



ORIGINAL ARTICLE 

Year : 2020  Volume
: 4
 Issue : 2  Page : 6064 

Validation of COVID19 spread model by early cases from Spain
Isack E Kibona, Jeremiah J Ruhere, Violet G Saria
Department of Mathematics and Statistics, Mbeya University of Science and Technology, Mbeya, Tanzania
Date of Submission  21Sep2020 
Date of Acceptance  24Sep2020 
Date of Web Publication  30Nov2020 
Correspondence Address: Prof. Isack E Kibona Department of Mathematics and Statistics, Mbeya University of Science and Technology, P. O. Box: 131, Mbeya 00255 Tanzania
Source of Support: None, Conflict of Interest: None
DOI: 10.4103/MTSP.MTSP_12_20
This article intends to illustrate the coronavirus disease 2019 (COVID19) model if strict restriction is not enforced. Early COVID19 cases from Spain have been considered an example. Thus, this article is for the estimation of specific parameter particularly to one of the most hit countries in April 2020. Our essence is the possibility to spot a natural model of COVID19. The cases between March 1 and 15, 2020 have been taken to validate and estimate the parameter of the model. Parameters were estimated by a nlinfit function from MATLAB developed by Levenberg–Marquardt, and thus, so is the reproduction number (R_{0}). R_{0}was found greater than the unit, which is catastrophic. Cases of COVID19 between March 1 and 15 have been chosen to validate the model because in this earlier stage of the pandemic, Spain restrictions against the spread were assumed not enough to impede the pace of natural spread to the pandemic. Had it not been the lockdown that followed after the mentioned dates, by April 15, 2020, Spain would have been in a more catastrophic situation by >3,400,000 COVID19 infection cases far worse from 180,695 cases that happened.
Keywords: Coronavirus disease 2019, infection cases, pandemic Spain restrictions
How to cite this article: Kibona IE, Ruhere JJ, Saria VG. Validation of COVID19 spread model by early cases from Spain. Matrix Sci Pharma 2020;4:604 
Introduction   
The word corona originates from a Latin word which means a crown. Most scientists and researchers concede that the virus was named so because the virus had phenotype figure that looks like a king's crown when observed under microscope.^{[1]} Max et al.^{[2]} explained that there are so many kinds of coronavirus that cause about 30% of common colds around the world. This latest coronavirus causing disease to human was initially named as the 2019novel coronavirus on January 12, 2020 by the World Health Organization (WHO). The WHO officially named the disease as coronavirus disease 2019 (COVID19) and the Coronavirus Study Group of the International Committee proposed to name the new coronavirus as severe acute respiratory syndrome coronavirus 2 (SARSCoV2), both issued on February 11, 2020.^{[3]}
COVID19 outbreak was once noted in Wuhan, Hubei Province, China, on December 31, 2019. TianMu Chen et al.^{[4]} suggested that this SARSCoV2 might have been originated from bats and transmitted to humans by eating seafood market products (hosts) of the virus. At the time of writing this article, coronavirus infecting people has now been manifested all over the continents within >212 territories. The World Health Systems spend most of the resources into learning about treating and preventing COVID19 infections in all countries by enforcing mass conference cancellations, travel restrictions, social distancing, and other unscientific prevention measures like prayers.^{[5]}
This article intends to formulate and justify a mathematical model that represents the natural spread of COVID19 using Spain data. Visconti, Eduardo Enrique Tal, justified that there was a natural spread of coronavirus infections among people in Spain from the first onset of the infections that was confirmed on January 31, 2020.^{[6]} In Spain, COVID19 was discovered when a German tourist was confirmed to be positive of COVID19 in La Gomera, Canary Islands. Until March 13, 2020, COVID19 cases were confirmed in almost 50 provinces of Spain, followed by a state of alarm and national lockdown that was imposed exactly on March 14, 2020. This study is about the verification of the COVID19 deterministic model by early infection cases of COVID19. In so doing parameters of the model particularly for Spain are going to be estimated. Presumably, Spain COVID19 cases that range from March 1 to 15, 2020 were spreading naturally.
Model Development   
A population of size N at time, t with constant inflow of susceptible at a rate of Λ is considered. The population is divided into five compartments. There is susceptible compartment (S), refers to individuals not infected, but is liable to infections. Exposed compartment (E), infectious class (I), and recovery compartment (R) are individuals who have recovered from COVID19, and we have included compartment of death (D) which is a collection of individuals dying due to induced death by COVID19. It is as well been assumed that individuals recovering from COVID19 are not reinfected.
β is an infection rate resulting from interaction of S with infected individuals, I or by contaminated environment. μ is the natural death rate. [Table 1] is a summary of parameters and their definitions. The model can be expressed in a flow diagram as in [Figure 1]. Therefore, the system of equations 1 is a proposed model of COVID19 in a no or ignorable lockdown against the spread of the epidemic population.
Basic Reproduction Number and Equilibrium of the Model   
In order to analytically understand the spread of pandemic, basic reproduction number (R_{0}) is the most vital quantity in epidemiology. R_{0} is basically the average number of new infectious cases caused by a single infected individual in a totally susceptible population. R_{0} from equation 1 has been estimated by the nextgeneration method. Thus
The diseasefree equilibrium point (DFE), Q^{0} is simply evaluated from the system of equations 1, by substituting E^{0} = I^{0} = R^{0} = 0 and S t = E t = I t = R t = 0, then solve for Q^{0}. Thus, the DFE point exists as follows:
