Asymptomatic Plasmodium falciparum infections may not be shortened by acquired immunity

Malaria Journal

Table 1 Mathematical models for clearance of infections

Survival distribution	Scale	Shape	Mean	Variance	PDF	CDF
Exponential	\(1/\mu > 0\)	-	\(1/\mu\)	\(1/\mu ^2\)	\(\mu e^{-\mu x}\)	\(1-e^{-\mu x}\)
Weibull	\(\lambda > 0\)	\(k > 0\)	\(\lambda \Gamma \left( 1+\frac{1}{k}\right)\)	\(\lambda ^2\Gamma \left( 1+\frac{2}{k}\right) - \mu ^2\)	\(\frac{k}{\lambda }\left( \frac{x}{\lambda }\right) ^{k-1}e^{-(x/\lambda )^{k}}\)	\(1- e^{-(x/\lambda )^k}\)
Log-Normal	\(\mu\)	\(\sigma > 0\)	\(e^{\mu +\sigma ^2/2}\)	\((e^{\sigma ^2}-1) e^{2\mu +\sigma ^2}\)	\(\frac{1}{x \sigma \sqrt{2 \pi }}e^{-\frac{(\ln x - \mu )^2}{2\sigma ^2}}\)	\(\frac{1}{2} + \frac{1}{2} \mathrm {erf}\left[ \frac{\ln x-\mu }{\sigma \sqrt{2}}\right]\)
Gamma	\(\theta > 0\)	\(k>0\)	\(k\theta\)	\(k\theta ^2\)	\(x^{k-1} \frac{\exp {\left( -x/\theta \right) }}{\Gamma (k)\,\theta ^k}\)	\(\frac{\gamma (k, x/\theta )}{\Gamma (k)}\)

Infection durations were modelled using parametric survival distributions. The exponential distribution is specified by a single scale parameter (the mean duration of infection). All the others have increased flexibility due to an additional shape parameter (distribution-specific parameter names are ignored). The following abbreviations are used in the table: for the gamma function \(\Gamma (z) = \int _0^\infty t^{z-1} e^{-t}\,dt\), for the lower incomplete gamma function \(\gamma (s,x) = \int _0^x t^{s-1}\,e^{-t}\,dt\), and for the error function \({\text {erf}}(x) = \frac{2}{\sqrt{\pi }}\int _0^x e^{-t^2} dt\).

ISSN: 1475-2875