# American Institute of Mathematical Sciences

• Previous Article
On the existence of axisymmetric traveling fronts in Lotka-Volterra competition-diffusion systems in ℝ3
• DCDS-B Home
• This Issue
• Next Article
Coexistence and competitive exclusion in an SIS model with standard incidence and diffusion
May 2017, 22(3): 1145-1166. doi: 10.3934/dcdsb.2017056

## Individual based models and differential equations models of nosocomial epidemics in hospital intensive care units

 Department of Mathematics, Vanderbilt University, Nashville, TN 37240, USA

This paper is an invited contribution to the special issue in honor of Steve Cantrell

Received  September 2015 Revised  December 2016 Published  January 2017

Mathematical models of antibiotic resistant infection epidemics in hospital intensive care units are developed with two modeling methods, individual based models and differential equations based models. Both models dynamically track uninfected patients, patients infected with a nonresistant bacterial strain not on antibiotics, patients infected with a nonresistant bacterial strain on antibiotics, and patients infected with a resistant bacterial strain. The outputs of the two modeling methods are shown to be complementary with respect to a common parameterization, which justifies the differential equations modeling approach for very small patient populations present in an intensive care unit. The model outputs are classified with respect to parameters to distinguish the extinction or endemicity of the bacterial strains. The role of stewardship of antibiotic use is analyzed for mitigation of these nosocomial epidemics.

Citation: Glenn F. Webb. Individual based models and differential equations models of nosocomial epidemics in hospital intensive care units. Discrete & Continuous Dynamical Systems - B, 2017, 22 (3) : 1145-1166. doi: 10.3934/dcdsb.2017056
##### References:

