Analysis of Covid-19 Virus Spreading Statistics by the Use of a New Modified Weibull Distribution

Abstract

Since the World Health Organization has declared Coronavirus a pandemic, researchers have given several interpretations on how this virus is spreads. In the present work, in anticipation of substantial fatal effects on health of people following this human-to-human spread, we aim to propose a new six parameter-modified Weibull distribution to analyze the spread of Covid-19 virus. We apply this model to study the cumulative cases infected in some countries, we give a global analysis of the statistical data of the pandemic, and we prove that our new distribution efficiently generalizes some existing models and fits correctly some data registered from February to June 2020. We use these results to assess the potential for human-to-human spread to occur around the globe.

Appendix

All the second partial derivatives of the log-likelihood function are obtained in this appendix as follows

$$\begin{aligned} {{\mathcal {L}}_{\alpha \alpha }}=-\sum \limits _{i=1}^{n}{\frac{h_{\alpha }^{2}}{{{h}^{2}}}}+2\lambda \sum \limits _{i=1}^{n}{t_{i}^{2\theta }}{{h}_{S\lambda }}, \end{aligned}$$

(74)

$$\begin{aligned} {{\mathcal {L}}_{\beta \beta }}=-\sum \limits _{i=1}^{n}{\frac{h_{\beta }^{2}}{{{h}^{2}}}}+2\lambda \sum \limits _{i=1}^{n}{t_{i}^{2\gamma }{{e}^{2\chi {{t}_{i}}}}}{{h}_{S\lambda }}, \end{aligned}$$

(75)

$$\begin{aligned} {{\mathcal {L}}_{\theta \theta }}=\sum \limits _{i=1}^{n}{\frac{\left( {{h}_{\theta \theta }}h-{{\alpha }^{2}}h_{\alpha \theta }^{2} \right) }{{{h}^{2}}}}-\alpha \sum \limits _{i=1}^{n}t_{i}^{\theta }{{{\ln }^{2}}\left( {{t}_{i}} \right) }-2\alpha \lambda \sum \limits _{i=1}^{n}t_{i}^{\theta }h_{S\lambda }^{\alpha }{\ln \left( {{t}_{i}} \right) }, \end{aligned}$$

(76)

$$\begin{aligned} {{\mathcal {L}}_{\gamma \gamma }}=\sum \limits _{i=1}^{n}{\frac{\left( {{h}_{\gamma \gamma }}h-{{\beta }^{2}}h_{\beta \gamma }^{2} \right) }{{{h}^{2}}}}-\beta \sum \limits _{i=1}^{n}{t_{i}^{\gamma }}{{e}^{\chi {{t}_{i}}}}{{\ln }^{2}}({{t}_{i}})-2\beta \lambda \sum \limits _{i=1}^{n}{t_{i}^{\gamma }{{e}^{\chi {{t}_{i}}}}}h_{S\lambda }^{\beta \chi }{{\ln }^{2}}({{t}_{i}}), \end{aligned}$$

(77)

$$\begin{aligned} {{\mathcal {L}}_{\chi \chi }}=\sum \limits _{i=1}^{n}{\frac{\left( {{h}_{\chi \chi }}h-h_{\beta \chi }^{2} \right) }{{{h}^{2}}}}-\beta \sum \limits _{i=1}^{n}{t_{i}^{\gamma +2}}{{e}^{\chi {{t}_{i}}}}-2\beta \lambda \sum \limits _{i=1}^{n}{t_{i}^{\gamma +2}{{e}^{\chi {{t}_{i}}}}}h_{S\lambda }^{\beta \chi }, \end{aligned}$$

(78)

$$\begin{aligned} {{\mathcal {L}}_{\lambda \lambda }}=-\sum \limits _{i=1}^{n}{h_{\lambda }^{2}}, \end{aligned}$$

