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# Chapter 5 : Mathematical Models of Colonization and Persistence in Bacterial Infections

^{1}, Rolf Freter

^{1}

Category: Bacterial Pathogenesis

Mathematical Models of Colonization and Persistence in Bacterial Infections, Page 1 of 2

< Previous page | Next page > /docserver/preview/fulltext/10.1128/9781555818104/9781555811594_Chap05-1.gif /docserver/preview/fulltext/10.1128/9781555818104/9781555811594_Chap05-2.gif**Abstract:**

This chapter focuses on mathematical models of colonization and persistent bacterial infections. It reviews the modeling method and the state of the field and then focuses on three key areas where modeling has, and will continue to have, an impact: the ecology of the indigenous microflora and its plasmids, Helicobacter pylori colonization, and host-pathogen interactions with Mycobacterium tuberculosis. Mathematical models of host-pathogen dynamics are formulated on the basis of specific assumptions regarding the system's components and their interactions. Models of persistent viral infections, namely, human immunodeficiency virus (HlV)-host models, also have a successful recent history. H. pylori induces chronic gastric inflammation that results in peptic ulcer disease or gastric cancer in a small set of infected persons. This high ratio of mucus-living to adherent bacteria, although characteristic, is not necessary for colonization, as low concentrations of H. pylori may be present in the mucus during persistence. Incorporation of the dynamic host response into a model of H. pylori colonization is critical if one understands the initial features of the interactions between microbe and host, as well as the phenomena that permit persistence to develop. If we consider long-term associations between bacteria and humans a question of bacterial ecology, such as persistent infections or the homeostasis of an indigenous microflora, it becomes more logical to consider mathematical approaches to understanding these associations, as modeling has long been used by ecologists.

**Citation:**

**Kirschner D, Freter R.**2000. Mathematical Models of Colonization and Persistence in Bacterial Infections, p 79-100.

*In*Nataro J, Blaser M, Cunningham-Rundles S (ed),

*Persistent Bacterial Infections*. ASM Press, Washington, DC. doi: 10.1128/9781555818104.ch5

## Figures

**Citation:**

**Kirschner D, Freter R.**2000. Mathematical Models of Colonization and Persistence in Bacterial Infections, p 79-100.

*In*Nataro J, Blaser M, Cunningham-Rundles S (ed),

*Persistent Bacterial Infections*. ASM Press, Washington, DC. doi: 10.1128/9781555818104.ch5

(A) Classical complete growth curve for bacteria. (B) Model of exponential growth phase only. (C) Model of exponential and stationary phases of growth.

##### FIGURE 1

(A) Classical complete growth curve for bacteria. (B) Model of exponential growth phase only. (C) Model of exponential and stationary phases of growth.

**Citation:**

**Kirschner D, Freter R.**2000. Mathematical Models of Colonization and Persistence in Bacterial Infections, p 79-100.

*In*Nataro J, Blaser M, Cunningham-Rundles S (ed),

*Persistent Bacterial Infections*. ASM Press, Washington, DC. doi: 10.1128/9781555818104.ch5

Passage of E. coli invaders through mouse cecum. The symbols represent experimental data: the circles are the bacteria suspended in the lumen, and the triangles are the adherent population. The curves represent the best-fit estimates generated by the mathematical model for each o f the two experimental populations. (Reprinted from Microecology and Therapy [ 25 ] with permission from publisher.)

##### FIGURE 2

Passage of E. coli invaders through mouse cecum. The symbols represent experimental data: the circles are the bacteria suspended in the lumen, and the triangles are the adherent population. The curves represent the best-fit estimates generated by the mathematical model for each o f the two experimental populations. (Reprinted from Microecology and Therapy [ 25 ] with permission from publisher.)

