A Mathematical and Statistical Estimation of Potential Transmission and Severity of COVID-19: A Combined Study of Romania and Pakistan

During the outbreak of an epidemic, it is of immense interest to monitor the effects of containment measures and forecast of outbreak including epidemic peak. To confront the epidemic, a simple SIR model is used to simulate the number of affected patients of coronavirus disease in Romania and Pakistan. The model captures the growth in case onsets, and the estimated results are almost compatible with the actual reported cases. Through the calibration of parameters, forecast for the appearance of new cases in Romania and Pakistan is reported till the end of this year by analysing the current situation. The constant level of number of patients and time to reach this level is also reported through the simulations. The drastic condition is also discussed which may occur if all the preventive restraints are removed.


Introduction
In December (2019), the Wuhan Municipal Health Commission (Hubei Province, China) informed to the World Health Organization (WHO) about a group of 27 cases of unknown etiology pneumonia, who were commonly exposed to a fish and live animal market in Wuhan City. It was also notified that seven of these patients were critically serious. The symptoms of the first case began on December 8, 2019. On January 7, 2020, Chinese authorities identified a new type of family virus as the agent causing the outbreak. The causative agent of this pneumonia was identified as a new virus in the Coronaviridae family that has since been named SARS-CoV-2. The clinical picture associated with this virus has been named COVID-19. On march 11, WHO declared the global pandemic [1]. The worldwide reported cases of COVID-19 are ∼3 million with nearly 0.2 million deaths.
Coronaviruses are a family of viruses that cause infection in humans and some animals. Diseases by coronavirus are zoonotic; that is, they can be transmitted from animals to humans [2]. Coronaviruses that affect humans (HCoV) can produce clinical symptoms from the common cold to serious ones like those caused by the severe acute respiratory syndrome (SARS) viruses and Middle East respiratory syndrome (MERS-CoV) [3]. The transmission mechanisms of SARS-COV-2 are animal-human and human-human. The first one is still unknown, but some researchers affirm that it could be through respiratory secretions and/or material from the digestive system. The second one is considered similar for other coronaviruses through the secretions of infected people, mainly by direct contact with respiratory drops and hands or fomites contaminated with these secretions, followed by contact with the mucosa of the mouth, nose, or eyes [4].
Modeling is a science of creative capabilities connected with a profound learning in a variety of strategies to represent physical phenomena in the form of mathematical relations. In the prevailing situation, agencies, which control the diseases and maintain all the data of diseases, are publishing data of COVID-19 on daily bases. This data includes number of people having positive corona test, number of deaths, number of recoveries and active number of cases, and also commulative data from all over the world. So, the appropriate model, with much accuracy, is needed at this level. Low dimensional models, with small number of compartments and having parameters which can be determined with the real data with good precision, are better to study and forecast the pandemic [5]. A high dimension model requires a huge number of parameters to describe it but this huge number of parameters cannot be found with enough precision [6]. In the absence of details, compartmental epidemic models describing the average behavior of the system can be a starting point. Even the simplest models contain several variables, which are hard to determine from the available data. The minimal SIR model describes the behavior of the susceptible SðtÞ, the infected I ðtÞ, and the removed (recovered or deceased) RðtÞ populations [7,8]. Numerous models have been published on COVID-19 [9][10][11][12][13][14]. To the best of our knowledge, it has not been focused on the implications of the mathematical model to guess the future trend of COVID-19 disease in Romania as well as in Pakistan. Thus, the present study is taken to fill this gap.
To estimate the early dynamics of the COVID-19 infection in Romania and Pakistan, we modeled the transmission through a deterministic SIR model. We are choosing the SIR model because in the present situation, worldwide data contains the infectious patients, recovered, and deaths only; so, from that data, we can have the average death rate and recovery. We estimate the size of the epidemic for both countries. We also forecast the maximum level of COVID-19 patients and the time period for approaching the endemic level through model simulations. The dreadful effects of the pandemic, if precautionary measures or social distancing were ended, has also been analysed. We also perform the sensitivity analysis of the parameters by varying the values of transmission rate, disease-related death rate, recovery rate, and the inhibition effect.

