Bifurcation analysis and global dynamics of a mathematical model of antibiotic resistance in hospitals

Abstract

Antibiotic-resistant bacteria have posed a grave threat to public health by causing a number of nosocomial infections in hospitals. Mathematical models have been used to study transmission dynamics of antibiotic-resistant bacteria within a hospital and the measures to control antibiotic resistance in nosocomial pathogens. Studies presented in Lipstich et al. (Proc Natl Acad Sci 97(4):1938–1943, 2000) and Lipstich and Bergstrom (Infection control in the ICU environment. Kluwer, Boston, 2002) have provided valuable insights in understanding the transmission of antibiotic-resistant bacteria in a hospital. However, their results are limited to numerical simulations of a few different scenarios without analytical analyses of the models in broader parameter regions that are biologically feasible. Bifurcation analysis and identification of the global stability conditions can be very helpful for assessing interventions that are aimed at limiting nosocomial infections and stemming the spread of antibiotic-resistant bacteria. In this paper we study the global dynamics of the mathematical model of antibiotic resistance in hospitals considered in Lipstich et al. (2000) and Lipstich and Bergstrom (2002). The invasion reproduction number ${{\mathcal {R}}}_{ar}$ of antibiotic-resistant bacteria is derived, and the relationship between ${{\mathcal {R}}}_{ar}$ and two control reproduction numbers of sensitive bacteria and resistant bacteria (${{\mathcal {R}}}_{sc}$ and ${{\mathcal {R}}}_{rc}$) is established. More importantly, we prove that a backward bifurcation may occur at ${{\mathcal {R}}}_{ar}=1$ when the model includes superinfection, which is not mentioned in Lipstich and Bergstrom (2002). More specifically, there exists a new threshold ${{\mathcal {R}}}_{ar}^c$, such that if ${{\mathcal {R}}}_{ar}^c<{{\mathcal {R}}}_{ar}<1$, then the system can have two positive interior equilibria, which leads to an interesting bistable phenomenon. This may have critical implications for controlling the antibiotic-resistance in a hospital.

Appendix

Proof of Proposition 2.1

Consider the directions of the orbits of system (1.2) on the boundary of $\Omega$.

Firstly, along the R-axis, it follows from the first equation of (1.2) that $S'(t)|_{S=0}=m\mu \ge 0$. Hence, either R-axis is an invariant line of system (1.2) ($m=0$), or the vector field of system (1.2) at each point on R-axis points to the interior of $\Omega$ ($m>0$).

Secondly, consider the boundary $S+R=1$. By system (1.2), we get that

$$\begin{aligned} \frac{d(S+R)}{dt}\Big |_{R=1-S}=-(1-m)\mu -\tau _2-\gamma -\tau _1 S<0. \end{aligned}$$

Therefore, any orbit starting from a point on the boundary $S+R=1$ enters the interior of $\Omega$ as t increases.

Note that S-axis is an invariant line of system (1.2). By the discussions above, we conclude that $\Omega$ is a positively invariant set. $\square$

Proof of Theorem 2.3

The Jacobian matrix at the resistance-free equilibrium $E_{0}^*$ is:

$$\begin{aligned} \left( \begin{array}{cc} -\sqrt{(\beta -\tau _{1}-\tau _{2}-\gamma -\mu )^{2}+4\beta m\mu } &{}\quad -\beta (1-c\sigma )S_0\\ 0 &{}\quad (\tau _{2}+\gamma +\mu +\sigma \beta S_{0})({\mathcal {R}}_{ar}-1) \end{array}\right) . \end{aligned}$$

Thus, $E_0^*$ has a negative eigenvalue $-\sqrt{(\beta -\tau _{1}-\tau _{2}-\gamma -\mu )^{2}+4\beta m\mu }$, and the other eigenvalue depends on ${\mathcal {R}}_{ar}$:

$$\begin{aligned} (\tau _{2}+\gamma +\mu +\sigma \beta S_{0})({\mathcal {R}}_{ar}-1). \end{aligned}$$

Obviously, if ${\mathcal {R}}_{ar}<1$, then $E_0^*$ is a l.a.s. node, and if ${\mathcal {R}}_{ar}>1$, then $E_0^*$ is an unstable saddle. When ${\mathcal {R}}_{ar}=1$, the eigenvalue dependent on ${\mathcal {R}}_{ar}$ equals to 0, and thus we need to apply center manifold theorem (Zhang et al. 1992) to judge the stability of $E_0^*$. First, translate $E_0^*$ to the origin, and do the affine transformation

$$\begin{aligned} (S,R)\rightarrow \left( S+\dfrac{\beta (1-\sigma )S_0 R}{\sqrt{(\beta -\tau _{1}-\tau _{2}-\gamma -\mu )^{2}+4\beta m\mu }},R\right) . \end{aligned}$$