**Citation:**

**Kirschner D, Freter R.**2000. Mathematical Models of Colonization and Persistence in Bacterial Infections, p 79-100.

*In*Nataro J, Blaser M, Cunningham-Rundles S (ed),

*Persistent Bacterial Infections*. ASM Press, Washington, DC. doi: 10.1128/9781555818104.ch5

(Top) Prediction by the mathematical model of the fate of an E. coli strain that invades the large intestine o f an animal that already harbors an adherent E. coli resident strain. (Bottom) Concentration of limiting nutrient in the system. (Reprinted from reference 24 with permission.)

##### FIGURE 3

(Top) Prediction by the mathematical model of the fate of an E. coli strain that invades the large intestine o f an animal that already harbors an adherent E. coli resident strain. (Bottom) Concentration of limiting nutrient in the system. (Reprinted from reference 24 with permission.)

**Citation:**

**Kirschner D, Freter R.**2000. Mathematical Models of Colonization and Persistence in Bacterial Infections, p 79-100.

*In*Nataro J, Blaser M, Cunningham-Rundles S (ed),

*Persistent Bacterial Infections*. ASM Press, Washington, DC. doi: 10.1128/9781555818104.ch5

Modeling of plasmid transfer in the human gut, based on data published by Anderson ( 2 ). The symbols represent Anderson's data; the lines were calculated by the mathematical models based on parameters derived from computer-generated best-fit estimates for mice and from CF cultures of mouse intestinal floras. (Reprinted from reference 26 with permission.)

##### FIGURE 4

Modeling of plasmid transfer in the human gut, based on data published by Anderson ( 2 ). The symbols represent Anderson's data; the lines were calculated by the mathematical models based on parameters derived from computer-generated best-fit estimates for mice and from CF cultures of mouse intestinal floras. (Reprinted from reference 26 with permission.)

**Citation:**

**Kirschner D, Freter R.**2000. Mathematical Models of Colonization and Persistence in Bacterial Infections, p 79-100.

*In*Nataro J, Blaser M, Cunningham-Rundles S (ed),

*Persistent Bacterial Infections*. ASM Press, Washington, DC. doi: 10.1128/9781555818104.ch5

Theoretical model describing interactions of H. pylori with the host, incorporating positive and negative feedback regulation. Effectors released by H. pylori interact with the mucosa and induce inflammation. Inflammation leads to the release of nutrients that are taken up by H. pylori, allowing replication and further release o f effectors. The bacteria sense inflammation indicators and down-regulate effector production, while the host also down-regulates the inflammatory response. The interactions within this system are governed by the four parameters τ, C, β, and η, which are not presently measurable. Therefore, mathematical modeling can play the unique role of elaborating these host-pathogen interactions. (Adapted from reference 1 .)

##### FIGURE 5

Theoretical model describing interactions of H. pylori with the host, incorporating positive and negative feedback regulation. Effectors released by H. pylori interact with the mucosa and induce inflammation. Inflammation leads to the release of nutrients that are taken up by H. pylori, allowing replication and further release o f effectors. The bacteria sense inflammation indicators and down-regulate effector production, while the host also down-regulates the inflammatory response. The interactions within this system are governed by the four parameters τ, C, β, and η, which are not presently measurable. Therefore, mathematical modeling can play the unique role of elaborating these host-pathogen interactions. (Adapted from reference 1 .)

**Citation:**

**Kirschner D, Freter R.**2000. Mathematical Models of Colonization and Persistence in Bacterial Infections, p 79-100.

*In*Nataro J, Blaser M, Cunningham-Rundles S (ed),

*Persistent Bacterial Infections*. ASM Press, Washington, DC. doi: 10.1128/9781555818104.ch5

Mathematical model describing the interaction of H. pylori and host. Mucosal bacteria, M(t), grow proportionally to nutrient at rate rg M and are cleared continuously by peristalsis at the rate µ M . They also migrate to the adherent sites [at rate α(K – A(t))] and gain in numbers due to migration from the adherent sites (at rate δ). Adherent bacteria, A(t), follow a similar dynamic, with opposite migration. Nutrients, N(t), are produced proportionally to effector amounts (at rate β) and are taken up by the adherent and mucosal populations (at rates g M and g A , respectively). Effectors are produced by both mucosal and adherent bacteria [at rate τC/τ + N(t)] and degrade nonspecifically at rate η).

