COVID-19 proposed framework

We should know that how COVID-19 diseases change from one stochastic state to another during the infection period, as discussed in the COVID-19 Conceptual framework. The parameters taken for analysis are Suspected (S(t)), Infected (I(t)), Recovered (R(t)), and Deaths (D(t)) are the four major phases found in a group [Reference Nair, Soni, Bajpai, Dhiman and Sagayam34]. A susceptible person who comes into interaction with an infectious individual, as per this theory, is at risk of being infected. An individual who has been infected can either recover or die in case of the virus. So, the total number of (S(t)), (I(t)), ((R(t)), and (D(t)) is defined to be constant in this model. Moreover, it is considered anyone who is infected with the virus got infected instantly, without any latent period among stimulation and infection. There seems to be no compensation for the impacts of isolation or quarantine. The period throughout the research is indicated by the ‘t’. A proposed framework is used to represent the phases, shown in Figure 1.

Figure 1. Proposed framework of COVID-19 disease.

The Pakistan Ministry of Health has started to release a special bulletin about COVID-19 infections in Pakistan on 26 February 2020. Data will be taken from the Ministry of Health website. We will use the regression model in Poisson distribution to predict the spread of COVID-19 cases in Pakistan by using data from the beginning until 31 January 2021.

Poisson regression model

The PRM, which is a nonlinear regression analysis of the Poisson distribution, is often used to estimate a response variable with discrete or count data given several explanatory factors. Poisson regression is said to obtain over-dispersion if the variance value is higher than the mean value. Overdispersion has the same effects as if such data types appeared over-dispersion however Poisson regression was also applied, regression coefficients of model parameters may ensure consistency and although inefficient. This is mostly applied when solving issues where the result of a stochastic function can only take count numbers. The Poisson distribution is one of the distributions that fulfil this constraint. It belongs to the exponential distribution family. Assume that ‘B’ is the r.v (Coronavirus death ratio) and that the case findings are an event. The ‘B’ variable approaches a Poisson distribution [Reference Kianifard and Gallo31] with parameter omega ω > 0

(1) $$ P\;\left(\;B=b\;\right)=\frac{e^{-\omega }{\omega}^n}{B!}, $$

where n = 1, 2, 3 is the number of occurrences of an event and ‘ω’ is defined as ω = Exp[B].

The generalised linear model (GLM) can be stated as

(2) $$ {g}_l={\beta}_0+{\beta}_1{A}_{l1}+\dots +{\beta}_m. $$

The first function explains how the mean, $ \mathrm{Exp}\left[{B}_l\;\right]={\omega}_l $ which depends on the linear predictor $ \varphi \left({\omega}_l\right)={g}_l $ while the second function explains how the variance, $ V\left({B}_l\right) $ depends on the mean $ V\left({B}_l\right)=\sigma V\left({\omega}_l\right) $ . Where parameter ‘ $ \sigma $ ’ is a constant, assuming $ {B}_l $ is a Poisson distribution, so

$$ {B}_l\sim \mathrm{pois}\left({\omega}_l\right),\hskip0.2em \mathrm{Exp}\left[{B}_l\right]={\omega}_l\hskip0.5em \mathrm{and}\hskip0.5em V\left({B}_l\right)={\omega}_l. $$

The variance function is defined as follows:

$ V\left({\omega}_l\right)=\omega $ Ranges from (0, ∞).

A log function is given as

$$ \ell \left({\omega}_l\right)={\log}_e\left({\omega}_l\right) $$

Based on the general linear model (GLM), the linear regression must be associated with the dependent variables with a link variable, which is also the link variable between the linear regression in a matrix form and the Stochastic regression line in this case. Assuming a linear model is as follows:

(3) $$ {B}_l={\beta}_l{A}_l+{\epsilon}_l, $$

where ‘A’ is n* (m + 1) vectors of predictors, a column of 1’s β is a (m + 1) by 1 vector of uncertain parameter, as well as ϵ is a (n* 1) vectors of error values with

(4) $$ \mathrm{Exp}\left(B/A\right)= a\beta . $$

Consider the general linear regression, the linked value and its transportation ‘B’ are as below:

