Correction: An SIRS model with nonmonotone incidence and saturated treatment in a changing environment

Correction: Journal of Mathematical Biology (2022) 85:23 10.1007/s00285-022-01787-3

Unfortunately, some equations were incorrect in the originally published article. The correct equations are given below.

Lemma 2.3

Let f(x) and ${\overline{x}}_2$ be given by (2.10) and (2.13), respectively. System (2.2) has at most two positive equilibria. Moreover,

• (I)
  when $0<{\mathcal{R}}_0<1$, we have

  • (i)
    if $n\le \frac{1+q+v+ms}{s}$, or $n>\frac{1+q+v+ms}{s}$ and $f({\overline{x}}_2)>0$, then system (2.2) has no positive equilibrium;
  • (ii)
    if $n>\frac{1+q+v+ms}{s}$ and $f({\overline{x}}_2)=0$, then system (2.2) has a unique positive equilibrium $E_{\ast }(x_{\ast },y_{\ast })$, which is degenerate and $0<x_{\ast }<n$, $y_{\ast }=q{x}_{\ast }+\frac{v{x}_{\ast }}{1+s{x}_{\ast }}$;
  • (iii)
    if $n>\frac{1+q+v+ms}{s}$ and $f({\overline{x}}_2)<0$, then system (2.2) has two positive equilibria $E_1(x_1,y_1)$ and $E_2(x_2,y_2)$, which are all elementary and $E_1$ is a hyperbolic saddle. $0<x_1<x_2<n$, $y_1=q{x}_1+\frac{v{x}_1}{1+s{x}_1}$, $y_2=q{x}_2+\frac{v{x}_2}{1+s{x}_2}$;
• (II)
  when ${\mathcal{R}}_0=1$, we have

  • (i)
    if $n\le \frac{1+q+v+ms}{s}$, then system (2.2) has no positive equilibrium;
  • (ii)
    if $n>\frac{1+q+v+ms}{s}$, then system (2.2) has a positive equilibrium $E_2(x_2,y_2)$;
• (III)when ${\mathcal{R}}_0>1$, then system (2.2) has a positive equilibrium $E_2(x_2,y_2)$.

where the expressions of $c_{02}$, $d_{12}$, $e_{22}$, $f_{20}$, $f_{30}$, $f_{40}$, $g_{20}$, $g_{21}$, $h_{20}$ and $h_{31}$ are given in supplementary materials. Then system (2.23) becomes (still denote ${\tau }_3$ by t)

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill \frac{d{X}_8}{\mathit{dt}}& =Y_8,\hfill \\ \hfill \frac{d{Y}_8}{\mathit{dt}}& ={\mu }_1+{\mu }_2{Y}_8+{\mu }_3{X}_8{Y}_8+X_8^2-X_8^3{Y}_8+R_3(X_8,Y_8,r),\hfill \end{array}\end{array}$$ 2.26

Table 3 $p=0.35$, $n=8.2$, $s=1.9$, $v=7$ and $q=1.236$

m	Positive equilibria and types	Closed orbits and homoclinic orbits
1.58	No	No (Fig. 6a)
1.569	$E_1$(saddle), $E_2$(unstable focus)	No (Fig. 6b)
1.563985	$E_1$(saddle), $E_2$(unstable focus)	A homoclinic orbit (Fig. 6c)
1.563956	$E_1$(saddle), $E_2$(unstable focus)	A stable limit cycle (Fig. 6d)
1.56389	$E_1$(saddle), $E_2$(stable focus)	A stable limit cycle, unstable limit cycle (Fig. 6e)
1.562	$E_1$(saddle), $E_2$(stable focus)	No (Fig. 6f)

$$\begin{array}{cc}\hfill \frac{\mathit{dx}}{\mathit{dt}}=& y+s_{20}{x}^2+s_{11}xy+s_{02}{y}^2+s_{30}{x}^3+s_{21}{x}^{2}y\hfill \\ \hfill & +s_{12}x{y}^2+s_{03}{y}^3+s_{40}{x}^4+s_{31}{x}^{3}y+s_{22}{x}^2{y}^2\hfill \\ \hfill & +s_{13}x{y}^3+s_{04}{y}^4+s_{50}{x}^5+s_{41}{x}^{4}y+s_{32}{x}^3{y}^2\hfill \\ \hfill & +s_{23}{x}^2{y}^3+s_{14}x{y}^4+s_{05}{y}^5+{o(|x,y|}^5),\hfill \\ \hfill \frac{\mathit{dy}}{\mathit{dt}}=& -x+r_{20}{x}^2+r_{11}xy+r_{02}{y}^2+r_{30}{x}^3\hfill \\ \hfill & +r_{21}{x}^{2}y+r_{12}x{y}^2+r_{03}{y}^3+r_{40}{x}^4+r_{31}{x}^{3}y+r_{22}{x}^2{y}^2\hfill \\ \hfill & +r_{13}x{y}^3+r_{04}{y}^4+r_{50}{x}^5+r_{41}{x}^{4}y+r_{32}{x}^3{y}^2\hfill \\ \hfill & +r_{23}{x}^2{y}^3+r_{14}x{y}^4+r_{05}{y}^5+{o(|x,y|}^5),\hfill \end{array}$$ 2.29

The original article has been corrected.