A new mathematical model of COVID-19 using real data from Pakistan

We propose a new mathematical model to investigate the recent outbreak of the coronavirus disease (COVID-19). The model is studied qualitatively using stability theory of differential equations and the basic reproductive number that represents an epidemic indicator is obtained from the largest eigenvalue of the next-generation matrix. The global asymptotic stability conditions for the disease free equilibrium are obtained. The real COVID-19 incidence data entries from 01 July, 2020 to 14 August, 2020 in the country of Pakistan are used for parameter estimation thereby getting fitted values for the biological parameters. Sensitivity analysis is performed in order to determine the most sensitive parameters in the proposed model. To view more features of the state variables in the proposed model, we perform numerical simulations by using different values of some essential parameters. Moreover, profiles of the reproduction number through contour plots have been biologically explained.


Introduction
COVID-19 is characterized as a disease caused by a novel COVID-19 now known as Severe Acute Respiratory Syndrome COVID-19 2 (SARS-CoV-2; formerly known as 2019-nCoV), first identified as a result of an outbreak of respiratory disease in Wuhan City, Hubei, China [1][2][3]. The report was initially made to the WHO on 31st December 2019. The WHO announced the COVID-19 outbreak as a global health concern on 30th January 2020 [4,5]. COVID-19 is transmitted from human to human by direct contact with contaminated surfaces and by inhaling respiratory droplets from infected persons. Most infected people with the COVID-19 virus may develop mild or severe respiratory disease and recover without any special treatment being needed. Older people and those with underlying medical conditions such as cardiovascular disease, diabetes, chronic respiratory disease and cancer are more likely to experience serious illness.
COVID-19 can live for hours or days on a surface, due to factors such as sunlight, temperatures, and the type of surface. A person can get COVID-19 by touching a surface or object that has the virus on it and then touching their own mouth, nose, or eyes. However, this is not deemed to be the only way the virus spreads. Social distancing helps to minimize the chances of touching infected surfaces and contagious individuals outside the home [6]. There are currently no clear COVID-19 vaccines or medical treatments. There are, however, many ongoing clinical trials evaluating potential treatments. Governments implemented various control measures such as stringent and obligatory lockdowns, other measures such as social distancing, the avoidance of crowded meetings, the implementation of maximum numbers of people at every meeting and the use of face masks in order to effectively reduce COVID-19 transmission. Although the government has introduced a number of control initiatives. Contact tracing of reported infected cases has been stepped up across several nations to further help in curbing the spread of COVID-19, and confirmed cases are quickly put on isolation for proper care [7]. In Pakistan, the virus was first confirmed on 26th February 2020, when two cases were recorded, a student who just arrived from Iran and another person in the Islamabad Capital Territory. On 18th March, 2020, cases of COVID-19 have been recorded in all four provinces of Pakistan and Islamabad Capital Territory, and by 17th June, 2020 each district in Pakistan had recorded at least one confirmed case of COVID-19. As of October 10th, 2020, Pakistan had reported 324,077 cases of COVID-19, with 6673 deaths, 9384 active cases (undergoing treatment) and 308,020 recovered cases [8].
It is widely accepted that using mathematical models can predict the occurrence of infectious diseases. In this way, it is possible to find the likely outcome of an outbreak that is beneficial for the purposes of public health initiatives. By using compartmental models as a basic mathematical framework, the complex dynamics of epidemiological processes can be studied, as stated in [9]. Phenomenological expectations are the basis of the way these two populations interact and the models are thus constructed. To make up these models, ordinary differential equations (ODEs) have traditionally been investigated. Additionally, to make these models more functional, other populations, denoted by R, which is the picture of the immune/ removed/ recovered compartment, are also taken into account. A relevant point here is that, appropriate criteria for the specific disease under consideration should be obtained and that these criteria should be used to assess the effects of possible control steps, such as medication or vaccination. The problem, then, is how to incorporate such measures from an optimal point of view. Recently, many notable attempts have been made to implement various research for different types of integer compartmental model [10][11][12][13][14][15]. Several mathematical models and non-mathematical findings have also been performed on COVID-19 as shown by most recent researchers in [16][17][18][19][20][21][22][23][24][25][26][27][28][29][30][31].
This paper presents a deterministic mathematical model that assesses the impact of some control measures on the spread of COVID-19 in a human population; the study will focus on the situation in Pakistan, so real data from that country will be used. The model, parameterized using the total number of confirmed cases in COVID-19 and the number of active cases in Pakistan, would provide a reliable estimate of Pakistan's disease burden. We hope that this study will give governments and public health sectors some insight into how to increase their nonpharmaceutical prevention initiatives to minimize the spread of the disease.

