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Chapter 16 : Biomathematical Modeling of Infection and Disease

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Abstract:

There are many areas of research in which biomathematical modeling has the potential to make an important contribution. In this chapter the authors begin by reviewing the mathematical models of that have been published. The authors then outline a system of dynamical equations that can be used as a governing framework for examining many aspects of infection and pathology, including their calibration to experimental data and insights gained. Later some of the more advanced and influential models are discussed in greater detail. The chapter also talks about the likely future of biomathematical modeling of . The field of chlamydial biomathematical modeling has been developing over the past decade and has started to make contributions to the understanding of chlamydial development, infection, and disease. Epidemiological modeling also has relevance to research because of the insights it can provide about vaccine development and immunity. Although the innate response is important, it is the adaptive immune response that is vital to the resolution of chlamydial infection. A major goal in research is to understand what causes the pathology that can arise during a infection in order to determine how it can be reduced or prevented. Relatively simple mathematical frameworks can be developed for exploring mechanisms and possible implications of the data. Foundational models can then be extended both mathematically and in the laboratory.

Citation: Craig A, Bavoil P, Rank R, Wilson D. 2012. Biomathematical Modeling of Infection and Disease, p 352-379. In Tan M, Bavoil P (ed), Intracellular Pathogens I: . ASM Press, Washington, DC. doi: 10.1128/9781555817329.ch16

Key Concept Ranking

Clinical and Public Health
0.62045
Cell-Mediated Immune Response
0.56078976
Chlamydia trachomatis
0.42197016
Immune Response
0.42077723
0.62045
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Figures

Image of FIGURE 1
FIGURE 1

Experimental IFU measurements. Mean IFU measurements from swabs taken from guinea pigs inoculated in the eye with 10 IFUs of (Rank and Maurelli, unpublished). doi:10.1128/9781555817329.ch16.f1

Citation: Craig A, Bavoil P, Rank R, Wilson D. 2012. Biomathematical Modeling of Infection and Disease, p 352-379. In Tan M, Bavoil P (ed), Intracellular Pathogens I: . ASM Press, Washington, DC. doi: 10.1128/9781555817329.ch16
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Image of FIGURE 2
FIGURE 2

Comparison of data and basic model. The data from Fig. 1 are contrasted with curves produced via the basic equations of dynamics (without adaptive immune response). Target cell quantities are the numbers of target cells in the area corresponding to the swab from which IFU load was measured. Parameters used were as follows: = 100, = 0.0086, = 1.26 × 10, = 0.5, = 300, = 1.549, = 0.637, and = 5 at time = 0. doi:10.1128/9781555817329.ch16.f2

Citation: Craig A, Bavoil P, Rank R, Wilson D. 2012. Biomathematical Modeling of Infection and Disease, p 352-379. In Tan M, Bavoil P (ed), Intracellular Pathogens I: . ASM Press, Washington, DC. doi: 10.1128/9781555817329.ch16
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Image of FIGURE 3
FIGURE 3

Data and model without adaptive immune response. The curve was produced using the basic equations of dynamics contrasted with geometric means of IFU measurements from swabs taken from RAG mice inoculated intravaginally with 10 IFUs of . RAG mice have no adaptive immune response. Model parameters are as follows: = 12.1, = 0.0086, = 1.28 × 10, = 0.5, = 300, = 0.155, = 0.235, and = 1,000 at time = 0. doi:10.1128/9781555817329.ch16.f3

Citation: Craig A, Bavoil P, Rank R, Wilson D. 2012. Biomathematical Modeling of Infection and Disease, p 352-379. In Tan M, Bavoil P (ed), Intracellular Pathogens I: . ASM Press, Washington, DC. doi: 10.1128/9781555817329.ch16
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Image of FIGURE 4
FIGURE 4

Model with adaptive immune response. Shown are curves produced by the basic model of dynamics when it included terms for the adaptive immune response. Target cell quantities are the numbers of target cells in the area corresponding to the swab from which IFU load was measured. Parameters used were = 1.2 and = 1, with the other parameters the same as those used in Fig. 2 . doi:10.1128/9781555817329.ch16.f4