(5) $$ \hskip2.639999em G(B)={\log}_e(B). $$

Hence, it may be expressed as

(6) $$ \hskip2.52em {\log}_e\;E\left(\frac{B}{A}\right)= a\beta . $$

So that, regarding a Poisson regression with the function ‘β’ and ‘a’ is its input vector, an expected average of the aligned Poisson distribution is taken as

(7) $$ E\left(\frac{B}{A}\right)={e}^{a\beta}. $$

If ‘ $ {B}_l $ ’ is a response variable with the predictor variable $ `{A}_l $ ’, the parameter ‘β’ may be evaluated by MLE method. Mathematical models may be used to evaluate the system indicated in equation (7), which is obtained the conditional expectation of response and predictor variables by adopting the logarithm transform of the conditional expectation. Moreover, PRMs’ MLE of the probabilistic area is usually concave, Gradients-based techniques are used as appropriate estimation procedures.

Let ‘ $ {b}_l $ ’ be a random variable and has a positive integer like $ l=1,2,3,\dots, n $ where ‘n’ denotes the total number of observations. Hence $ {b}_l\sim $ Pois dist. [Reference Kianifard and Gallo31]. So, the Probability Mass Function (PMF) will be

(8) $$ P\left({B}_l={b}_l\right)=\frac{\omega_l^{b_l}{e}^{-{\omega}_l}}{b_l!}, $$

(9) $$ P\left(B=b\right)=f(b)=\frac{e^{-\omega }{\omega}^b}{b!}, $$

where $ {b}_l=0,1,2,\dots $ with mean and variance are given as

$$ \mathrm{Exp}\left[{b}_l\right]=V\left({b}_l\right)=\omega . $$

In equation (4), the estimated average of a Poisson Process is defined as

$$ \mathrm{Exp}\left(\frac{B}{A}\right)={e}^{a\beta}=\omega =\mathrm{Exp}\left({b}_l\right), $$

whereas the values of the independent factor $ \beta =\left({\beta}_1,\hskip0.2em ,{\beta}_2,\dots {\beta}_m\right)\hskip0.24em $ be k – dimension vectors of logistic coefficients are unknown, while the average of the estimated stochastic process is presented as $ \hskip0.24em \mathrm{Exp}\left(\frac{B}{A}\right)\hskip0.48em $ and $ \mathrm{Var}\left({B}_l\right) $ is given as $ V\left(\frac{B}{A}\right) $ .

The following three factors are indicated by using Poisson regression, which is designed to fit such a rate of fatalities in Pakistan:

A ₁ = Suspected cases

A ₂ = Infected cases

A ₃ = Recovered cases.

(10) $$ \log \left(\theta \right)={\beta}_0+{\beta}_1{A}_1+{\beta}_2{A}_2+{\beta}_3{A}_3. $$

The parameter ‘β’ shows the average variation in the independent factor ‘ $ {A}_l $ ’ in the logarithmic equation (8) that may be evaluated using the maximum likelihood estimator (MLE) approach.

Negative binomial regression

To cope with over-dispersion in count data, NBR is being used. As populations are usually varied, specifically distinct to the assumption involved in commonly use basic stochastic methods, over-dispersion is a usual feature of the real statistical approach. In this research, the NBR model [Reference Juarez‐Colunga and Dean32] is stated as follows:

$$ {\displaystyle \begin{array}{rcl}\mathrm{Prob}\left[{B}_l=b\right]& =& \sum \mathrm{Prob}\left({B}_l={b}_l/{\theta}_l\right)f\left({\theta}_l\right){d}_{\theta l}\\ {}& =& \frac{\Gamma \left({b}_l+\frac{1}{\alpha}\right)}{\Gamma \left({b}_l+1\right)\Gamma \left(\frac{1}{\alpha}\right)}{\left[\frac{1}{1+{\alpha \omega}_l}\right]}^{\frac{1}{\alpha }}{\left[\frac{{\alpha \omega}_l}{1+{\alpha \omega}_l}\right]}^{b_l},\end{array}} $$

(11) $$ \mathrm{for}\hskip0.24em l,0,1,2, $$

where mean is given as

$$ {\omega}_l=\mathrm{Exp}\left({b}_l\right)={V}_l\left[{e}^{\sum {\alpha a}_{lm}{\beta}_m}\;\right],\hskip0.84em \mathrm{for}\;l=1,2,3,\dots .,n. $$