Then, system (1.2) can be reduced to the norm form:

$$\begin{aligned} \dfrac{dS}{dt}= & {} -\sqrt{(\beta -\tau _{1}-\tau _{2}-\gamma -\mu )^{2}+4\beta m\mu }S+\mathcal F(S,R),\\ \dfrac{dR}{dt}= & {} {\mathcal {G}}(S,R), \end{aligned}$$

where

$$\begin{aligned} \mathcal F(S,R)= & {} -\beta S^2+\dfrac{\beta (1-c\sigma )[\beta -\tau _{1}-\tau _{2}-\gamma -\mu -\beta (1-c+c\sigma )S_0]SR}{\sqrt{(\beta -\tau _{1}-\tau _{2}-\gamma -\mu )^{2}+4\beta m\mu }}\\&+\dfrac{\beta ^2 c(1-\sigma )(1-c\sigma )[\tau _{1}+\tau _{2}+\gamma +\mu -\beta +\beta (1+c\sigma )S_0]S_0 R^2}{(\beta -\tau _{1}-\tau _{2}-\gamma -\mu )^{2}+4\beta m\mu },\\ {\mathcal {G}}(S,R)= & {} -\beta (1-c+c\sigma )SR\\&+\dfrac{\beta [(1-c)(\beta -\tau _{1}-\tau _{2}-\gamma -\mu )-\beta (1-c-c^2\sigma +c^2\sigma ^2)S_0]R^2}{\sqrt{(\beta -\tau _{1}-\tau _{2}-\gamma -\mu )^{2}+4\beta m\mu }} \end{aligned}$$

satisfy

$$\begin{aligned} \mathcal F(0,0)=0,\quad {\mathcal {G}}(0,0)=0,\quad D\mathcal F(0,0)=\mathrm {O},\quad D{\mathcal {G}}(0,0)=\mathrm {O}. \end{aligned}$$

It follows from the center manifold theorem that there exists a locally invariant manifold $S=\mathcal H(R)\in \mathcal {C}^{2}$, such that all the solutions on this manifold have the property:

$$\begin{aligned} \dfrac{dR}{dt}={\mathcal {G}}(\mathcal H(R),R)=h_2 R^2+h_3 R^3+O(|R|^4). \end{aligned}$$

Here

$$\begin{aligned} h_2= & {} \dfrac{\beta [(1-c)(\beta -\tau _{1}-\tau _{2}-\gamma -\mu )-\beta (1-c-c^2\sigma +c^2\sigma ^2)S_0]}{\sqrt{(\beta -\tau _{1}-\tau _{2}-\gamma -\mu )^{2}+4\beta m\mu }},\\ h_3= & {} \dfrac{\beta ^3 c(1-\sigma )(1-c\sigma )(1-c+c\sigma )[\beta -\tau _{1}-\tau _{2}-\gamma -\mu -\beta (1+c\sigma )S_0]S_0}{(\sqrt{(\beta -\tau _{1}-\tau _{2}-\gamma -\mu )^{2}+4\beta m\mu })^3}. \end{aligned}$$

Using

$$\begin{aligned} S_0=\frac{(1-c)({\mathcal {R}}_{rc}-1)}{(1-c+c\sigma ){\mathcal {R}}_{rc}}, \end{aligned}$$

which comes from ${\mathcal {R}}_{ar}=1$, we get

$$\begin{aligned} h_2= & {} \dfrac{\beta [\beta (1-c)\sigma (1-\sigma )c^2({\mathcal {R}}_{rc}-1)^2-m\mu (1-c+\sigma c)^2{\mathcal {R}}_{rc}^2]}{(1-c+\sigma c){\mathcal {R}}_{rc}({\mathcal {R}}_{rc}-1)\sqrt{(\beta -\tau _{1}-\tau _{2}-\gamma -\mu )^{2}+4\beta m\mu }}\\= & {} \dfrac{\beta [c\sqrt{\beta (1-c)\sigma (1-\sigma )}({\mathcal {R}}_{rc}-1)+(1-c+\sigma c)\sqrt{m\mu }{\mathcal {R}}_{rc}]f({\mathcal {R}}_{rc})}{(1-c+\sigma c){\mathcal {R}}_{rc}({\mathcal {R}}_{rc}-1)\sqrt{(\beta -\tau _{1}-\tau _{2}-\gamma -\mu )^{2}+4\beta m\mu }},\\ \end{aligned}$$

where $f({\mathcal {R}}_{rc})$ is defined in (2.8). Obviously, $h_2\le 0({>}0)$ iff $f({\mathcal {R}}_{rc})\le 0({>}0)$.