Structure of the Model
In an SIR type model, the total population is partitioned into three categories, the susceptible (S), the infectious (I), and the recovered (R). If the homogeneous mixing of people is assumed, the mathematical form of the model is given as In the above model, we assume that the birth and death rate is equal and is denoted by μ. The parameter β is the transmission rate as a result of the contact of susceptible individuals with the infected ones. The incidence term is assumed to be nonlinear and is represented as βIS/1 + νI. The parameter ν represents the inhibition effect or precautions that have been adopted to prevent the mixing of susceptible and infectious individuals. We assume that the recovery rate of infectious individuals is α, and δ is the disease-related death rate.

Case Study for Romania
The coronavirus 2019-20 (COVID-19) pandemic was affirmed to have arrived in Romania on 26 th February of this year [15]. Due to the spread of the coronary disease in Italy, the government of Romania reported two weeks of isolation, starting from 21 st February, for its residents which were coming back from the influenced regions [16]. On the very next day, the Romanian government declared a few preventive measures, including assignment of five clinics as separation habitats for new cases, arrangement of warm scanners on airport terminals, and uniquely assigned lines for travelers originating from zones influenced by the COVID-19 outbreak [17]. For avoiding the virus expansion, several steps were taken by the government like on 9 th March, and the authorities reported discontinuance of trips to and from Italy via all terminals [18] which also the Special National Emergency Situations Committee ordered to close all schools on the same day. Two days later, on 11 th March, the government distributed a rundown of the fifteen rules in regards to the mindful social conduct in forestalling the spread of COVID-19 [19]. Specialists have forced a prohibition on all religious, scientific, sports, social, or diversion occasions with more than 100 members for the next three weeks.
The number of affected people crossed the first hundred at the end of the second week of March. The first three deaths were announced in Romania on 22 nd March. All three deceased were already suffering from different diseases such as diabetes, dialysis, and lung cancer. [20]. Following a flood of new affirmed cases, on March 24, the administration declared military ordinance, establishing a national lockdown and bringing in the military to help police and the Gendarmerie in authorizing the new limitations. Developments outside the homes were strictly prohibited, with certain exemptions (work, purchasing nourishment or medication, and so forth.). Old people over 65 years were permitted to leave their homes just between 11 a.m. and 1 p.m. [21]. Two days after this, on March 26, the national airline also suspended all local flights [22]. . One can see from the model (1) that we are involving disease-related death and immunity, so we have to fit our model with active real cases, active means no disease-related death and no recovery. So, initially, we have 3 active cases on March 5,2020. Hence, our initial conditions are Ið0Þ = 3 and Rð0Þ = 3, and the rest are the susceptible. We have simulated our model and fit with the real cases. Figure 1 portrays the fitting of our model (1) with the real data given in Figure 2.
By observing Figure 1, one can compare the actual data reported by [25] and the data collected by the simple SIR model (1) given in section 2. We can see a number of active cases are almost matching with the actual ones. We also estimate the number of COVID-19 patients that will appear in the next duration. It can be observed, from Figure 1, that infection is continuously spreading until August, 2020. After     Figure 1, we can see that the number of patients will be ∼10091 by the 31 st May, on June 28 th patients will be ∼11127, and by the end of this year, number will reach at ∼12000.
Week-wise expected number of patients for the next months of this year is shown in Table 1.

