivation of mathematical model are provided in the methods section.

The number of infected cases is given by Equation 1 where ∅ is the ratio of total infected cases I, Table 1 . . 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 June 25, 2020. . https://doi.org/10.1101/2020.06.24.20139071 doi: medRxiv preprint 

We have demonstrated from Figure 2 It follows from Equation 2 that for a given country, a high value of indicates a more severe epidemic. is the ratio of total number of infected cases at the initial phase of epidemic over the total number of initial susceptible people . It can be seen from Equation 2 that as interaction factor → 0, the epidemic growth factor reduces, indicating that to minimize the spread of the disease, must be reduced by social distancing and the isolation of infected persons. Figure 3 shows the influence of on the trend of infected cases. It can be observed from Figure 3 that as the reduces the number infected cases reduces and the plateau is slightly delayed. This agrees with the strict social distancing and lockdown measures implemented by several countries in order to control the spread of the virus.

. 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 June 25, 2020. . https://doi.org/10.1101/2020.06.24.20139071 doi: medRxiv preprint Figure 3 : Describes the effect of on the trend of COVID-19 infected cases. The base case shows the France data.

One can see that with the subsequent reduction in which corresponds to social isolation of infected individual the growth of epidemic can be suppressed.

Interestingly, Equation 2 predicts that for a given , an increase in (parameter linked to mitigation strategy) will lead to a decrease in the growth factor . In contrast to , a relative increase in is favorable to optimally control the spread of an epidemic. Practically, this suggests widespread testing, contact tracing, and isolation of infected persons coupled with high quality medical resources will result in the deceleration of the spread of the epidemic. It is important to note that both increase in and will result in an increase in the total number of infected cases. Therefore, as represents the transmission of virus due to human-human contact, an increase will reflect an increase in the total number of both symptomatic and asymptomatic cases that may be left undetected. Conversely, an increase in will also result in an increase in the number of reported infected individuals, due to a robust public health mitigation strategy that includes widespread . 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 June 25, 2020. . https://doi.org/10.1101/2020.06.24.20139071 doi: medRxiv preprint testing and contact tracing. We have also shown the relationship between and for various countries (Figure 4) , affirming that high resulted in the isolation of the infected individuals and buffered transmission rates. 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 June 25, 2020. . distancing, widespread testing and contact tracing is the optimum strategy for fighting the pandemic.

Ignoring the advective flux the traditional fluid transport equation is written as 21 ;

relates to the rate of change in the concentration of species in fluid, is the diffusive flux of species, 2 is the Laplace indicating spatial coordinates and is the net growth of species.

Following our intuition that a fluid is a carrier of species such as salts, an infected individual could also be conceptualized as the carrier of the virus. Utilizing this approach we substitute with ∅ which represents the ratio of total infected cases I, over the initial number of susceptible people 

. 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 June 25, 2020. . We further simplify Equation 5 by equating ∅ +1 = 0, ∅ −1 = 0 and ∅ = 0. In order to nondimensionalize Equation 5 we substitute 2 ∆ [ 1 ∆ 2 + 1 ∆ 2 + 1 ∆ 2 ] and ∆ with and respectively. This results in the derivation of final equational form, which relates the increase in the infected cases ∅ − ∅ with the transfer of virus due to human-human interaction, ∅ , the growth or the activity of virus within infected individual, −∅ 0 , and mitigation strategies, ∅ .

For the derivation of the growth factor we sum the effects together reported in Equation 6 that are responsible for controlling the pandemic resulting in = ∅ + −∅ + ∅

As during the initial phase of pandemic ∅ ≈ 0, consequently, we can substitute ∅ as and equate −∅ = 1. This results in Equation 8.

The subsequent derivation of with respect to results in = + −1 (9) where is the growth factor as given in Equation 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 June 25, 2020. . https://doi.org/10.1101/2020.06.24.20139071 doi: medRxiv preprint

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The financial assistance of Qatar Foundation, the Qatar National Research Fund (NPRP10-0104-170104) and Texas Engineering Experimental Station are gratefully acknowledged by the authors.Dr. Harris Sajjad Rabbani would like to acknowledge Dr. Nida Jaleel for the insightful discuss on the topic.

The data presented in this manuscript will be available freely via sending a request to the corresponding author.

The MATLAB code employed for the fitting the data and predicting the course of COVID-19. 1); I = f(2); t = 200; phi = zeros(t,1); phi(1) = I/s; cases = zeros(t,1); cases(1) = phi(1)*s; for i = 2:t phi(i)= phi(i-1) + (C(2).*(phi(i-1))) + C(3).*exp(-phi(i-1)) -((phi(i-1)).^(C(1))); cases(i) = phi(i)*s; end % Optimization function used for fitting the model to data function [error] = optimization(C) d = readtable('Data.xlsx','basic',true); f = d.France; s = f(1); I = f(2); t = length(f(2:end)) -sum(isnan(f(2:end))); phi = zeros(t,1); phi(1) = I/s; cases = zeros(t,1); cases(1) = phi(1)*s; for i = 2:t phi(i)= phi(i-1) + (C(2).*(phi(i-1))) + C(3).*exp(-phi(i-1)) -((phi(i-1)).^(C(1))); cases(i) = phi(i)*s; end a = f(2:t+1); error = (cases-a).^2;