show all references

##### References:
Schematic diagram of the IBM patient compartments and parameters. All exiting patients are replaced immediately by an uninfected patient
An example of the IBM with parameters $N_H = 4$ $N_P$ = 10, $T_V = 4 hr$ $N_V = 2$, $\omega_{Noff}=0.9$, $\omega_{Non}=0.9$, $\omega_R=0.9$, $\pi_N=0.9$, $\pi_R=0.9$, $\beta_{on}=0.6$, $\beta_{off}=0.6$, $\mu_{Noff}=0.25$, $\mu_{on}=0.2$, $\mu_R=0.1667$
Example 2 of the IBM with parameters $N_H = 10,$ $N_P = 30$, $N_V = 16$, $\omega_{Noff}=0.1$, $\omega_{Non}=0.1$, $\omega_R=0.2$, $\pi_N=0.1$, $\pi_R=0.2$, $\beta_{on}=0.3$, $\beta_{off}=0.1$, $\mu_{Noff}=0.25$, $\mu_{on}=0.2$, $\mu_R=0.1667$. All three infected patient compartments extinguish
Graphs of 50 runs of the IBM with parameters and initial conditions in Example 2. The thicker curves represent the averages of the 50 runs in each compartment. The averages of the simulations approach extinction in 30 days, but some individual runs do not
Example 3 of the IBM with parameters $N_H = 10,$ $N_P = 30$, $N_V = 16$, $\omega_{Noff}=0.3$, $\omega_{Non}=0.1$, $\omega_R=0.2$, $\pi_N=0.2$, $\pi_R=0.2$, $\beta_{on}=0.4$, $\beta_{off}=0.1$, $\mu_{Noff}=0.25$, $\mu_{on}=0.2$, $\mu_R=0.1667$. Only the resistant strain is prevalent after 30 days
Graphs of 50 runs of the IBM with parameters and initial conditions in Example 3. The thicker curves represent the averages of the 50 runs in each compartment. The average of the resistant patient populations approaches extinction over 30 days, but the averages of the nonresistant patient populations do not
Example 4 of the IBM with parameters $N_H = 10,$ $N_P = 30$, $N_V = 16$, $\omega_{Noff}=0.2$, $\omega_{Non}=0.2$, $\omega_R=0.2$, $\pi_N=0.2$, $\pi_R=0.3$, $\beta_{on}=0.7$, $\beta_{off}=0.03$, $\mu_{Noff}=0.25$, $\mu_{on}=0.2$, $\mu_R=0.1$. Both the nonresistant and resistant strains are prevalent after 30 days
Graphs of 50 runs of the IBM with parameters and initial conditions in Example 4. The thicker curves represent the averages of the 50 runs in each compartment. The averages of both nonresistant and resistant strains are prevalent after 30 days
The DEM in the case that both strains extinguish. The parameters match the parameters for the IBM in Example 2: $N_P=30$, $N_H=10$, $N_V=16$, $\omega_{Noff}=0.1$, $\omega_{Non}=0.1$, $\omega_R=0.2$, $\mu_{Noff}=0.25=1/T_{Noff}$, $\mu_{Non}=0.2=1/T_{Non}$, $\mu_R=0.1667=1/T_R$, $\pi_N=0.1$, $\pi_R=0.2$, $\beta_{on}=0.3$, $\beta_{off}=0.1$
Phase portrait of the DEM trajectories $(PN_{off}(t), PN_{on}(t), PR(t))$ for an array of initial conditions with the parameters in Example 5. The thick black curve corresponds to the graphs in Fig. 9. Trajectories oscillate as they converge to $(0, 0, 0)$. An initial increase or decrease in infected patient populations could be misinterpreted as a long-term trend
The DEM in the case that only the nonresistant strain extinguishes. The parameters match the parameters for the IBM in Example 3: $N_P=30$, $N_H=10$, $N_V=16$, $\omega_{Noff}=0.3$, $\omega_{Non}=0.1$, $\omega_R=0.2$, $\mu_{Noff}=0.25=1/T_{Noff}$, $\mu_{Non}=0.2=1/T_{Non}$, $\mu_R=0.1667=1/T_R$, $\pi_N=0.2$, $\pi_R=0.2$, $\beta_{on}=0.4$, $\beta_{off}=0.1$
Phase portrait of the DEM trajectories $(PN_{off}(t), PN_{on}(t), PR(t))$ for an array of initial conditions with the parameters in Example 6. The thick black curve corresponds to the graphs in Fig. 11. All trajectories converge to $(5.56098, 11.122, 0)$
The DEM in the case that both strains become endemic. The parameters match the parameters for the IBM in Example 4: $N_P=30$, $N_H=10$, $N_V=16$, $\omega_{Noff}=0.2$, $\omega_{Non}=0.2$, $\omega_R=0.2$, $\mu_{Noff}=0.25=1/T_{Noff}$, $\mu_{Non}=0.2=1/T_{Non}$, $\mu_R=0.1=1/T_R$, $\pi_N=0.2$, $\pi_R=0.3$, $\beta_{on}=0.7$, $\beta_{off}=0.03$
Phase portrait of the DEM trajectories $(PN_{off}(t), PN_{on}(t), PR(t))$ for an array of initial conditions with the parameters in Example 7. The thick black curve corresponds to the graphs in Fig. 13. All trajectories converge with oscillations to the same limiting value
Graph of $R_{01}$ as a function of $\beta_{off}$ and $\pi_R$. All other parameters are as in Example 7. $R_{01}$ increases linearly with increasing $\pi_R$, but nonlinearly with increasing $\beta_{off}$. Consequently, there is advantage in stopping AB use in patients infected with the resistant strain as soon as possible. The yellow dot corresponds to the parameters in Example 7. The red plane corresponds to $R_{01}=1.0$
Graph of $R_{01}$ as a function of $T_{Non}$ and $T_R$. All other parameters are as in Example 7. $R_{01}$ increases sub-linearly with increasing $T_{Non}$ and $T_R$. Consequently, there is greater effect in reducing the LOS of patients infected with the nonresistant strain on AB and patients infected with the resistant strain (also on AB) as soon as possible. The yellow dot corresponds to the parameters in Example 7. The red plane corresponds to $R_{01}=1.0$