Citation: Craig A, Bavoil P, Rank R, Wilson D. 2012. Biomathematical Modeling of Infection and Disease, p 352-379. In Tan M, Bavoil P (ed), Intracellular Pathogens I: . ASM Press, Washington, DC. doi: 10.1128/9781555817329.ch16
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Image of FIGURE 5
FIGURE 5

Predicted effect of varying infectivity. IFU time courses were produced by varying the infectivity parameter, . The effect on the relative timing of the peaks is greater than the effect on the peaks' magnitudes. The baseline value of is 1.26 × 10. All other parameters are the same as those used in Fig. 4 . doi:10.1128/9781555817329.ch16.f5

Citation: Craig A, Bavoil P, Rank R, Wilson D. 2012. Biomathematical Modeling of Infection and Disease, p 352-379. In Tan M, Bavoil P (ed), Intracellular Pathogens I: . ASM Press, Washington, DC. doi: 10.1128/9781555817329.ch16
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Image of FIGURE 6
FIGURE 6

Target cell percentages when infectivity is varied. The percentages of target cells that are infected during the infections are shown in Fig. 5 . doi:10.1128/9781555817329.ch16.f6

Citation: Craig A, Bavoil P, Rank R, Wilson D. 2012. Biomathematical Modeling of Infection and Disease, p 352-379. In Tan M, Bavoil P (ed), Intracellular Pathogens I: . ASM Press, Washington, DC. doi: 10.1128/9781555817329.ch16
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Image of FIGURE 7
FIGURE 7

Predicted effect of varying innate immunity. Shown are curves produced by the model when the innate immune response is varied. For the normal innate immune response, = 1.549 and μ 0.637; for the doubled innate immune response magnitude, = 2 × 1.549 and μ 2 × 0.637; and the curve for no innate immune response is produced with = 0 and μ 0 (other parameters are the same as those used in Fig. 4 ). doi:10.1128/9781555817329.ch16.f7

Citation: Craig A, Bavoil P, Rank R, Wilson D. 2012. Biomathematical Modeling of Infection and Disease, p 352-379. In Tan M, Bavoil P (ed), Intracellular Pathogens I: . ASM Press, Washington, DC. doi: 10.1128/9781555817329.ch16
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Image of FIGURE 8
FIGURE 8

Predicted effect of varying adaptive immunity. The IFU curve from Fig. 4 compared to the IFU time course when the magnitude of the adaptive immune response is doubled (i.e., by setting = 2.4 and = 2); other parameters for both curves are the same as those used in Fig. 4 . doi:10.1128/9781555817329.ch16.f8

Citation: Craig A, Bavoil P, Rank R, Wilson D. 2012. Biomathematical Modeling of Infection and Disease, p 352-379. In Tan M, Bavoil P (ed), Intracellular Pathogens I: . ASM Press, Washington, DC. doi: 10.1128/9781555817329.ch16
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Image of FIGURE 9
FIGURE 9

Time courses for different values. (A) Theoretical IFU time courses associated with different values when the adaptive immune response is not considered. Note that time course with < 1 resolves, while the time courses with > 1 do not resolve. (B) IFU time courses when the adaptive immune response is included; note how the reproduction number changes with all other factors remaining unchanged. There are initially 5 IFUs for all time courses. For the time courses with = 13.7, = 100, = 0.0086, = 1.26 × 10, = 0.5, = 300, = 1.549, and = 0.637 (for panel B, = 1.2 and = 1). For the time courses with = 5.7, = 80, = 0.03, = 2 × 10, = 0.5, = 300, = 1.549, and = 0.637 (for panel B, = 2.4 and = 2). For the time courses with = 1.2, = 80, = 0.03, = 1.26 × 10, = 0.5, = 100, = 1.549, and = 0.637 (for panel B, = 2.4 and = 2). For the time courses with = 0.8, = 100, = 0.0086, = 1.26 × 10, = 0.5, = 150, = 1.549, and = 0.637 (for panel B, = 2.4 and = 2). doi:10.1128/9781555817329.ch16.f9

