Sufficient conditions for stability of linear differential equations with distributed delay (Q5936445)

The authors study the stability of the linear equation NEWLINE\[NEWLINE \dot{x}(t)=-\alpha x(t)- \beta\int_{0}^{\infty}x(t-\tau)f(\tau) d\tau, \tag{1}NEWLINE\]NEWLINE where \(\alpha\) and \(\beta\) are constants, \(\beta\geq|\alpha|\), \(f\) is the density of the distribution of maturation delays, i.e., \(f(\tau)\geq 0\) and \(\int_{0}^{\infty}f(\tau) d\tau=1\). A method to parametrically determine the boundary of the region of stability is developed. The authors obtain the following sufficient conditions for the stability based on the expectation \(E=\int_{0}^{\infty}\tau f(\tau) d\tau\) and the skewness \(B(f)\) of density \(f\) and prove the theorem: Suppose \(\beta>|\alpha|\). Then in equation (1): NEWLINENEWLINENEWLINE(i) the zero solution is asymptotically stable if \(E<\frac{\pi(1+\frac{\alpha}{\beta})} {c\sqrt{\beta^{2}-\alpha^{2}}}\), where \(c=\sup\{c:cos(x)=1-cx/\pi,x>0\}\approx 2.2764\); NEWLINENEWLINENEWLINE(ii) if the skewness \(B(f)=0\) then the stronger sufficient condition holds for the asymptotic stability: \(E<\frac{\arccos(-\alpha/\beta)} {\sqrt{\beta^{2}-\alpha^{2}}}\); NEWLINENEWLINENEWLINE(iii) if \(B(f)>0\) then there is a \(\delta>0\) such that the condition in (ii) holds for \(|\alpha|<\delta\). NEWLINENEWLINENEWLINEThe theorem is applied to a model for the peripheral regulation of neutrophil production. NEWLINENEWLINENEWLINEThe reviewed paper is written by well-known researchers, and it presents new and useful results for applied mathematicians dealt with mathematical modelling in biology and medicine, but the text of the paper contains some misprints and erroneous calculations.