# American Institute of Mathematical Sciences

January  2020, 19(1): 203-220. doi: 10.3934/cpaa.2020011

## The continuous morbidostat: A chemostat with controlled drug application to select for drug resistance mutants

 1 Department of Mathematics, Swinburne University of Technology, Melbourne VIC 3122, Australia 2 Department of Mathematics and National Center of Theoretical Science, National Tsing Hua University, Hsinchu, Taiwan 3 Department of Electrical Engineering, National Tsing Hua University, Hsinchu, Taiwan

* Corresponding author

Received  October 2018 Revised  March 2019 Published  July 2019

The morbidostat is a bacteria culture device that progressively increases antibiotic drug concentration and maintains a constant challenge for study of evolutionary pathway. The operation of a morbidostat under serial transfer has been analyzed previously. In this work, the global dynamics for the operation of a morbidostat under continuous dilution is analyzed. The device switches between drug on and drug off modes according to a simple threshold algorithm. We prove the extinction and uniform persistence of all species with both forward and backward mutations. Numerical simulations for the case of logistic growth and the Hill function for drug inhibition are also presented.

Citation: Zhenzhen Chen, Sze-Bi Hsu, Ya-Tang Yang. The continuous morbidostat: A chemostat with controlled drug application to select for drug resistance mutants. Communications on Pure & Applied Analysis, 2020, 19 (1) : 203-220. doi: 10.3934/cpaa.2020011
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Schematic of a continuous morbidostat. There is no drug injection when the total microbes are less than threshold $U$. There is continuous drug injection once the total microbes reach the threshold $U$
Forward mutations between species. Mutant $v_{i}$ mutates to mutant $v_{i+1}$ with a forward mutation rate $q_{ii+1}$, and there is no backward mutations. We have $v_{0} = u$ and $i = 0, 1,2,\cdots, N-1.$
Forward-backward mutations between species. Mutant $v_{i}$ mutates to mutant $v_{i+1}$ with a forward mutation rate $q_{ii+1}$, while mutant $v_{i+1}$ mutates to mutant $v_{i}$ with a backward mutation rate $\tilde{q}_{ii+1}$. We have $v_{0} = u$ and $i = 0, 1,2,\cdots, N-1.$
Cell, substrate, and inhibitor densities of system (3) when $U = 8$. The wild type $u$, mutants $v_{1}$, $v_{2}$ and inhibitor $P$ go extinction in the drug on drug off model, while mutant $v_{3}$ and substrate $S$ persist at fixed values in the long-term. In this figure, we take $S^{0} = 10$, $D = 0.9$, $P^{0} = 10,$ $q_{01} = q_{02} = q_{03} = q_{12} = q_{13} = q_{23} = 10^{-4},$ $m = 0.3$, $r = 0.5$, $a = 0.5$, $L = 1$, $K_{1} = 1$, $K_{2} = 3$, $K_{3} = 10$ and $K_{4} = 30$
Cell, substrate, and inhibitor densities of system (3) when $U = 2$. The wild type $u$ and mutants $v_{1}$, $v_{2}$ go extinction in the drug on drug off model, while mutant $v_{3}$, substrate $S$, and inhibitor $P$ persist at fixed values in the long-term. In this figure, we take $S^{0} = 10$, $D = 0.9$, $P^{0} = 10,$ $q_{01} = q_{02} = q_{03} = q_{12} = q_{13} = q_{23} = 10^{-4},$ $m = 0.3$, $r = 0.5$, $a = 0.5$, $L = 1$, $K_{1} = 1$, $K_{2} = 3$, $K_{3} = 10$ and $K_{4} = 30$
Cell, substrate, and inhibitor densities of system (3) when $U = 6.5$. The wild type $u$ and mutants $v_{1}$, $v_{2}$ go extinction in the drug on drug off model, while mutant $v_{3}$, substrate $S$, and inhibitor $P$ oscillate in the long-term. The inset figure shows the density of mutant $v_{3}$ (green) and concentration of the Substrate $S$ (blue) in the long-term. In this figure, we take $S^{0} = 10$, $D = 0.9$, $P^{0} = 10,$ $q_{01} = q_{02} = q_{03} = q_{12} = q_{13} = q_{23} = 10^{-4},$ $m = 0.3$, $r = 0.5$, $a = 0.5$, $L = 1$, $K_{1} = 1$, $K_{2} = 3$, $K_{3} = 10$ and $K_{4} = 30$
Cell, substrate, and inhibitor densities of system (3) when $U = 6.1$. The wild type $u$ and mutants $v_{1}$, $v_{2}$ go extinction in the drug on drug off model, while mutant $v_{3}$, substrate $S$, and inhibitor $P$ persist at fixed values in the long-term. In this figure, we take $S^{0} = 10$, $D = 0.9$, $P^{0} = 10,$ $q_{01} = q_{02} = q_{03} = q_{12} = q_{13} = q_{23} = 10^{-4},$ $m = 0.3$, $r = 0.5$, $a = 0.5$, $L = 1$, $K_{1} = 1$, $K_{2} = 3$, $K_{3} = 10$ and $K_{4} = 30$
Extinction of all the microbes of system (4). In this case, all the cells and inhibitor go to extinction in the drug on drug off model, while the substrate persists at a fixed level. In this figure, we take $S^{0} = 10$, $D = 0.9$, $P^{0} = 10,$ $U = 5$, $q_{01} = q_{02} = q_{03} = q_{12} = q_{13} = q_{23} = 10^{-4}$, $\tilde{q}_{01} = \tilde{q}_{02} = \tilde{q}_{03} = \tilde{q}_{12} = \tilde{q}_{13} = \tilde{q}_{23} = 10^{-4}$, $m = 0.08$, $r = 0.5$, $a = 0.5$, $L = 1$, $K_{1} = 1$, $K_{2} = 3$, $K_{3} = 10$ and $K_{4} = 30$
Persistence of the all the microbes of system (4). In this case, all the microbes persist in the drug on drug off model. However, the most resistant microbe dominates all the species. In this figure, we take $S^{0} = 10$, $D = 0.9$, $P^{0} = 10,$ $U = 6$, $q_{01} = q_{02} = q_{03} = q_{12} = q_{13} = q_{23} = 0.005$, $\tilde{q}_{01} = \tilde{q}_{02} = \tilde{q}_{03} = \tilde{q}_{12} = \tilde{q}_{13} = \tilde{q}_{23} = 0.005$, $m = 0.3$, $r = 0.5$, $a = 0.5$, $L = 1$, $K_{1} = 1$, $K_{2} = 3$, $K_{3} = 10$ and $K_{4} = 30$
Persistence of the all the microbes of system (4). In this case, all the microbes persist in the drug on drug off model. In this figure, we take $S^{0} = 10$, $D = 0.9$, $P^{0} = 10,$ $U = 6$, $q_{01} = q_{02} = q_{03} = q_{12} = q_{13} = q_{23} = 0.05$, $\tilde{q}_{01} = \tilde{q}_{02} = \tilde{q}_{03} = \tilde{q}_{12} = \tilde{q}_{13} = \tilde{q}_{23} = 0.05$, $m = 0.3$, $r = 0.5$, $a = 0.5$, $L = 1$, $K_{1} = 1$, $K_{2} = 3$, $K_{3} = 10$ and $K_{4} = 30$
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