ENGINEERING MATHEMATICAL MODELLING OF CORONA VIRUS (COVID-19) TRANSMISSION IN IRAQ

Since its rise in December 2019, the outbreak of novel Coronavirus Disease 2019 (COVID-19) flare-up has tainted more than 140 000 individuals all-inclusive with almost 5 400 deaths (Zhao et al., 2000). COVID-19, brought about by the extreme intense respiratory disorder coronavirus 2 (SARS-CoV-2), creates a respiratory and foundational disease that advances to a severe type of pneumonia in 10-15% of patients (Mission et al. 2020). Extreme COVID-19 can prompt primary ailment, with intense respiratory misery (ARDS) and multi-organ disappointment (MOF) as its essential difficulties, in the long run, followed by intravascular coagulopathy (Mattiuzzi et al. 2020). To streamline quiet consideration and asset distribution during this pandemic, biomarkers are desperately required to stratify patients' hazards and effectively check disease seriousness. Platelet tally is a primary and promptly accessible biomarker, which is freely connected with sickness seriousness and danger of mortality in the emergency unit (Khurana et al. 2017, Vanderschueren et al. 2000, Hui et al. 2011). Besides, low platelet checks associates with higher malady seriousness scores, for example, Multiple Organ Dysfunction Score (MODS), Simplified Acute Physiology Score (SAPS) II, and Acute Physiology and Chronic Health Evaluation (APACHE) II (Vanderschueren et al. 2000).


INTRODUCTION
Since its rise in December 2019, the outbreak of novel Coronavirus Disease 2019 (COVID-19) flare-up has tainted more than 140 000 individuals all-inclusive with almost 5 400 deaths (Zhao et al., 2000). COVID-19, brought about by the extreme intense respiratory disorder coronavirus 2 (SARS-CoV-2), creates a respiratory and foundational disease that advances to a severe type of pneumonia in 10-15% of patients (Mission et al. 2020). Extreme COVID-19 can prompt primary ailment, with intense respiratory misery (ARDS) and multi-organ disappointment (MOF) as its essential difficulties, in the long run, followed by intravascular coagulopathy (Mattiuzzi et al. 2020). To streamline quiet consideration and asset distribution during this pandemic, biomarkers are desperately required to stratify patients' hazards and effectively check disease seriousness. Platelet tally is a primary and promptly accessible biomarker, which is freely connected with sickness seriousness and danger of mortality in the emergency unit (Khurana et al. 2017, Vanderschueren et al. 2000, Hui et al. 2011). Besides, low platelet checks associates with higher malady seriousness scores, for example, Multiple Organ Dysfunction Score (MODS), Simplified Acute Physiology Score (SAPS) II, and Acute Physiology and Chronic Health Evaluation (APACHE) II (Vanderschueren et al. 2000).
In the serious, intense respiratory disorder (SARS) flare-up, thrombocytopenia was accounted for to happen in up to 55% of patients and was distinguished as a huge hazard factor for mortality (Yang et al. 2005, He et al. 2005 (Stroup et al. 2000). This is the situation for the clinical, research centre, and picture highlights of COVID-19.

COVID-19 as an Epidemic Phenomenon
As a result of a complex parameters interference in the society of the Wuhan city in China, a novel virus is created in a strange and rapid manner, that makes this virus has a distinguishing feature, and it is able to reform its abilities and genetic structure based on the ambient of existence and the employed anti-biotics, so that makes this novel virus to be rapidly spread not only in the Wuhan city no but in many countries around the world, as indicated in Figure 1 At the same, with the first case of infection, there was no effective medical cure for this novel virus, so that makes the controlling situation is very difficult. The only solution was the immunity system of an infected human body only. There were many cases of the total confirmed, total deaths, and totally recovered cases as indicated in Table 1 below, in addition to the top five highest rates countries until March 12, 2020, 2:47 pm GMT.   This paper will concern about the situation on earth for the republic of Iraq, where the first recorded confirmed case was on Monday 24 th . February 2020, then by direct contact with this foreigner infected man from Iran, many cases have been confirmedly infected, and to get a clear idea about some statistical view about the overall situation in Iraq till the 9 th . of March 2020, take a look at Table 2 below:  As indicated in Table 2 above, the governorate of Baghdad is occupying the highest infection level comparing with the other zones within Iraq, so the proposed engineering mathematical model will completely concern the capital Baghdad in analyzing this critical medical situation as in the coming paragraph.