When $h_2\ne 0$, i.e., $f({\mathcal {R}}_{rc})\ne 0$, $E_0^*$ is a saddle-node. However, we are only interested in the directions of the trajectories near $E_0^*$ in the region $\Omega$ (This is because that $E_0^*$ is a boundary equilibrium). Therefore, we get that when $f({\mathcal {R}}_{rc})>0$, $E_0^*$ is unstable, when $f({\mathcal {R}}_{rc})<0$, $E_0^*$ is l.a.s., and when $f({\mathcal {R}}_{rc})=0$, $h_3$ is simplified as

$$\begin{aligned} h_3=-\dfrac{\beta ^4c^2\sigma (1-\sigma )(1-c+c\sigma )(1-c\sigma )^2S_0^2}{(1-c)(\sqrt{(\beta -\tau _{1}-\tau _{2}-\gamma -\mu )^{2}+4\beta m\mu })^3}<0, \end{aligned}$$

and thus, $E_0^*$ is a l.a.s. node.

Proof of Theorem 2.8

(a) When ${\mathcal {R}}_{ar}>1$, there are two cases.

If $c\sigma (1-\sigma )=0$, then $\mathrm {det}J(E^{*})=\beta (1-c)m\mu R^{*}/S^{*}>0$. With (2.22) and (2.23), we get that Jacobian matrix $J(E^{*})$ has two negative eigenvalues, i.e., the unique interior equilibrium is a node and it is l.a.s..

If $c\sigma (1-\sigma )>0$, then the unique zero of $\Phi (S)$ in $(0,a^{*})$ is $S^{*}=(-\phi _{1}-\sqrt{\Delta })/(2\phi _{2})$. Thus

$$\begin{aligned} \mathrm {det}J(E^{*})=\frac{\beta R^{*}(\phi _{0}-\phi _{2}S^{*2})}{S^{*}}=-\dfrac{\sqrt{\Delta }(\phi _{1}+\sqrt{\Delta })}{2\phi _{2}}\cdot \frac{\beta R^{*}}{S^{*}}=\beta R^{*}\sqrt{\Delta }>0. \end{aligned}$$

The same conclusion can be obtained using (2.22) and (2.23).

(b)  (i) In this case, $\Phi (S)$ has two zeros $S_{1}^{*}=(-\phi _{1}-\sqrt{\Delta })/(2\phi _{2})$ and $S_{2}^{*}=(-\phi _{1}+\sqrt{\Delta })/(2\phi _{2})$. Denote the corresponding interior equilibria by $E_1^*(S_1^*,R_1^*)$ and $E_2^*(S_2^*,R_2^*)$, respectively. Thus,

$$\begin{aligned} \mathrm {det}J(E_1^{*})=\frac{\beta R_1^{*}(\phi _{0}-\phi _{2}S_1^{*2})}{S_1^{*}}=\beta R_1^{*}\sqrt{\Delta }>0, \end{aligned}$$

and

$$\begin{aligned} \mathrm {det}J(E_2^{*})=\frac{\beta R_2^{*}(\phi _{0}-\phi _{2}S_2^{*2})}{S_2^{*}}=-\beta R_2^{*}\sqrt{\Delta }<0. \end{aligned}$$

We have $E_{1}^{*}$ is a l.a.s. node, and $E_{2}^{*}$ is an unstable saddle by (2.22) and (2.23).

(b)  (ii) Here $\Phi (S)$ has a zero $S^{*}=-\phi _{1}/(2\phi _{2})$ in $(0,a^{*})$ with multiplicity 2. Further calculus shows that

$$\begin{aligned} \mathrm {det}J(E^{*})=\dfrac{\beta R^{*}(4\phi _{0}\phi _{2}-\phi _{1}^{2})}{4\phi _{2}S^{*}}=0. \end{aligned}$$

Therefore, $\mathrm {det}J(E^{*})$ has an eigenvalue equal to zero. We will determine the stability by using the center manifold theory to system (1.2). Translate $E^{*}$ to the origin and take the affine transformation

$$\begin{aligned}&(S,R)\rightarrow \left( \dfrac{(1-c+c\sigma )S+(1-c)R}{(1-c\sigma )(1-c+c\sigma )S^{*}+(1-c)^{2}R^{*}},\right. \\&\left. \dfrac{-(1-c)R^{*}S+(1-c\sigma )S^{*}R}{(1-c\sigma )(1-c+c\sigma )S^{*}+(1-c)^{2}R^{*}}\right) . \end{aligned}$$