While the variance of $ {b}_l $ is given as

$$ V\left({b}_l\right)={\omega}_l+\alpha {\omega_l}^2. $$

If $ \alpha $  > 0, the method might be called a dispersion parameter. The PRM may be treated with a limited form of the NBR model with $ \alpha \propto 0 $ . The MLE statistic is written as

(12) $$ \log \left(\beta, \alpha \right)={\displaystyle \begin{array}{l}\sum \left(\sum \limits_{r=1}^{b_l}\log \left(1+\alpha r\right)\right)-{b}_l\log \left(\alpha \right)-\log \left({b}_l!\right)\\ {}+\hskip2px {b}_l\log \left({\alpha \omega}_l\right)-\left({b}_l+{\alpha}^{-1}\right)\log \left(1+{\alpha \omega}_1\right).\end{array}} $$

A partially differential of the MLE w.r.t (β, α) can be used to evaluate the parameter (β, α). The NBR may not involve the mean = variance but it controls over-dispersion, which happens whenever the value of Variance is larger than the mean value.

The objective is to determine a model with the lowest number of parameters that minimises the Negative likelihood, as shown in the equation below:

(13) $$ \mathrm{AIC}=2R-2 lnL, $$

in which ‘L’ denotes the likelihood ratio and ‘R’ is the number of model parameters.

Finding and discussion

The methods used in this analysis are PRM and NBR which comply mostly with the study’s particular target. The count variables are the total number of fatalities that are the response data, also with events of total suspected, infected, and recovered COVID-19 occurrences as the predictor variables. Continuous count data is used for all the variables.

Application

From 26 February 2020 to 1 January 2021 secondary information is attained from the National Command and Operation Center (NCOC) website for total suspected, infected, recovered, and deaths COVID-19 occurrences.

A set of count methods, like the PRM and the NBR model have been used to estimate the total number of COVID-19 suspected, infected, recovered, and deaths cases in Pakistan. The data set is taken from 26 February 2020 to 1 January 2021.

Table 1 shows that mean and standard deviation at Mean = 5,166.33 and SD = 4,888.679 for each suspected because of coronavirus. Same as the mean and standard deviation of infected occurrences of coronavirus infection is Mean = 57,580.18 and SD = 60,088.140. The standard deviation mean and of the recovered coronavirus occurrences in the study reveals that Mean = 51,266.59 and SD = 55,753.951. Thus, the mean and standard deviation of the death occurrences of coronavirus is Mean = 1,233.06 and SD = 1,223.407. According to the findings thus far away, the large mean estimate of Covid-19 infected and recovered cases in Pakistan throughout the situation under analysis has resulted in comparatively large cases of coronavirus-associated fatalities.

Table 1. Summary statistics of COVID-19 cases in Pakistan

Figure 2 shows the graphical representation of suspected cases of COVID-19 infection in different provinces of Pakistan, such as, Punjab, Baluchistan, Sindh and KPK.

Figure 2. Suspected cases in Punjab, Baluchistan, Sindh and KPK.

Figure 3 shows the graphical representation of infected cases of COVID-19 infection in different provinces of Pakistan, such as, Punjab, Baluchistan, Sindh and KPK.

Figure 3. Infected cases in Punjab, Baluchistan, Sindh and KPK.

Figure 4 shows the graphical representation of recovered cases of COVID-19 infection in different provinces of Pakistan, such as, Punjab, Baluchistan, Sindh and KPK.

Figure 4. Recoveries in Punjab, Baluchistan, Sindh and KPK.

Figure 5 shows the graphical representation of death cases of COVID-19 infection in different provinces of Pakistan, such as, Punjab, Baluchistan, Sindh and KPK.

Figure 5. Deaths in Punjab, Baluchistan, Sindh and KPK.

The omnibus test shows the actual model to the null model that used a test of likelihood-ratio chi-square. The present framework overcomes the null hypothesis since its p-value is less than 0.05. It is a likelihood ratio test to perceive all the explanatory variables (suspected, infected, and recovered COVID-19 cases) are significantly outperforming the intercept-only analysis.

Table 2 shows the explanatory variables (i.e., suspected, infected, and recovered cases) have 0.000 p-value (p < 0.05), in Poisson regression (PR) and NBR models, which indicate an overall statistically significant model.