##### FIGURE 6

Mathematical model describing the interaction of H. pylori and host. Mucosal bacteria, M(t), grow proportionally to nutrient at rate rg M and are cleared continuously by peristalsis at the rate µ M . They also migrate to the adherent sites [at rate α(K – A(t))] and gain in numbers due to migration from the adherent sites (at rate δ). Adherent bacteria, A(t), follow a similar dynamic, with opposite migration. Nutrients, N(t), are produced proportionally to effector amounts (at rate β) and are taken up by the adherent and mucosal populations (at rates g M and g A , respectively). Effectors are produced by both mucosal and adherent bacteria [at rate τC/τ + N(t)] and degrade nonspecifically at rate η).

**Citation:**

**Kirschner D, Freter R.**2000. Mathematical Models of Colonization and Persistence in Bacterial Infections, p 79-100.

*In*Nataro J, Blaser M, Cunningham-Rundles S (ed),

*Persistent Bacterial Infections*. ASM Press, Washington, DC. doi: 10.1128/9781555818104.ch5

Simulation o f colonization model showing H. pylori persistence. The four populations shown are the mucosal bacteria, the adherent bacteria, and the effector and nutrient concentrations. Notice that within a year the populations enter a steady-state in which they will remain indefinitely unless there is some perturbation in the system.

##### FIGURE 7

Simulation o f colonization model showing H. pylori persistence. The four populations shown are the mucosal bacteria, the adherent bacteria, and the effector and nutrient concentrations. Notice that within a year the populations enter a steady-state in which they will remain indefinitely unless there is some perturbation in the system.

**Citation:**

**Kirschner D, Freter R.**2000. Mathematical Models of Colonization and Persistence in Bacterial Infections, p 79-100.

*In*Nataro J, Blaser M, Cunningham-Rundles S (ed),

*Persistent Bacterial Infections*. ASM Press, Washington, DC. doi: 10.1128/9781555818104.ch5

Model dynamics. (A) Initial transient dynamics and development of persistent colonization. The results are obtained by numerically solving the modified model system. (B) Transient colonization that results from a much larger host response (due to an increase in the host carrying capacity). Note how this larger host response causes a timely elimination o f the bacteria.

##### FIGURE 8

Model dynamics. (A) Initial transient dynamics and development of persistent colonization. The results are obtained by numerically solving the modified model system. (B) Transient colonization that results from a much larger host response (due to an increase in the host carrying capacity). Note how this larger host response causes a timely elimination o f the bacteria.

**Citation:**

**Kirschner D, Freter R.**2000. Mathematical Models of Colonization and Persistence in Bacterial Infections, p 79-100.

*In*Nataro J, Blaser M, Cunningham-Rundles S (ed),

*Persistent Bacterial Infections*. ASM Press, Washington, DC. doi: 10.1128/9781555818104.ch5

Our hypothetical cytokine-mediated immune response network in M. tuberculosis infection. The progression of disease to either active or latent TB may depend on the balance of the TH1 and TH2 cytokines that are generated during the expression of disease.

##### FIGURE 9

Our hypothetical cytokine-mediated immune response network in M. tuberculosis infection. The progression of disease to either active or latent TB may depend on the balance of the TH1 and TH2 cytokines that are generated during the expression of disease.

**Citation:**

**Kirschner D, Freter R.**2000. Mathematical Models of Colonization and Persistence in Bacterial Infections, p 79-100.

*In*Nataro J, Blaser M, Cunningham-Rundles S (ed),

*Persistent Bacterial Infections*. ASM Press, Washington, DC. doi: 10.1128/9781555818104.ch5