(79)

$$\begin{aligned} {{\mathcal {L}}_{\theta \alpha }}=\sum \limits _{i=1}^{n}{\frac{\left( {{h}_{\alpha \theta }}h-\alpha {{h}_{\alpha \theta }}{{h}_{\alpha }} \right) }{{{h}^{2}}}}-\sum \limits _{i=1}^{n}{t_{i}^{\theta }\ln \left( {{t}_{i}} \right) }-2\lambda \sum \limits _{i=1}^{n}{t_{i}^{\theta }h_{S\lambda }^{\alpha },} \end{aligned}$$

(80)

$$\begin{aligned} {{\mathcal {L}}_{\beta \alpha }}=-\sum \limits _{i=1}^{n}{\frac{{{h}_{\alpha }}{{h}_{\beta }}}{{{h}^{2}}}+2\lambda \sum \limits _{i=1}^{n}{t_{i}^{\theta +\gamma }{{e}^{\chi {{t}_{i}}}}{{h}_{S\lambda }}}}, \end{aligned}$$

(81)

$$\begin{aligned} {{\mathcal {L}}_{\gamma \alpha }}=-\beta \sum \limits _{i=1}^{n}{\frac{{{h}_{\alpha }}{{h}_{\beta \gamma }}}{{{h}^{2}}}+2\beta \lambda \sum \limits _{i=1}^{n}{t_{i}^{\theta +\gamma }{{e}^{\chi {{t}_{i}}}}{{h}_{S\lambda }}\ln \left( {{t}_{i}} \right) }}, \end{aligned}$$

(82)

$$\begin{aligned} {{\mathcal {L}}_{\chi \alpha }}=-\sum \limits _{i=1}^{n}{\frac{{{h}_{\alpha }}{{h}_{\beta \chi }}}{{{h}^{2}}}+2\beta \lambda \sum \limits _{i=1}^{n}{t_{i}^{\theta +\gamma +1}{{e}^{\chi {{t}_{i}}}}{{h}_{S\lambda }}}}, \end{aligned}$$

(83)

$$\begin{aligned} {{\mathcal {L}}_{\lambda \alpha }}=-2\sum \limits _{i=1}^{n}{t_{i}^{\theta }\frac{{{e}^{S}}}{S_{\lambda }^{2}}}, \end{aligned}$$

(84)

$$\begin{aligned} {{\mathcal {L}}_{\theta \beta }}=-\alpha \sum \limits _{i=1}^{n}{\frac{{{h}_{\alpha \theta }}{{h}_{\beta }}}{{{h}^{2}}}+2\alpha \lambda \sum \limits _{i=1}^{n}{t_{i}^{\theta +\gamma }{{e}^{\chi {{t}_{i}}}}{{h}_{S\lambda }}\ln \left( {{t}_{i}} \right) }}, \end{aligned}$$

(85)

$$\begin{aligned} {{\mathcal {L}}_{\gamma \theta }}=-\alpha \beta \sum \limits _{i=1}^{n}{\frac{{{h}_{\alpha \theta }}{{h}_{\beta \gamma }}}{{{h}^{2}}}+2\alpha \beta \lambda \sum \limits _{i=1}^{n}{t_{i}^{\theta +\gamma }{{e}^{\chi {{t}_{i}}}}{{h}_{S\lambda }}{{\ln }^{2}}\left( {{t}_{i}} \right) }}, \end{aligned}$$

(86)

$$\begin{aligned} {{\mathcal {L}}_{\chi \theta }}=-\alpha \sum \limits _{i=1}^{n}{\frac{{{h}_{\alpha \theta }}{{h}_{\beta \chi }}}{{{h}^{2}}}+2\alpha \beta \lambda \sum \limits _{i=1}^{n}{t_{i}^{\theta +\gamma +1}{{e}^{\chi {{t}_{i}}}}{{h}_{S\lambda }}\ln \left( {{t}_{i}} \right) }}, \end{aligned}$$