Table 2. Omnibus test of COVID-19 cases in Pakistan

The condition of equi-dispersion in Poisson regression analysis, the mean should be equal to the variance. Therefore, the claim is hardly observed, over-dispersion is frequent in many situations. Though over-dispersion is perhaps the most common issue, it is really essential to examine this number firstly. The deviance for all of the models is near to ‘1.2’, which is admissible. Whenever the amount of deviance is reduced by the degree of freedom (df), it should be larger than ‘1.2’ to identify over-dispersion; meanwhile, under-dispersion occurs whenever the amount of deviance is lowered with df is smaller than ‘1.2’.

Table 3 shows the deviance/df and Pearson chi-square/df are both more than ‘1.2’, implying that the numbers of COVID-19 occurrences in Pakistan are over-dispersed, based on the PRM. The analysis is performed by NBR to get the better solution as this approach may handle the dispersion parameter, and the resulting log-likelihood and Akaike informatics criteria (AIC) results are (−209,814.870 and 419,637.741) respectively.

Table 3. Association of variables with model selection (PRM)

Table 4 indicates both amounts of Pearson chi-square/df and deviance/df are below ‘1.2’ (i.e., 1.069 and 0.687, respectively), implying that the amount of COVID-19 occurrences in Pakistan are under-dispersion, and have log-likelihood and −9,664.914 and 19,337.828, respectively.

Table 4. Association of variables with model selection (NBR)

The reported COVID-19 death cases are a count, and then this form of data seems to have a Poisson distribution. To analyse the variables that impact the number of COVID-19 cases of deaths, the Poisson regression modelling is used. The maximum likelihood estimator (MLE) technique is commonly used to predict the parameters of Poisson regression.

Table 5 shows p-values for all parameters that are smaller than 0.05, indicating that the parameters $ {\beta}_1,{\beta}_2\;\mathrm{and}\;{\beta}_3 $ have a significant impact on results. Consequently, the PRM is given as

$$ {b}_l=\exp \left(5.941+0.00009605{A}_1+0.00003278{A}_2-0.000003883{A}_3\right). $$

Explanatory factors that determine the coronavirus fatalities in Pakistan are the coronavirus suspected ( $ {A}_1 $ ), infected ( $ {A}_2 $ ) and recovered $ \Big({A}_3 $ ) occurrences.

Table 5. Multiple terms test between diagnose disease and respondence in Pakistan (PRM)

The NBR is often used to predict count observation as well as obtain dispersion measurement to tackle the limitation of over-dispersion related to the PRM. Consequently, the NBR model is given as

$$ {b}_l=\exp \left(5.370+0.00008899{A}_1+0.00001533{A}_2-0.000002329{A}_3\right). $$

Table 6 reveals that basically, factors are statistically significant at 5% level, also p < 0.05. With exception of the recovered cases, the NBR model shows that two of the COVID-19 factors are positive. This indicates that when using the NBR model to estimate the parameter, a 1% rise in suspected and infected COVID-19 instances might result in a 0.009% and 0.002% higher in COVID-19-related deaths in Pakistan. COVID-19 patients that are suspected and infected have a positive effect on the deaths related to coronavirus in Pakistan. Consequently, at the 5% level, the recovered occurrences have a negative value (−0.000002329), which implies a 1% increase in recovered coronavirus occurrences in Pakistan throughout the time period under consideration may occur in a −0.0002% reduction in the number of deaths caused by infectious disease in Pakistan and outcome often reveals that recovered occurrences may have an adverse impact of the coronavirus fatalities.

Table 6. Multiple terms test between diagnose disease and respondence in Pakistan (NBR)

Table 7. Selected count data model for COVID-19 cases

Table 7 shows the association of variables with a model selection of the two counts data, specifically PRM and NBR. The AIC is often used to evaluate approaches. AIC is a way to estimate the likelihood of a model that can forecast results depending on in-sample fits. As compared to another method, a model performs best if its AIC is the lowest.

Table 8. Descriptive statistics of COVID-19 (female) cases in Pakistan

The NBR has the smallest AIC and likelihood values, as stated by the consequence: (5,244.866, −2,618.433) in Punjab, (3,491.162, −1,741.581) in Baluchistan, (5,252.483, −2,622.242) in Sindh, (4,971.439, −2,481.719) in KPK and (19,337.828, −9,664.914) in Pakistan, and is a selected count model for modelling suspected, infected and recovered coronavirus epidemic occurrences in all Provinces of Pakistan.