Engineering Mathematical Modelling
There are many procedures employed to analyze the similar phenomenon of spreading of the novel Corona virus, scientifically named as COVID-19, during the past decades, and in this paper, the Suspected, Infected, and Recovered (SIR) model that has been developed by Kermack and McKendrick will be a theoretical basis of this investigation. Simply, any conservative system must have three main elements, as illustrated in Figure 3 below, and they are some input parameters, process(s) or operation(s), and infected output parameters. This principle of the conservative system was representing a core idea of the general formula of the SIR model, so Figure 4 below is representing the main schematic of the SIR model with a flow line for indicating the direction of each input, process, and output for all of the action within this proposed conservative system. Where µ is the natural rate of death, P is the input rate of suspected persons, q and r are the infected and recovered cases rates, respectively, and d is the total death rate due to the infection under consideration.
SIR mathematical model may be represented by a set of first-order, first-degree ordinary differential equations as in the coming lines: Figure 6 shows the proposed model for the Suspected cases (S) icon: With the same technique, the Recovered cases formulation with being as indicated in Figure 8: where the SIR epidemic model is in equilibrium, yields: ………. (4) ………. (5) ………. (6) and according to the logic interaction between the four rates around the I icon, see Figure 9, it yields: Figure 9: The mutual interaction of the four rates around the I icon The principal reproduction value , where is representing a reproduction rate value and may be defined as the average gained the total number of secondarily infected peoples generated by one primary infected case within the total number of a suspected population and will equal to: ………. (7) The most important icon in the SIR epidemic model is the I because it represents a critical three intersected results, either natural death, total recovery, or death due to the COVID-19 virus infection, so it is essential to deeply analysis this central I icon as follows: so to reform Eq. (11, 12, and 13), it is better to use matrix notation as follows:  Table 3 below summarizes this set of essential equations as follows: Table 3: Summary of a set of essential SIR epidemic equations model Based on Eqs. (4, 5, and 6), and to examine the regional stability of the epidemic COVID-19 virus-free equilibrium state (E), it is essential to construct the Jacobian matrix in association with the major assumption of as indicated in the following matrix: Now the next step is to find the determine of the Jacobian matrix, which must equal to zero as follows: , or ………. (15) so, since the novel COVID-19 virus is still spreading rapidly in many countries without the presence of any effective vaccine around the globe, it is impossible to say that the novel communicable disease-free equilibrium is stable! Furthermore, in order to say that the disease-free equilibrium is stable as the ratio must be less than the summation of , where is the total infected cases within the total population. Table 4 gives a clear idea about the total situation in some of the Iraqi governments, so to conduct a numerical application and statistics based on the above SIR epidemic mathematical model, three governorates with the highest rates of infection has been nominated are: Baghdad, Sulaimani, and Karbala, in addition, Iraq is occupying the highest infection to death ratio around the world (Shilani. H, 2020), where this ratio is equal to 11.2% followed by Italy 6.6% and Iran 3.9%. Furthermore, Table 5 illustrates to total population in the above mentioned three governorates:  Now in return to Eq. (9) , this equation consists of two terms; the first one is impossible to be equal to zero, i.e. if it is equal to zero, then this will lead to no infection or no infected cases, but actually, the situation on earth is a presence of tens of the infected cases at the above-mentioned governorates with variant infections levels; therefore, the second term must be equal zero, hence:

Numerical Application
………. (16) or ………. (17) where K is an arbitrary numerical constant, and for conducting the required calculations for finding the expected Reproduction ratios, R 0 ) K , let , then: Regarding the Baghdad situation: so, table (6) gives the gained Reproduction ratios R 0 ) K :   Regarding Karbala situation: Table 8 summarizes the obtained values of the Reproduction ratios R 0 ) K : The statistical treatment of the above-tabulated data for the Reproduction ratios as a function of the recently updated populations for the three nominated governorates is indicated in Figure 11(a, b, c, d, and e), where the x-axis represents the selected three governorates and the y-axis is representing the associated values of the Reproduction ratios, taking into account variation of the arbitrary constant K.  After few days, the infection situation is starting to vary according to the increase of the infected cases, as shown in Figure 11 (b) below: Again, due to increment of the active and infected cases, which is highly affected by the increment of the suspected cases, the acquired values R 0 ) K are also changing as shown in Figure 11(c) below: This comparison in each one of the graphical sketches must be based on the same value of the arbitrary constant K to conduct comparing process between the three governorates at the same value of K in each sketch, Figure 11(d) is approximately giving the same values as in Figure 11(c), but Karbala is still occupying the highest level, then Sulaimani, followed by Baghdad.

Figure 11(d):
Reproduction ratio as a function of the governorate, K=0.8 Figure 11(e) below illustrates a constancy of the Reproduction ratio with three selected zones with a peak value of the arbitrary constant K, which is an equal one, and Karbala is still at the top-ranking comparing with the other two zones. It is important to mention that the total population N of Karbala is lee than the other two cities, but the total number of the suspected cases and then the primary infected cases are relatively times higher than the remaining cities; this is confirming the positivity of the Reproduction ratio (it is a good idea to return to the scientific definition of the R 0 ) K ), and this will be deeply explained in the coming paragraphs.  March 14, 2020, where the black, green, and yellow lines are representing the total number of deaths due to the COVID-19 virus recovered and infected cases, respectively. As per this trusted data, Iraq is at the international highest top-ranking level of the ratio of deaths to infected cases, with about 11%, so that stimulates us to conduct this investigation to at least present a contribution in this regards to safe humanity from this epidemic disease.

RESULTS AND DISCUSSION
According to the achieved analyzed results, the Reproduction function values have positive values, i.e. R 0 ) K > 0, so that supports the overall situation in the three investigated governorates and positively comply with the following achieved points: 1. Total suspected cases are in a continuous increase and consequently, cause an increment in the positively tested (infected) cases population as the days go on.