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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
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##### 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: $N_P=30$, $N_H=10$, $N_V=16$, $\omega_{Noff}=0.1$, $\omega_{Non}=0.1$, $\omega_R=0.2$, $\mu_{Noff}=0.25=1/T_{Noff}$, $\mu_{Non}=0.2=1/T_{Non}$, $\mu_R=0.1667=1/T_R$, $\pi_N=0.1$, $\pi_R=0.2$, $\beta_{on}=0.3$, $\beta_{off}=0.1$
. 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">Figure 10.  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: $N_P=30$, $N_H=10$, $N_V=16$, $\omega_{Noff}=0.3$, $\omega_{Non}=0.1$, $\omega_R=0.2$, $\mu_{Noff}=0.25=1/T_{Noff}$, $\mu_{Non}=0.2=1/T_{Non}$, $\mu_R=0.1667=1/T_R$, $\pi_N=0.2$, $\pi_R=0.2$, $\beta_{on}=0.4$, $\beta_{off}=0.1$
. All trajectories converge to $(5.56098, 11.122, 0)$">Figure 12.  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: $N_P=30$, $N_H=10$, $N_V=16$, $\omega_{Noff}=0.2$, $\omega_{Non}=0.2$, $\omega_R=0.2$, $\mu_{Noff}=0.25=1/T_{Noff}$, $\mu_{Non}=0.2=1/T_{Non}$, $\mu_R=0.1=1/T_R$, $\pi_N=0.2$, $\pi_R=0.3$, $\beta_{on}=0.7$, $\beta_{off}=0.03$
. All trajectories converge with oscillations to the same limiting value">Figure 14.  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 $R_{01}=1.0$
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