Graph of $R_{01}$ as a function of $T_V / 48$ days = visit time intervals in hours and $N_P$. All other parameters are as in Example 7. $R_{01}$ increases super-linearly with decreasing $T_V$ and $N_P$. Increasing either $T_V$ or $N_P$ results in fewer patient-HCW visits. The yellow dot corresponds to the parameters in Example 7. The red plane corresponds to $R_{01}=1.0$
 [1] Deqiong Ding, Wendi Qin, Xiaohua Ding. Lyapunov functions and global stability for a discretized multigroup SIR epidemic model. Discrete & Continuous Dynamical Systems - B, 2015, 20 (7) : 1971-1981. doi: 10.3934/dcdsb.2015.20.1971 [2] Robert E. Beardmore, Rafael Peña-Miller. Rotating antibiotics selects optimally against antibiotic resistance, in theory. Mathematical Biosciences & Engineering, 2010, 7 (3) : 527-552. doi: 10.3934/mbe.2010.7.527 [3] Hermann Brunner, Chunhua Ou. On the asymptotic stability of Volterra functional equations with vanishing delays. Communications on Pure & Applied Analysis, 2015, 14 (2) : 397-406. doi: 10.3934/cpaa.2015.14.397 [4] Michael Schönlein. Asymptotic stability and smooth Lyapunov functions for a class of abstract dynamical systems. Discrete & Continuous Dynamical Systems - A, 2017, 37 (7) : 4053-4069. doi: 10.3934/dcds.2017172 [5] Nobuyuki Kato. Linearized stability and asymptotic properties for abstract boundary value functional evolution problems. Conference Publications, 1998, 1998 (Special) : 371-387. doi: 10.3934/proc.1998.1998.371 [6] Janusz Mierczyński, Wenxian Shen. Formulas for generalized principal Lyapunov exponent for parabolic PDEs. Discrete & Continuous Dynamical Systems - S, 2016, 9 (4) : 1189-1199. doi: 10.3934/dcdss.2016048 [7] Yukihiko Nakata, Yoichi Enatsu, Yoshiaki Muroya. On the global stability of an SIRS epidemic model with distributed delays. Conference Publications, 2011, 2011 (Special) : 1119-1128. doi: 10.3934/proc.2011.2011.1119 [8] Zhen Jin, Zhien Ma. The stability of an SIR epidemic model with time delays. Mathematical Biosciences & Engineering, 2006, 3 (1) : 101-109. doi: 10.3934/mbe.2006.3.101 [9] Michele L. Joyner, Cammey C. Manning, Whitney Forbes, Michelle Maiden, Ariel N. Nikas. A physiologically-based pharmacokinetic model for the antibiotic ertapenem. Mathematical Biosciences & Engineering, 2016, 13 (1) : 119-133. doi: 10.3934/mbe.2016.13.119 [10] Mudassar Imran, Hal L. Smith. A model of optimal dosing of antibiotic treatment in biofilm. Mathematical Biosciences & Engineering, 2014, 11 (3) : 547-571. doi: 10.3934/mbe.2014.11.547 [11] Linda J. S. Allen, B. M. Bolker, Yuan Lou, A. L. Nevai. Asymptotic profiles of the steady states for an SIS epidemic reaction-diffusion model. Discrete & Continuous Dynamical Systems - A, 2008, 21 (1) : 1-20. doi: 10.3934/dcds.2008.21.1 [12] Wanbiao Ma, Yasuhiro Takeuchi. Asymptotic properties of a delayed SIR epidemic model with density dependent birth rate. Discrete & Continuous Dynamical Systems - B, 2004, 4 (3) : 671-678. doi: 10.3934/dcdsb.2004.4.671 [13] Chufen Wu, Peixuan Weng. Asymptotic speed of propagation and traveling wavefronts for a SIR epidemic model. Discrete & Continuous Dynamical Systems - B, 2011, 15 (3) : 867-892. doi: 10.3934/dcdsb.2011.15.867 [14] Dashun Xu, Xiao-Qiang Zhao. Asymptotic speed of spread and traveling waves for a nonlocal epidemic model. Discrete & Continuous Dynamical Systems - B, 2005, 5 (4) : 1043-1056. doi: 10.3934/dcdsb.2005.5.1043 [15] Peter E. Kloeden. Asymptotic invariance and the discretisation of nonautonomous forward attracting sets. Journal of Computational Dynamics, 2016, 3 (2) : 179-189. doi: 10.3934/jcd.2016009 [16] Avner Friedman, Najat Ziyadi, Khalid Boushaba. A model of drug resistance with infection by health care workers. Mathematical Biosciences & Engineering, 2010, 7 (4) : 779-792. doi: 10.3934/mbe.2010.7.779 [17] C. Connell McCluskey. Global stability of an $SIR$ epidemic model with delay and general nonlinear incidence. Mathematical Biosciences & Engineering, 2010, 7 (4) : 837-850. doi: 10.3934/mbe.2010.7.837 [18] Xiaomei Feng, Zhidong Teng, Kai Wang, Fengqin Zhang. Backward bifurcation and global stability in an epidemic model with treatment and vaccination. Discrete & Continuous Dynamical Systems - B, 2014, 19 (4) : 999-1025. doi: 10.3934/dcdsb.2014.19.999 [19] Gang Huang, Edoardo Beretta, Yasuhiro Takeuchi. Global stability for epidemic model with constant latency and infectious periods. Mathematical Biosciences & Engineering, 2012, 9 (2) : 297-312. doi: 10.3934/mbe.2012.9.297 [20] Geni Gupur, Xue-Zhi Li. Global stability of an age-structured SIRS epidemic model with vaccination. Discrete & Continuous Dynamical Systems - B, 2004, 4 (3) : 643-652. doi: 10.3934/dcdsb.2004.4.643

2017 Impact Factor: 0.972