Formulation of the model
The model under consideration subdivides the population of human at time t into five compartments. That is, susceptible S(t), exposed E(t), infected I(t), quarantine Q(t) and recovered R(t). We assumed that newborns can become infected with the virus that causes COVID-19 during childbirth or by exposure to sick caregivers after delivery. Therefore, the recruitment into the susceptible class is either by birth or immigration at a rate θ. The progression of exposed individual to infected class is at a rate ω. Individual in each of the classes can die a natural death at the rate μ . We also assumed that recovery from COVID-19 does not confers permanent immunity, therefore, the recovered individuals can returned back to susceptible class at a rate σ. The infected and quarantine individuals are reduced as a result of COVID-19 related complication at the rate δ. τ and ϕ represent the treatment rate for the infectious and quarantine individuals respectively. The force of infection where ψ is the proportion of individuals that maintain social distancing to prevent the spread of COVID-19 (at least two meter apart) this lies in the inequality 0⩽ψ⩽1. Also, υ represents fraction of the total population that effectively make use of the face mask and use of hand sanitizer. We assumed that face mask and use of hand sanitizer are highly effective when in public gathering so that 0⩽υ⩽1.Thus, putting the above descriptions and assumptions together gives the following COVID-19 model, given by system of ordinary differential equations below as . The proposed model's flowchart and parameters' description are well explained in the Fig. 1 and the Table 1, respectively.

The invariant region
The invariant region defines the domain in which solutions to the model are of both biological and mathematical importance. All the variables and parameters of the model are assumed to be non-negative. To this end, we find first of all the total human population M(t), where By differentiating with respect to t on both side, we obtain the following: In the absence of COVID-19 induced death (δ = 0), the above equation becomes Integrating on both side of (2) yields the following ∫ dM This simplifies into, with the initial condition M(0) = M and B is a constant with the initial condition on eq. (4), we get the following As t→∞in eq. (5), we obtain Thus, the region in (6) is the feasible solution sets P for the total population in eq.(1).
Therefore, the proposed COVID-19 model (1) is well posed and are both epidemiologically and mathematically meaningful. It is therefore sufficient to examine the dynamics of the basic model in the domain P.

Basic reproduction ratio (R 0 )
The basic reproductive ratio is a baseline measure in epidemiology that shows the total number of related diseases caused during an infectious period by a single infected person in a fully susceptible population. The matrices produced for the new disease and the condition for the disease transition are denoted by F and V. We have used the same strategy as described in [32].
The Jacobian matrix of F and V computed at the COVID-19 free equilibrium expressed as F and V such that, The spectral radius of the matrix FV − 1 gives the basic reproduction number

Disease free equilibrium state and its global stability
The equilibrium state in the absence of COVID-19 is known as disease free equilibrium.This is obtained by equating the right hand side of Eq.
(1) to zero. Thus, is found to be the COVID-19 free equilibrium state. Proof. Lyapunov function is commonly used to proof the Global Stability of the Disease Free Equilibrium [33,34]. Consider the formed Lyapunov function of the type where a 1 = (μ + δ + ρ + τ), a 2 = (μ + ω). It is easy to establish that a 1 and a 2 are positive. Let's differentiate L with respect to time to have Solving eq.(8) to obtain Simplifying eq.(8) with Eq. (9) to obtain We obtain from the eq.(10) that L = 0 when E = I = 0 and R 0 . This means that the highest invariance set in {( S, E, I, Q, R) ∈ R 5 + } is the  singleton DFE (E 0 ) and by LaSalle's Invariance Principle [35], DFE (E 0 ) is globally asymptotically stable in R 5 + .

Parameters fitting
The concept related with modeling a data set is ubiquitous almost in all applied areas of study. Making sure repetition of the data is an essential task because it would make predictions easier about items related to our data. In a curve fitting approach what we do is to take existing data and try to fit that data to a curve or a line. If we look at the data for fatalities or for number of cases or we look at the data for hospitalizations with respect to the coronavirus then we see from first row of the Table 2 that the curve has an upward trend from 01 July, 2020 to 14 August, 2020 in the country of Pakistan. Various approaches are available to adjust differential systems' parameters to fit data so commonly we have a model of the system that might be something like an ordinary differential equation and then we also collect some data at particular time such as the one depicted by first rows in the Table 2. The table represents data versus model predictions along with absolute percentile errors wherein one can observe that the maximum error comes in second predicted value as about 11% whereas majority of errors are comparatively smaller. In this present section, we have used nonlinear curve fitting strategy to obtain unknown parameters of the proposed model (1) as listed in the Table 3. The objective was to obtain smallest possible sum of squared residuals defined as: where x j and k(t j ) stand for observational data values and predicted values from simulations of the infectious compartment in the model, respectively. Here, total number of data points are n = 45 which demonstrate number of cases that occurred in Pakistan on a regular basis in Pakistan from 01 July, 2020 to 14 August, 2020. After achieving the objective of getting minimum error possible, we have obtained average absolute relative error to be equal to about 2.025841e − 02 which is considered to relatively small error whereas the basic reproductive number as shown in the Eq. (7) is computed as R 0 ≈ 4.8406e +00 after using fitted parameters from Table 3. In addition to this, best fitting curve in comparison with the real observational data of COVID-19 cases is shown in the Fig. 2 wherein residuals have also been graphically depicted. One can observe from the plot how efficient the nonlinear least-squares curve fitting approach is in our model settings that ultimately helped to obtain unknown important parameters for validation of the proposed model of COVID-19 as shown by the Eq. (1).