(87)

$$\begin{aligned} {{\mathcal {L}}_{\lambda \theta }}=-2\alpha \sum \limits _{i=1}^{n}{t_{i}^{\theta }\frac{{{e}^{S}}}{S_{\lambda }^{2}}\ln \left( {{t}_{i}} \right) }, \end{aligned}$$

(88)

$$\begin{aligned} {{\mathcal {L}}_{\gamma \beta }}=\sum \limits _{i=1}^{n}{{{h}_{\beta \gamma }}\frac{\left( h-\beta {{h}_{\beta }} \right) }{{{h}^{2}}}}-\sum \limits _{i=1}^{n}{t_{i}^{\gamma }{{e}^{\chi {{t}_{i}}}}\ln \left( {{t}_{i}} \right) }-2\lambda \sum \limits _{i=1}^{n}{t_{i}^{\gamma }{{e}^{\chi {{t}_{i}}}}h_{S\lambda }^{\beta \chi }\ln \left( {{t}_{i}} \right) }, \end{aligned}$$

(89)

$$\begin{aligned} {{\mathcal {L}}_{\chi \beta }}=\sum \limits _{i=1}^{n}{\frac{\left( {{h}_{\chi \gamma }}h-{{h}_{\beta \chi }}{{h}_{\beta }} \right) }{{{h}^{2}}}}-\sum \limits _{i=1}^{n}{t_{i}^{\gamma +1}}{{e}^{\chi {{t}_{i}}}}-2\lambda \sum \limits _{i=1}^{n}{t_{i}^{\gamma +1}{{e}^{\chi {{t}_{i}}}}}h_{S\lambda }^{\beta \chi }, \end{aligned}$$

(90)

$$\begin{aligned} {{\mathcal {L}}_{\lambda \beta }}=-2\sum \limits _{i=1}^{n}{t_{i}^{\gamma }{{e}^{\chi {{t}_{i}}}}h_{S}^{\lambda }}, \end{aligned}$$

(91)

$$\begin{aligned} \begin{aligned}&{{\mathcal {L}}_{\chi \gamma }}=\beta \sum \limits _{i=1}^{n}{\frac{\left( h_{\beta \gamma }^{\chi }h-{{h}_{\beta \chi }}{{h}_{\beta \gamma }} \right) }{{{h}^{2}}}}-\beta \sum \limits _{i=1}^{n}{t_{i}^{\gamma +1}{{e}^{\chi {{t}_{i}}}}\ln \left( {{t}_{i}} \right) } \\&\,\,\,\,\,\,\,\,\,\,\,\,-2\beta \lambda \sum \limits _{i=1}^{n}{t_{i}^{\gamma +1}{{e}^{\chi {{t}_{i}}}}h_{S\lambda }^{\beta \chi }\ln \left( {{t}_{i}} \right) },\\ \end{aligned} \end{aligned}$$

(92)

$$\begin{aligned} {{\mathcal {L}}_{\lambda \gamma }}=-2\beta \sum \limits _{i=1}^{n}{t_{i}^{\gamma }{{e}^{\chi {{t}_{i}}}}h_{S}^{\lambda }\ln \left( {{t}_{i}} \right) }, \end{aligned}$$

(93)

and

$$\begin{aligned} {{\mathcal {L}}_{\lambda \chi }}=2\beta \lambda \sum \limits _{i=1}^{n}{t_{i}^{\gamma +1}{{e}^{\chi {{t}_{i}}}}\frac{{{e}^{S}}}{S_{\lambda }^{2}}}, \end{aligned}$$

(94)

where

$$\begin{aligned} {{h}_{\alpha }}=\theta t_{i}^{\theta -1}, \end{aligned}$$

(95)

$$\begin{aligned} {{h}_{S\lambda }}=\frac{\left( {{S}_{\lambda }}-2\lambda {{e}^{S}} \right) {{e}^{S}}}{S_{\lambda }^{2}}, \end{aligned}$$