Then system (1.2) is reduced to the norm form:

$$\begin{aligned} \dfrac{dS}{dt}= & {} (-\beta S^{*}-m\mu /S^{*}-\beta (1-c)R^{*})S+F(S,R),\\ \dfrac{dR}{dt}= & {} G(S,R), \end{aligned}$$

where F(S, R) and G(S, R) are homogenous quadratic polynomials and thus they satisfy

$$\begin{aligned} F(0,0)=0,\quad G(0,0)=0,\quad DF(0,0)=\mathrm {O},\quad DG(0,0)=\mathrm {O}. \end{aligned}$$

It follows from the center manifold theorem (Zhang et al. 1992) that there exists a locally invariant manifold $S=h(R)\in \mathcal {C}^{2}$ such that all the solutions on this center manifold have the property

$$\begin{aligned} \dfrac{dR}{dt}=-\dfrac{\beta \sigma (1-\sigma )c^{2}(1-c)^{2}R^{*}}{(1-c\sigma )(1-c+c\sigma )S^{*}+(1-c)^{2}R^{*}}R^{2}+O(|E|^{3}). \end{aligned}$$

(4.1)

Since the coefficient of $R^{2}$ in (4.1) is negative, we have $E^{*}$ is a saddle-node and it is unstable.

(b) (iii) In this case $\Delta =(\phi _1+2\phi _2a^*)^2>0$ and $\Phi (S)$ has a unique zero $S^{*}=(-\phi _{1}-\sqrt{\Delta })/(2\phi _{2})$ in $(0,a^{*})$. Hence

$$\begin{aligned} \mathrm {det}J(E^{*})=\frac{\beta R^{*}(\phi _{0}-\phi _{2}S^{*2})}{S^{*}}=-\dfrac{\sqrt{\Delta }(\phi _{1}+\sqrt{\Delta })}{2\phi _{2}}\cdot \frac{\beta R^{*}}{S^{*}}=\beta R^{*}\sqrt{\Delta }>0. \end{aligned}$$

In the same way, the unique interior equilibrium is a l.a.s. node. $\square$

Global dynamics of system (1.2) for the case $m=0$.

First, we give the existence theorem of equilibria.

Theorem 4.1

Let ${\mathcal {R}}_{sc}$ and ${\mathcal {R}}_{rc}$ be the control reproduction numbers defined in (2.2) and (2.3), respectively.

• (a) When ${\mathcal {R}}_{sc}\le 1$ and ${\mathcal {R}}_{rc}\le 1$, system (1.2) only has one boundary equilibrium O(0, 0), which is the unique disease-free equilibrium;

• (b) when ${\mathcal {R}}_{sc}>1$ and ${\mathcal {R}}_{rc}\le 1$, system (1.2) has a disease-free equilibrium O(0, 0) and a resistance-free equilibrium $E_0(S_0,0)$;

• (c) when ${\mathcal {R}}_{sc}\le 1$ and ${\mathcal {R}}_{rc}>1$, system (1.2) has a disease-free equilibrium O(0, 0) and a resistant equilibrium $U_0(0,R_0)$;

• (d) when ${\mathcal {R}}_{sc}>1$ and ${\mathcal {R}}_{rc}>1$, system (1.2) always has a disease-free equilibrium O(0, 0), a resistance-free equilibrium $E_0(S_0,0)$, and a resistant equilibrium $U_0(0,R_0)$, in addition,

  • (i) if $c\sigma (1-\sigma )>0$, $(1-c+c\sigma ){\mathcal {R}}_{rc}/(1-c+c\sigma {\mathcal {R}}_{rc})<{\mathcal {R}}_{sc}<{\mathcal {R}}_{rc}/(1-c\sigma +c\sigma {\mathcal {R}}_{rc})$, then system (1.2) has a unique interior equilibrium $E_1(S_1,R_1)$,

  • (ii) if $c\sigma =0$ and ${\mathcal {R}}_{sc}={\mathcal {R}}_{rc}$, then system (1.2) has a singular line $S_0-R-S=0$; and if $\sigma =1$ and ${\mathcal {R}}_{sc}={\mathcal {R}}_{rc}/(1-c+c{\mathcal {R}}_{rc})$, then system (1.2) has a singular line $S_0-(1-c)R-S=0$,

  where

  $$\begin{aligned} S_0= & {} 1-\dfrac{1}{{\mathcal {R}}_{sc}},\quad \,\,\, R_0=1-\dfrac{1}{{\mathcal {R}}_{rc}},\nonumber \\ S_1= & {} \dfrac{(1-c)(-S_0+(1-c\sigma )R_0)}{c^2\sigma (1-\sigma )},\nonumber \\ R_1= & {} \dfrac{(1-c+c\sigma )S_0-(1-c)R_0}{c^2\sigma (1-\sigma )}. \end{aligned}$$