Sensitivity of parameters
This section employs the principle of sensitivity analysis to investigate the robust significance of the standard parameters found in the unique specific reproduction number. Furthermore, with the aid of parameter values, both analytical and numerical values of the various parameters in R 0 are determined from reliable assumptions. The analytical expressions obtained can be used to shed some light on how to track the onset of COVID-19 in different locations if the dynamics obey the model. R 0 is a quantity that is considered the key means of minimizing and aborting the spread of victimization by decreasing the number to less than unity. The sensitivity index approach is used to calculate the model's most sensitive parameters; those with nonnegative signs are considered to be extremely and proportionally sensitive to increasing the value of Ro, while those with negative signs are less susceptible to lowering Ro, and the remaining category is neutrally sensitive (with zero relative sensitivity). The cause of raping transmission is widely understood to be directly correlated with the basic reproduction number R 0 . We measure the sensitivity indices for the state variables as follows: where X i represents the points at which the disease persist and P i are the corresponding parameters in X i . In a similar way, the elasticity indices of the basic reproduction number to the associated parameters are described as: where R 0 represents the basic reproduction number and P i is as stated above. Thus, after some computation we reach:  The numerical values showing the relative importance of the R 0 parameters are shown in Table 4. It can be shown that, based on the findings shown in Table 4, αc and θ have a clear effect on the virus' stability and a very similar relationship. Therefore, by increasing αc and θ by 1%, R 0 would have an increase of 1% in influence. As shown in Table 4, there are parameters with a positive relation and those with a negative relation. A negative relationship suggests that an increase in the values of those metrics would help to mitigate the pandemic's brutality. While a positive relationship indicates that the frequency of the pandemic en mass would be greatly influenced by an increase in the values of those parameters. The numerical signs in Table 4 are shown in the Fig. 3. Fig. 3 depicts the powerful parameters that easily allow the virus to spread quickly.

Numerical dynamics
This is the position where we get deep insights into the complex behavior of the model. Under various computational simulations of the proposed model for state variables of interest, the transmission dynamics of an infectious disease can be adequately understood. In this section, we have conducted numerous simulations to understand the complex behavior of asymptomatically infectious individuals and the number of reproductive control R 0 . Various forms of time series graphs are shown by changing the parameter values under the parameters obtained by the nonlinear minimum-squares fitting approach in Section 5. Moreover, for the most significant parameters of the model, the dynamics of R 0 are achieved through contour plots.The Susceptible S(t), exposed E(t), infected I(t), quarantined Q(t) and recovered R(t) populations are investigated with different values of the parameters. As shown in Fig. 4,5 over the interval [0 44], one can see that the S(t) is decreasing, E(t) is increasing, I(t) is increasing, Q(t) is decreasing with slight signal of an increase whereas the R(t) is strictly increasing. By changing the interval to [0 200], the behavior of the aforementioned state variables changes with strong impact. Fig. 6(a) depicts the dynamical behavior of the I(t) with some values of the τ (treatment rate for infectious people) while Fig. 6(b) depicts the I(t) individuals with some values of α c (effective transmission rate). In Fig. 7(a) and (b), I(t) is plotted by varying ρ (rate of recovery from infection) and δ (COVID-19 death rate), respectively. Moreover, the impact of ψ (proportion of individuals that maintain social distancing) and υ (fraction of the total population that effectively Make use of the face mask and use of hand sanitizer) on the I(t) individuals is depicted respectively in Fig. 8(a) and (b). The profile of R 0 versus some parameters via 2-dimentional and contour plots is shown in Figs. 9 and 10, respectively. Based on the illustrated figures, it can be noted that the methods used by the public health sector and the government concerned to resolve the situation leading to a rise in the pandemic would be more successful with a controlling transmission rate (see Fig. 5).

Conclusion
COVID-19 pandemic is one of the disturbing viruses that has confused the entire world. It has a subtle nature that causes delay of the emergence of its proper vaccine. Several measures and precautions by the concerned governments and medical practitioners have been suggested to combat the spread of the virus. One of the helpful ways to facilitate the understanding of the virus is the mathematical approach. In this regard, we proposed a new mathematical model on the disturbing contemporary pandemic called COVID-19 to comprehend its transmission dynamics. Several important properties of the new model such as the invariant region, equilibrium state, basic reproduction number, global stability of the free disease equilibrium have been investigated in the present research work. Moreover, real statistical data set from 01 July, 2020 to 14 August, 2020 in the country of Pakistan is also used to obtain best fitted values of some parameters using nonlinear least squares estimation technique with MATLAB function lsqcurvefit. Sensitivity analysis of the parameters has been conducted in order to determine the most sensitive parameters of the new proposed model. Numerical simulations confirmed these observations with best fitted curve obtained for the proposed model. In this research, it was shown that with a regulated transmission rate, the methods used by the public health sector and the government to address the situation leading to an increase in the pandemic will be more effective. Following the strategy employed in recently published researches [36][37][38][39][40][41][42][43][44][45], the proposed COVID-19 model will be investigated in more detail in future research work.

Availability of data and material
All the data used in this manuscript can be found on WHO website.