(96)

$$\begin{aligned} {{h}_{\beta }}=\left( \gamma +\chi {{t}_{i}} \right) t_{i}^{\gamma -1}{{e}^{\chi {{t}_{i}}}}, \end{aligned}$$

(97)

$$\begin{aligned} {{h}_{\beta \gamma }}={{e}^{\chi {{t}_{i}}}}t_{i}^{\gamma -1}\left[ 1+\left( \gamma +\chi {{t}_{i}} \right) \ln \left( {{t}_{i}} \right) \right] , \end{aligned}$$

(98)

$$\begin{aligned} {{h}_{\chi \chi }}=\beta t_{i}^{\gamma +1}{{e}^{\chi {{t}_{i}}}}\left( 2+\gamma +\chi {{t}_{i}} \right) , \end{aligned}$$

(99)

$$\begin{aligned} {{h}_{\gamma \gamma }}=\beta {{e}^{\chi {{t}_{i}}}}t_{i}^{\gamma -1}\left[ 2+\left( \gamma +\chi {{t}_{i}} \right) \ln \left( {{t}_{i}} \right) \right] \ln \left( {{t}_{i}} \right) , \end{aligned}$$

(100)

$$\begin{aligned} {{h}_{\alpha \theta }}=t_{i}^{\theta -1}\left( 1+\theta \ln \left( {{t}_{i}} \right) \right) , \end{aligned}$$

(101)

$$\begin{aligned} {{h}_{\theta \theta }}=\alpha t_{i}^{\theta -1}\left( 2+\theta \ln \left( {{t}_{i}} \right) \right) \ln \left( {{t}_{i}} \right) , \end{aligned}$$

(102)

$$\begin{aligned} h_{S\lambda }^{p\chi }=\frac{\left[ {{S}_{\lambda }}-pt_{i}^{\gamma }{{e}^{\chi {{t}_{i}}}}\left( {{S}_{\lambda }}-2\lambda {{e}^{S}} \right) \right] {{e}^{S}}}{S_{\lambda }^{2}}, \end{aligned}$$

(103)

$$\begin{aligned} h_{S\lambda }^{p}=\frac{\left[ {{S}_{\lambda }}-pt_{i}^{\theta }\left( {{S}_{\lambda }}-2\lambda {{e}^{S}} \right) \right] {{e}^{S}}\ln \left( {{t}_{i}} \right) }{S_{\lambda }^{2}}, \end{aligned}$$

(104)

$$\begin{aligned} {{h}_{\beta \chi }}=\beta t_{i}^{\gamma }{{e}^{\chi {{t}_{i}}}}\left( 1+\gamma +\chi {{t}_{i}} \right) , \end{aligned}$$

(105)

$$\begin{aligned} {{h}_{\chi \gamma }}=t_{i}^{\gamma }{{e}^{\chi {{t}_{i}}}}\left[ 1+\left( \gamma +\chi {{t}_{i}} \right) \right] , \end{aligned}$$

(106)

$$\begin{aligned} {{h}_{\lambda }}=\frac{\left( {{S}_{\lambda }}-1 \right) }{\lambda {{S}_{\lambda }}}, \end{aligned}$$

(107)

$$\begin{aligned} {{h}_{S}}=\frac{\left( 2{{e}^{S}}-1 \right) {{e}^{S}}}{S_{\lambda }^{2}}, \end{aligned}$$

(108)

$$\begin{aligned} h_{S}^{\lambda }=\frac{\left[ {{S}_{\lambda }}-\lambda \left( 2{{e}^{S}}-1 \right) \right] {{e}^{S}}}{S_{\lambda }^{2}} \end{aligned}$$

(109)

and

$$\begin{aligned} h_{\beta \gamma }^{\chi }=t_{i}^{\gamma }{{e}^{\chi {{t}_{i}}}}\left[ 1+\ln \left( {{t}_{i}} \right) +\left( \gamma +\chi {{t}_{i}} \right) \ln \left( {{t}_{i}} \right) \right] . \end{aligned}$$

(110)