Table 2 Summary of existing models and proposed models.

1

FDP: G-O model¹

FCP: G-O model with constant time delay²⁵

\(m_d(t)=a(1-e^{-bt})\)

\(m_c(t)=a(1-e^{-b(t-c)})\)

2

FDP: G-O model

FCP: G-O model with time-dependent delay²⁵

\(m_d(t)=a(1-e^{-bt})\)

\(m_c(t)=a(1-(1+ct)e^{-bt})\)

3

FDP: G-O model

FCP: G-O model with exponential distributed delay²⁵

\(m_d(t)=a(1-e^{-bt})\)

\(m_c(t)=a(1-\frac{c}{c-b}e^{-bt}+\frac{b}{c-b}e^{-ct})\)

4

FDP: G-O model

FCP: G-O model with normally distributed time delay²⁶

\(m_d(t)=a(1-e^{-bt})\)

\(m_c(t)=-ae^{-bt+\mu b+b^2 {\sigma }^2/2}\big (\phi (t,b{\sigma }^2+\mu ,\sigma )-\phi (0,b{\sigma }^2+\mu ,\sigma )\big )+a(\phi (t,\mu ,\sigma )+\phi (0,\mu ,\sigma ))\)

5

FDP: G-O model

FCP: G-O model with gamma distributed time delay²⁵

\(m_d(t)=a(1-e^{-bt})\) \(m_c(t)=a\Gamma (t,\alpha ,\beta )-\frac{ae^{-bt}}{(1-b\beta )^\alpha }\Gamma (t,\alpha ,\frac{\beta }{1-b\beta })\)

6

FDP: Li et al.⁴² FCP: Li et al.⁴²

\(m_d(t)=\frac{a}{1-\alpha }\bigg (1-e^{-c(1-\alpha )t^r}\bigg )\) \(m_c(t)=\frac{a}{1-\alpha }\bigg (1-e^{-c(1-\alpha )t^r}\bigg )\frac{1}{1+be^{-\beta t}}\)

7

FDP: Li et al.⁴²

FCP: Li et al.⁴²

\(m_d(t)=\frac{a}{1-\alpha }\bigg (1-(1+rt)^{1-\alpha }e^{-r(1-\alpha )t}\bigg )\)

\(m_c(t)=\frac{a}{1-\alpha }\bigg (1-(1+rt)^{1-\alpha }e^{-r(1-\alpha )t}\bigg )\frac{1}{1+be^{-\beta t}}\)

8

FDP: Li et al.⁴²

FCP: Li et al.⁴²

\(m_d(t)=\frac{a}{1-\alpha }\Bigg (1-\bigg (\frac{(1+c)e^{-rt}}{1+ce^{-rt}}\bigg )^{1-\alpha }\Bigg )\)

\(m_c(t)=\frac{a}{1-\alpha }\Bigg (1-\bigg (\frac{(1+c)e^{-rt}}{1+ce^{-rt}}\bigg )^{1-\alpha }\Bigg )\frac{1}{1+be^{-\beta t}}\)

9

Proposed FDP model

Proposed FCP model

\(m_d(t)=\frac{pq\Big \{(1+bt)^p-e^{-(q+p(\alpha -1)bt)}(1+bt)^{q+p\alpha }\Big \}}{q(1+bt)^p-p(1-\alpha )e^{-(q+p(\alpha -1)bt)}(1+bt)^{q+p\alpha }}\)

\(m_c(t)= {\left\{ \begin{array}{ll} m_d(t)\Big (\frac{1}{1+\beta e^{-r t}}\Big ), & 0\le t \le \tau \\ m_d(t)\Big (\frac{1}{1+\gamma e^{-\sigma t}}\Big ), & t > \tau \end{array}\right. }\)