The endemic equilibrium point, Q^{∗}(S = S^{∗}, E = E^{∗}, I = I^{∗}, R = R^{∗}) is obtained by solving for I^{∗}ƒ = 0 at the equilibrium point of the system 1. Thus Q^{∗}(S^{∗}, E^{∗}, I^{∗}, R^{∗}) has been found, where:
from which it can be proved that the endemic equilibrium exists if R_{0}> 1.
Simulation of the Model   
The COVID19 infection cases for Spain have carefully been observed and recorded from March 1 to 15, 2020 as listed in [Table 2].^{[7]} Between March 1 and 15, 2020, COVID19 cases in Spain demonstrate a typical COVID19 deterministic model.  Table 2: Number of coronavirus disease 19 cases in Spain from March 1 to 15, 2020
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In order to simulate a situation in Spain, the information in [Table 2] with a corresponding curve of COVID19 infection cases [Figure 2]a. Were employed with the aid of the Levenberg–Marquardt algorithm for parameter estimation. The parameters were estimated as listed in [Table 3].  Figure 2: (a and b) Plots of coronavirus disease 2019 cases both original and simulated cases
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 Table 3: Estimated parameters of the proposed coronavirus disease 19 model
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Simulations have depicted that for R_{0}> 1 leads to the existence of the epidemic for longer time than when R_{0}< 1, this is true if recovered individuals are not reinfected, [Figure 3]b.  Figure 3: (a and b) Simulations of coronavirus disease 2019 cases for R0 > 1
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Otherwise, with R_{0}< 1, the spread of the epidemic immediately slows and goes to extinction, [Figure 4]b.  Figure 4: (a and b) Simulations of coronavirus disease 2019 cases for R0 < 1
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Impact of lockdown to the spread of COVID19 in Spain
The estimation of R_{0} for Spain between March 1 and 15, 2020 has been found to be greater than the unit according to estimated parameters of [Table 3].
Had it been no lockdown after this period, then Spain would have suffered a more catastrophic state than what happened, [Figure 5]a and b.  Figure 5: (a and b) Coronavirus disease 2019 infection cases as of April 15, 2020
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That is the impact of lockdown as it was insisted by the government particularly after this period, the number of COVID19 infection cases accumulated to 180,695, which is far lower than 3,419,000, [Figure 5]a from simulation. Therefore, lockdown saved 3,238,305 possible infection cases from COVID19. Moreover, extending the projection to June 14, 2020, Spain shall have reached its peak, and therefore, the number of infections in 24 h will keep lowering, [Figure 6].  Figure 6: Projection of coronavirus disease 2019 infection cases by June 14, 2020
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Discussion and Conclusion   
COVID19 is a pandemic like other infectious diseases; it can be modeled, simulated, and can be controlled. One of the immediate control methods is the hygienic approach including lockdown. Lockdown suggests minimization of contact presumably between infected individuals and susceptible group, as well with contaminated environment. Basing on the consequences to economy one side of COVID19 fighters is suggesting a highly restricted hygienic environment would suffice to combat COVID19 instead of lockdown. The hygienic approach only is far more advantages compared to lockdown because people continue with not only economic activities but also social, and so on.
Future researches
The study of COVID19 in this context has not analytical been fully analyzed including and not limited to equilibrium stability. Moreover, the study has not included intervention by lockdown, or what if vaccination is invented. Not only that but also, we think that a model could be more appealing by explicitly analyzing how contaminated environment by the COVID19 virus directly impact the spread of COVID19. Good parameter estimation other than the nlinfit algorithm may help to determine the possible number of deaths a community or the world is likely to face. Basing on all these, more insight for future research is open for further refinement of the study.
Acknowledgments
The Mbeya University of Science and Technology is vigilant and dynamic striving to expand both in the number of students and curricula in science and technology. The university is also a front liner promoting excellence not only in teaching and consultancy but has also put emphasize on research activities. Through the Department of Mathematics and Statistics, academic staff are sorting out to research as healthy as mathematicians. Mathematical modeling of COVID19 is one of an emergent concern to be addressed in this department. The university is financing all necessities to enhance researches be able to address the COVID19 pandemic.  Figure 7: Nlinfit algorithm sample for Estimation of the Model Parameter
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Financial support and sponsorship
Nil.
Conflicts of interest
There are no conflicts of interest.
References   
1.  Alexander OF, Ludmila VC, Andrew VG. A new pelobiont protist pelomyxa corona sp. n. (peloflagellatea, pelobiontida). Protistology 2004;3:23341. 
2.  Max R, Hannah R, Esteban OO, Joe H. Coronavirus Disease (COVID19), Our World in data; 2020. 
3.  YanRong G, QingDong C, ZhongSi H, YuanYang T, ShouDeng C, HongJun J, et al. The origin, transmission and clinical therapies on coronavirus disease 2019 (COVID19) outbreak–an update on the status. Mil Med Res 2020;1:110. 
4.  TianMu C, Jia R, QiuPeng W, ZeYu Z, JingAn C, Ling Y, A mathematical model for simulating the phasebased transmissibility of a novel coronavirus. Infect Dis Poverty 2020;1:18. 
5.  John W. Preventing a covid19 Pandemic; 2020. 
6.  Eduardo ETV, Guía de prevención de riesgos laborales en el ámbito de la empresa ante la crisis sanitaria del covid19., Gestión práctica de riesgos laborales: Integración y desarrollo de la gestión de la prevención. 2020;180:1021. 
7.  
[Figure 1], [Figure 2], [Figure 3], [Figure 4], [Figure 5], [Figure 6], [Figure 7]
[Table 1], [Table 2], [Table 3]