Variation in the Number of Patients with the Variation of
Parameters. According to reported data, it has been observed that average weekly recovery rate and disease-related death rate vary. The maximum average recovery rate happened between (1−7) March, and it is 5.71%. During the week (29 March-4 April), the minimum average recovery rate has been observed, and its value is 3.5%. Similarly, the average diseaserelated death rate varies every week. Its minimum value occurred between (12 April and 18 April) which is 0.32%. The maximum average number of deaths per day appeared during the week (29 March-4 April) and its value is 0.7%.  We vary the values of recovery and disease-related death rates by observing this pattern and estimate the number of patients that will appear in the later weeks of this year. Similarly, we increase and decrease the values of the transmission rate and inhibition effect up to 25% and 50% and also estimate the number of COVID-19 cases. The effect of the transmission rate (β), the death rate due to COVID-19 (δ), recovery rate (α), and the inhibition or precautionary measures (ν) on the number of COVID-19 patients have been calculated and shown in Figure 3.
In Figure 3(a), we present the dependency of the number of patients on the transmission rate β. The transmission rate is measured by the number of people that get infected due to a source of COVID-19. For example, β = 0:1 means every 10% people, per day, get infected. We can see from Figure 3(a) that the number of patients accelerates as β increases. The model fitted value for β is 0.396 and for that value, the number of patients by the end of this year will be ∼12000. Since the transmission rate may vary for the next duration, so we have estimated the number of patients by varying the value of β up to 25% and 50%. For β = 0:2, the number of patients by the end of this year decreases to ∼2364. For β = 0:3, this number will be ∼4046. For β = 0:5, the number of patients will be ∼7400 and for β = 0:6, the  Week-wise number of patients for each value of β is given in Table 2. We next present our results, in Figure 3(c), for the death rate dependence (δ) of the total number of COVID-19 patients. δ is the total number of patients who died, per day, due to COVID-19 disease. δ = 0:001 means one patient dies, per day, in every thousand patients. Since all the other parameters are fixed, the trend of δ dependence is as follows: the higher the δ, the lower the number of active patients. As we know that δ varies day by day, so we have plotted for five different values of δ ranging from 0.003 to 0.006 as the model fitted value of δ which turns out to be 0.003. The total number of active patients by the end of this year ranges from 7000 to 6000 for this range of δ. Week-wise number of active patients for the different values of δ is given in Table 3.
In Figure 3(b), we present our results for the change in the total number of active patients as a function of the recovery rate of infected patients α. As for the β and δ, α is also measured as a ratio per day. α = 0:01 means everyone out of hundred COVID-19 patients get recovered, per day. Definition of α infers the trend of the number of patients as a function of α: the higher the value of α means lower the number of active COVID-19 patients. The model fitted value of α is 0.056. In Figure 3 Table 4.
In Figure 3(d), we present our results for the number of patients as a function of the inhibitory effect ν. The model   Figure 3(d). Since ν is proportional to the precautionary measures adopted by the COVID-19 patients along with the general population, higher values of ν mean lower the number of active patients. The values that we have chosen for ν other than the model fitted value are ν = 9509:6, 14264.3, and 23733.9. We can see in Figure 3(d) that the total number of COVID-19 patients ranges from 4591 to 11395. Weekly data for the number of COVID-19 patients as a function of five different values of ν is given in Table 5.

Dreadful Effects of Removal of Social Distancing and Precautionary
Measures. According to the present recovery rate, disease-related death rate, and estimated values of the transmission rate, we observe that if we remove the social distancing and adopted precautionary measures, then the worst effects appear in the population. Almost ∼55% of the population will be infected up to 31 st May, and then infected people will begin to decrease. Note that this situation will according to the current position. It means that it will happen only according to the current transmission rate, recovery rate, and disease-related death rate. However, the situation    BioMed Research International may vary with the variation of these parameters. The epidemic curve without any barrier is shown in Figure 4, and calculated results are given in Table 6.