  (4.2)

Fig. 9

The phase portraits of system (1.2) when $m=0$. a Case (a), b case (b), c case (c), d cases d(iv) and d(vi), e cases d(iii) and d(v), f case d(i), g case d(ii)

Next, we determine the stability of the equilibria described in Theorem 4.1.

Theorem 4.2

Consider the equilibria described in Theorem 4.1 , the following results hold.

• (a) When ${\mathcal {R}}_{sc}\le 1$ and ${\mathcal {R}}_{rc}\le 1$, the disease-free equilibrium O(0, 0) is locally asymptotically stable (l.a.s.);

• (b) when ${\mathcal {R}}_{sc}>1$ and ${\mathcal {R}}_{rc}\le 1$, the disease-free equilibrium O(0, 0) is unstable, and the resistance-free equilibrium $E_0(S_0,0)$ is l.a.s.;

• (c) when ${\mathcal {R}}_{sc}\le 1$ and ${\mathcal {R}}_{rc}>1$, the disease-free equilibrium O(0, 0) is unstable, and the resistant equilibrium $U_0(0,R_0)$ is l.a.s.;

• (d) when ${\mathcal {R}}_{sc}>1$ and ${\mathcal {R}}_{rc}>1$, the disease-free equilibrium O(0, 0) is unstable,

  • (i) if $c\sigma (1-\sigma )>0$ and $(1-c+c\sigma ){\mathcal {R}}_{rc}/(1-c+c\sigma {\mathcal {R}}_{rc})<{\mathcal {R}}_{sc}<{\mathcal {R}}_{rc}/(1-c\sigma +c\sigma {\mathcal {R}}_{rc})$, then both the resistance-free equilibrium $E_0(S_0,0)$ and the resistant equilibrium $U_0(0,R_0)$ are l.a.s., and the unique interior equilibrium $E_1(S_1,R_1)$ is unstable,

  • (ii) if $c\sigma =0$ and ${\mathcal {R}}_{sc}={\mathcal {R}}_{rc}$, then all the solution orbits of system (1.2) converge to the singular line $S_0-R-S=0$, and if $\sigma =1$ and ${\mathcal {R}}_{sc}={\mathcal {R}}_{rc}/(1-c+c{\mathcal {R}}_{rc})$, then all the solution orbits of system (1.2) converge to the singular line $S_0-(1-c)R-S=0$,

  • (iii) if ${\mathcal {R}}_{sc}<(1-c+c\sigma ){\mathcal {R}}_{rc}/(1-c+c\sigma {\mathcal {R}}_{rc})$, then the resistance-free equilibrium $E_0(S_0,0)$ is unstable, and the resistant equilibrium $U_0(0,R_0)$ is l.a.s.,

  • (iv) if ${\mathcal {R}}_{sc}>{\mathcal {R}}_{rc}/(1-c\sigma +c\sigma {\mathcal {R}}_{rc})$, then the resistance-free equilibrium $E_0(S_0,0)$ is l.a.s., and the resistant equilibrium $U_0(0,R_0)$ is unstable,

  • (v) if $c\sigma (1-\sigma )>0$ and ${\mathcal {R}}_{sc}=(1-c+c\sigma ){\mathcal {R}}_{rc}/(1-c+c\sigma {\mathcal {R}}_{rc})$, then the resistance-free equilibrium $E_0(S_0,0)$ is unstable and the resistant equilibrium $U_0(0,R_0)$ is l.a.s.,

  • (vi) if $c\sigma (1-\sigma )>0$ and ${\mathcal {R}}_{sc}={\mathcal {R}}_{rc}/(1-c\sigma +c\sigma {\mathcal {R}}_{rc})$, then the resistance-free equilibrium $E(S_0,0)$ is l.a.s. and the resistant equilibrium $U_0(0,R_0)$ is unstable,

  where $S_0,R_0,S_1$ and $R_1$ are given by (4.2).

By Proposition 2.1, Theorems 4.1, 4.2 and Proposition 2.10, we obtain the global dynamics of system (1.2) for the case $m=0$, see Fig. 9. The phase portraits correspond to the cases of Theorem 4.2.