Table 1.

Fitting of different distributions for different infectious diseases.

S. no.	References	Conclusion drawn
1	Meyer and Held (62)	The authors have studied the short-time human travel behavior through power Law (Pareto, Uniform, Cauchy, etc.) concerning the distance. They used extended space-time models for influenza infectious disease surveillance data to better capture the dynamics of disease spread. They have studied the statistical properties of the best-fitted distribution for a better explanation and prediction of influenza.
2	Virlogeux et al. (63)	In this work a novel avian influenza virus, influenza A(H7N9) emerged in China was studied. The authors have fitted different parametric and non-parametric distribution for A(H7N9) incubation periods and studied the properties of the fitted distributions. The best fitted parametric distribution observed was Weibull distribution and the mean incubation period was 3.4 days with a 95% confidence interval [3.0 3.7] and the variance was 2.9 days. The results were very similar for the non-parametric Turnbull estimate as well.
3	Virlogeux et al. (64)	The authors studied Middle East Respiratory Syndrome coronavirus (MERS) disease in the Arabian Peninsula and in South Korea in 2015. They examined the incubation period distribution of MERS coronavirus infection using parametric (Lognormal, Gamma, Weibull, Exponential, Log-logistic) and non-parametric (turnbull) methods. They have shown that Gamma and Weibull are best-fitted distributions for South Korea while Lognormal and Log-logistic are the best fitted for Saudi Arabia and estimated a mean incubation period of 6.9 days with 95% credibility interval as [6.3 7.5] for cases in South Korea and 5.0 days with 95% credibility interval as [4.0 6.6] among cases in Saudi Arabia.
4	Hanel et al. (65)	The authors worked on the most standard methods based on maximum likelihood (ML) estimates of power-law function which is an exponential distribution. The best-fitted power function distribution based on the fitting measures was observed after that the appropriate ML estimator was derived for arbitrary exponents of power-law distributions on bounded discrete sample spaces. They had shown that a similar estimator was also working for continuous data. This ML estimator was implemented and its performance was compared with previous works. Further, a general protocol was given on how it could be used for estimating the spread of the infections.
5	Li et al. (66)	In this study, prediction and parameter estimation of infections were studied using noisy case reporting data. A simple stochastic, discrete-time, discrete-state epidemic model was established with both process and observation errors and was used to characterize the efficiency of different flavors of Bayesian Markov chain Monte Carlo (MCMC). They fitted different parametric distributions with ceilings (binomial and beta-binomial distributions) and without ceilings (Poisson and negative binomial) and the best-fitted distribution were studied for the statistical properties to explain and prediction of the nature of the infections.
6	De-Souza et al. (67)	The authors inferred that climate change has a high impact on governing the health and death rates due to respiratory system diseases and remained poorly understood by probability distribution modeling. They fitted the Burr, Inverse Gaussian, Lognormal, Pert, Rayleigh, and Weibull distributions to respiratory diseases, and the shape and scale parameters of the distributions were determined to verify the quality of fit through fitting measures. The lognormal and Rayleigh are best observed fit for hospital admissions.
7	Valvo (68)	The author studied the epidemiological model for the prediction of the time trends of COVID-19 deaths worldwide. They have taken a bimodal distribution function as a mixture of two lognormal distributions to model the time distribution of deaths in a country. They mentioned that an asymmetric lognormal distribution is better fitted in comparison to symmetric distribution functions. Based on the best model, they have further analyzed and predicted the future behavior of the spread of COVID-19 and was extrapolated until the end of the year 2020.
8	Vazquez (69)	The author has shown that infection spreads are expected to grow exponentially in time but their initial kinetics is not well understood. In this study, derivation of the analytical expressions was carried out for the kinetic behavior with a gamma distribution of generation intervals. Omitting the exponential distribution, the spread of the infection grows as a power law at short times. At long times, the kinetics is exponential with a growth rate estimated by the reproductive number and the parameters of the generation interval distribution. These kinetic derivations can be deployed to do better estimates of parameters used for infection spread.
9	El-Monsef (70)	The author has fitted finite mixture of m-Erlang distributions to analyze the COVID-19 dissemination. The author has derived different moments and shape parameters estimate for the suggested model and shown that it has a bound hazard function. A special case of the suggested distribution has also been discussed along with the predictive technique to estimate the parameters of the fitted distribution. In this fitted distribution, the data of the COVID-19 cases from Egypt was used to examine the flexibility of the proposed model.
10	Almetwally et al. (71)	The authors suggested a model for fitting the COVID 19 mortality rates in the UK and Canada using optimal statistical technique. They have suggested a new two-parameter lifetime distribution by combining inverted Topp-Leone (ITL) and modified Kies inverted Topp-Leone (MKITL) distributions. They have shown that the suggested model has various important properties as simple linear representation, hazard rate function, and moment function. They have used various methods of estimation for the estimation of parameters of the suggested distribution. They have shown through the data simulation study on COVID-19 cases that the suggested model is better than the traditional methods.
11	Mubarak and Almetwally (72)	The authors have introduced a new extended three-parameter exponential distribution and studied the survival function and hazard function. They have also used the maximum likelihood estimation (MLE) and maximum product spacing (MPS) methods for to evaluate the parameters of this distribution. An empirical study is carried out to judge the superiority of the suggested model over some well-known distributions using COVID-19 data and it was concluded that the suggested distribution is better fitted over competing distributions.
12	Gonçalves et al. (73)	In the presented work, authors have concluded that the inaccurate epidemiological concepts are being used during COVID-19 pandemic. They pointed out about social media and scientific journals regarding wrong references for “normal epidemic curve” and “log-normal curve/distribution” and the textbooks and courses of reputed institutions have spread slightly incorrect information. Most of them have shown histogram as epidemic curve or using epidemic data as Gaussian distribution, ignoring its property of temporal indexing. The authors have further observed that epidemic curve may be of Gaussian curve type and be modeled from Gauss function but it could not be a perfect normal distribution or a log-normal, as some of the previous studies have shown. Further, they have mentioned that a pandemic gives highly-complex data and to handle it effectively, there is need to go beyond the “one-size-fits-all solution” of statistical and mathematical modeling. Finally they suggested that the classical textbooks should be updated on pandemics and epidemiology should give reliable information to policy making and implementation.