COVID-19 Case Study in Pakistan
The novel coronavirus (COVID-19) pandemic was affirmed to have arrived at Pakistan on February 26, 2020. The first patient has been observed in Sindh Province, and the second is in the federal territory of the country [26]. Within a week of appearance of initial two cases, this pandemic started to increase other areas of the country. On 29th April 2020, the quantity of affirmed cases in the nation is 15759, with 4052 (25.7% of the commulative cases) recuperation and 346 (2.2% of the commulative cases) deceased, and Punjab is, right now, the area with the most elevated number of cases at over 6000 [27]. In Figure 5, we have plotted only active cases with recovered and deaths from 26 of Feb, 2020 to 29 of April, 2020.
Currently, Pakistan has, approximately, a total population of 220 million [28], and life expectancy is 67 years [29]. As we have included the disease-related death and immunity in our proposed model (1), so this is telling us that we have to fit our model with the active cases of real data (deaths and recoveries are excluded), and Figure 6 is portraying the fitting of our model with real data, given in Figure 5, from 1 st of March, 2020 to 29 of April, 2020. The initial values are Ið0Þ = 4 and Rð0Þ = 0, and the rest of the population is susceptible. In the figure, we have compared weekwise data and then extended this week-wise data till 31 Dec., 2020 to forecast the COVID-19 cases in Pakistan. According to Figure 6, there will be ∼ 0000 by the end of May, 2020 and at the end of August, this number would be ∼ 50000. Week-wise expected number of patients for the next months of this year is shown in Table 7. Next, we will check the dependence of number of active cases on the recovery rate, α. It is the rate which tells that how many people are getting immunity from this disease. For example, if α = 0:001, then it means that out of 1000 people, one person is recovered per day. We have taken four different values of α, one is our model fitted value which is α = 0:015 and three from the real data [27]; by observing the real data, we perceived that the average recovery rate is maximum for the week 19 th − 25 th April, 2020 which is 0.037 and minimum for the week 15 th − 21 st April, 2020 which is 0.001, so we have considered these two values and fourth is the average of 0.037 and 0.001. Figure 7(b) represents the trend of active cases depending on α, and we can see that number of COVID-19 cases is inversely proportional to the recovery rate α, which makes sense. All the possible number of cases for all these values of α are given in Table 9.
Next, we will see that how the death rate δ affects the number of COVID-19 cases. It is the rate which tells that how many people die from this disease. For example, if δ = 0:007, then it means that out of 1000 people, seven people die per day. We have taken four different values of δ, one is our model fitted value which is δ = 0:00703844071 and three from the real data [27]. We have seen that the average death rate is minimum for the week 19 th − 25 th April, 2020 which is 0.004 and maximum for the week 15 th − 21 st April, 2020 which is 0.00122985. Fourth is 0.0008, and it is the average of 0.004 and 0.001. Figure 7(c) is depicting the number of active cases as a function of δ. In Table 10, we have calculated the number of COVID-19 cases for all these values of δ. In Figure 7(d), we present our results for the number of patients as a function of the inhibition effect ν. The model fitted value of ν is 30072. Since this number can also vary, we have taken four other values of ν in Figure 7(d). Since ν is proportional to the precautionary measures adopted by the COVID-19 patients along with the general population, higher values of ν mean lower the number of active patients. The values that we have chosen for ν other than the model fitted value are ν = 15036:1, 22554.2, 37590.2, and 45108.3. We can see in Figure 7(d) that the total number of COVID-19 patients ranges from 5500 to 8000. The per day data for number of COVID-19 patients as a function of five different values of ν is given in Table 11.

Dreadful Effects of Removal of Social Distancing and Precautionary
Measures. We know that the major factor to avoid from the COVID-19 is social distancing and precautionary measures; in our model, we have considered ν as this major factor. Now, if we have the present scenario and we consider do not take care of ν, then we can see from the figure that almost 33% of the population of the whole country will be infected till 19 th of July, 2020, and this is the peak of infection; after this, it will start decreasing, and we have shown that the epidemic curve in Figure 8 and calculated results are given in Table 12.

Conclusion
In this study, we used a mathematical model to assess the feasibility of the appearance of COVID-19 cases in Romania and Pakistan as well as the ultimate number of patients according to the current situation. By comparing model outcomes with the confirmed cases, it has been observed that our estimated values have good correspondence with the confirmed numbers. If the current pattern is going on, then according to our estimate, there will be ∼12000 infectious individuals in Romania by the end of this year. Pakistan will bear the burden of ∼55800 till the end of December, 2020. The situation will vary by the variation of the transmission rate, death rate, recovery rate, and further implementation of social distancing in both countries. It has been observed that the average weekly recovery rate and average weekly disease-related death vary for both countries.
If the transmission rate in Romania increases 50% and recovery rate and disease-related death rate are taken for 30 th April, according to reported data, then there will be ∼9000 persons carrying Corona malady and if this rate decreases 50%, then 2364 infected persons will exist in the Romanian community by the end of this year. If we take the previous average maximum weekly recovery rate and disease-related death rate, then there will be ∼5613 and ∼5301, patients, respectively, in Romania. Similarly, by assuming the minimum weekly average recovery and disease-related death rate will result in ∼23812 and ∼5724, if precautionary measures decrease to 50%. The worst effects of the disease appear in the community, if we remove all the barriers. In such case, this infection may increase by effecting ∼33% of the population till the end of this month. This number will start to decrease after May, 2020. Although these estimates may vary with the passage of time, it will really help us to observe the most influential factors that cause to increase the epidemic. On the basis of this analysis, competent authorities may design the most effective strategies in order to control the epidemic.

Data Availability
The data used to support the findings of this study are available from the corresponding author upon request.

Conflicts of Interest
The authors declare that they have no conflicts of interest.