Comparison of COVID-19 cases between male and female

The study analysed the effect of the COVID-19 death rates on gender. However, females have a better immune system as compared to males. So, male occurrences are more severe than female occurrences. Males with COVID-19 are at a higher risk for adverse outcomes and deaths. The statistical analysis of the COVID-19 data set shows that men suffer a significantly higher fatality rate than women.

Table 8 shows the mean and standard deviation of females at (Mean = 1,446.59 and SD = 1,368.817) for each suspected because of COVID-19. Same as the mean and standard-deviation of infected occurrences of coronavirus (Mean = 16,122.44 and SD = 16,824.680). The mean and standard deviation of the recovered occurrences of coronavirus in the study reveal that Mean = 13,840.42 and SD = 15,559.324. Thus, the mean and standard deviation of the death occurrences of COVID-19 are Mean = 345.26 and SD = 342.552. According to the findings thus far away, the large mean estimate of COVID-19-infected and recovered (female) cases in Pakistan during the period under analysis has resulted in comparatively large cases of COVID-19-associated deaths.

Table 9. Omnibus test of COVID-19 (female) cases in Pakistan

Table 9 shows that the explanatory factors (i.e., suspected, infected, and recovered occurrences of coronavirus) have a 0.000 p-value (i.e., p < 0.05), showing a statistically significant appropriate assessment in PRM and NB).

Table 10. Association of variables with model selection (PRM) of COVID-19 (female) cases in Pakistan

Table 10 demonstrates that the amount of deviance/df and Pearson chi-square/df are both more than ‘1.2’, implying that the number of COVID-19 (female) occurrences in Pakistan is over-dispersed, based on PRM. The analysis is performed by NBR to get the better solution as this approach may handle the dispersion parameter, and the resulting log-likelihood and AIC results are −66,865.403 and 133,738.807, respectively.

Table 11. Association of variables with model selection (NBR) of COVID-19 (female) cases in Pakistan

Table 11 indicates that the amounts of deviance/df and Pearson chi-square/df are both below ‘1.2’ (i.e., 1.036 and 0.775, respectively), implying that the amount of COVID−19 (female) occurrences in Pakistan are under-dispersion, having log-likelihood and AIC numbers −8,253.522 and 16,515.044, respectively.

Table 12. Multiple terms test between diagnose disease and respondence (female) in Pakistan (PRM)

Table 12 shows the p-value of all parameters are lower than 0.05, indicating the parameters $ {\beta}_1,{\beta}_2\;\mathrm{and}\;{\beta}_3 $ have a significant impact on results. Therefore, the PRM is given as

$$ {b}_l=\exp \left(4.615+0.0007{A}_1+0.00004415{A}_2-0.00001190{A}_3\right). $$

Table 13. Multiple terms test between diagnose disease and respondence (female) in Pakistan (NBR)

Explanatory factors which determine the coronavirus fatalities in Pakistan are the coronavirus suspected ( $ {A}_1 $ ), infected ( $ {A}_2 $ ) and recovered ( $ {A}_3 $ ) occurences.

Consequently, the NBR is given as

$$ {b}_l=\exp \left(3.997+0.0004569{A}_1+0.00001589{A}_2-0.00001538{A}_3\right). $$

Table 13 reveals that almost all factors are significant statistically at a 0.05 level of significance, and p-values of all paramaters are lower than 0.05. Except for recovered cases, the NBR model shows that two of the COVID-19 (female) factors are positive. This indicates that when using the NBR model to estimate the parameter, a 1% rise in suspected and infected COVID-19 (female) instances might result in a 0.05% and 0.002% higher in COVID-19-related deaths in Pakistan. COVID-19 patients that are suspected and infected have a direct positive impact on deaths related to COVID-19 in Pakistan. Consequently, at the 5% level, the recovered occurrences have a negative value (−0.00001538), which implies that a 1% rise in recovered coronavirus occurrences in Pakistan under the period of analysis may occur in a (−0.002%) reduction in deaths caused by COVID-19 in Pakistan, and the outcome often reveals that recovered occurrences may have a negative impact of the COVID-19 (female) deaths.

Table 14. Descriptive statistics of COVID-19 (male) cases in Pakistan

Table 14 shows that there is a mean and standard deviation of males at Mean = 3,719.74 and SD = 3,519.862 for each suspected because of coronavirus infection. Same as the mean and standard deviation of infected occurrences of coronavirus are Mean = 41,457.74 and SD = 43,263.461. The mean and standard deviation of the recovered occurrences of coronavirus disease upon the study reveals that Mean = 35,589.65 and SD = 40,009.649. Thus, the mean and standard deviation of the death occurrences of coronavirus infection are Mean = 887.79 and SD = 880.855. According to the findings thus far away, the large mean estimate of coronavirus infected and recover (male) occurrences in Pakistan during the period under analysis resulted in a comparatively large amount of COVID-19-associated deaths.

Table 15. Omnibus test of COVID-19 (male) cases in Pakistan

Table 15 shows that the explanatory factors (i.e., suspected, infected, and recovered occurrences of coronavirus) have a 0.000 p-value (i.e., p < 0.05), showing a statistically significant appropriate assessment in PRM and NBR.

Table 16. Association of variables with model selection (PRM)

Table 16 demonstrates that amount of Pearson chi-square/df and deviance/df are both more than ‘1.2’, implying that the number of COVID-19 (male) occurrences in Pakistan is over-dispersed, based on PRM. The analysis is performed by NBR to get a better solution as this approach may handle the dispersion parameter, and the resulting log-likelihood and AIC results are −165,882.729 and 331,773.459, respectively.

Table 17. Association of variables with model selection (NBR)

Table 17 indicates that amounts of deviance/df and Pearson chi-square/df are both below ‘1.2’ (i.e., 1.064 and 0.783), respectively, then it is implying that the number of COVID-19 (male) occurrences in Pakistan are under-dispersion, having log-likelihood and AIC numbers are (−9,488.507, and 18,985.015) respectively.

Table 18. Multiple terms test between diagnose disease and respondence (male) in Pakistan (PRM)

Table 18 shows the parameters of the p-values are lower than 0.05, indicating that the parameters $ {\beta}_1,{\beta}_2,\mathrm{and}\;{\beta}_3 $ have a significant impact on results. Consequently, the PRM is given as

$$ {b}_l=\exp \left(5.559+0.00087{A}_1+0.000005503{A}_2-0.000004631{A}_3\right). $$

Explanatory factors which determine the coronavirus fatalities in Pakistan are the coronavirus suspected occurrences ( $ {A}_1 $ ), infected occurrences ( $ {A}_2 $ ), and recovered occurrences ( $ {A}_3 $ ).

Table 19. Multiple terms test between diagnose disease and respondence (male) in Pakistan (NBR)

Therefore, the NBR is given as

$$ {b}_l=\exp \left(4.937+0.00067{A}_1+0.000002579{A}_2-0.000006080{A}_3\right). $$

Table 19 reveals that almost all the parameters are significant statistically at the 5% level of significance, and the parameters of the p-values are all less than 0.05. Apart from the recovered cases, the NBR model shows that two of the COVID-19 factors are positive. This indicates that when using the NBR model to estimate the parameter, a 1% rise in suspected and infected COVID-19 (male) instances might result in a 0.067% and 0.003% higher in COVID-19 related deaths (male) in Pakistan. COVID-19 patients that are suspected and infected have a direct positive impact on deaths related to COVID-19 in Pakistan. Consequently, at the 5% level, the recovered occurrences have a negative value (−0.000006080) which implies that a 1% rise in recovered COVID-19 occurrences in Pakistan under the period of analysis may occur in a (−0.0006%) reduction in deaths caused by COVID-19 in Pakistan, and the outcome often reveals that recovered occurrences may have a negative impact of the Covid19 (male) deaths.

Table 20. Selected count data model for COVID-19 cases between male and female

Table 20 shows that the NBR has the smallest AIC and log-likelihood values, as stated by the consequence: (18,985.015, −9,488.507) in males, respectively, while (11,515.044, −6,053.522) in females and is the selected count data model for modelling the suspected, infected and recovered Coronavirus widespread occurrences between males and females in Pakistan. Hence the resulted AIC and log-likelihood shows that males are more prone than females.