Citation: Craig A, Bavoil P, Rank R, Wilson D. 2012. Biomathematical Modeling of Infection and Disease, p 352-379. In Tan M, Bavoil P (ed), Intracellular Pathogens I: . ASM Press, Washington, DC. doi: 10.1128/9781555817329.ch16
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Image of FIGURE 10
FIGURE 10

Two chlamydial strains in competition. Each graph in the upper row compares IFU time course for model simulation against ocular infection of guinea pigs for mono- or coinfection of wild-type (SP) and mutant azithromycin-resistant (SPAZ) strains. In the lower graphs, the predicted numbers of target cells are shown. Peak numbers of infected cells are highest in the mutant-only infection, but in all cases the number of target cells drops by at least 97%. The model was first optimized to the data for days 2 and 5 without considering the adaptive immune response, and then those terms were introduced and the model was further optimized to fit the data for day 8. The number of IFUs at time = 0 was 300 in each of the wild-type-only and mutant-only infections, and there were 300 IFUs of each strain for the coinfection. For the wild-type-only infection, ϕ = 296.2; for the mutant-only infection, ϕ = 158.4; for the coinfection, ϕ = 242.5. Other parameters used were π = 681, δ = 0.0025, = 3.61 × 10, = 0.5, 0.432, = 300, = 50.2, ω = 0.101, ω= 0.0101, μ = 0.96, υ = 1.06, ρ = 0.0058, and ρ = 5.75 × 10. doi:10.1128/9781555817329.ch16.f10

Citation: Craig A, Bavoil P, Rank R, Wilson D. 2012. Biomathematical Modeling of Infection and Disease, p 352-379. In Tan M, Bavoil P (ed), Intracellular Pathogens I: . ASM Press, Washington, DC. doi: 10.1128/9781555817329.ch16
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Image of FIGURE 11
FIGURE 11

The contact-dependent T3S-mediated hypothesis of chlamydial development. The developmental cycle is represented graphically in an infected host cell. Following EB adherence to and entry into a susceptible cell, early differentiation takes place, yielding the first RB. In the contact-dependent hypothesis, at this developmental stage, there is maximum contact between the RB surface and the inclusion membrane surface because the inclusion is small, as well as biosynthesis of additional surface projections (injectisomes). As the developmental cycle progresses, the surface contact between an individual RB and the inclusion membrane is reduced with each round of RB division, and the number of surface projections per RB gradually decreases. By the midstage of the developmental cycle, the RBs are arrayed along, and still in contact with, the inner circumference of the inclusion membrane. With transition to the late stage of the cycle, the surface contact between individual RBs and the inclusion membrane may be reduced beyond a threshold, at which time the RB is no longer able to “stick” and becomes untethered. The contact-dependent hypothesis predicts that physical detachment of a RB in a normally growing inclusion is the trigger, or is closely associated with the trigger, for the late differentiation of the noninfectious RB into the infectious EB. A direct implication of the hypothesis is that the key step in RB-to-EB differentiation is the disruption of the T3S injectisome, whereby effectors are no longer translocated to their cytosolic targets, may accumulate in the chlamydial cytoplasm, or may be secreted into the inclusion lumen. Several T3S effectors (Tarp, CT694, IncA, and CopN) are represented in the figure according to their developmental expression and proposed subcellular location (“?” indicates other effectors). These are described in additional detail in chapter 6, “Initial interactions of chlamydiae with the host cell,” and chapter 9, “Protein secretion and pathogenesis. (Adapted from .) doi:10.1128/9781555817329.ch16.f11

Citation: Craig A, Bavoil P, Rank R, Wilson D. 2012. Biomathematical Modeling of Infection and Disease, p 352-379. In Tan M, Bavoil P (ed), Intracellular Pathogens I: . ASM Press, Washington, DC. doi: 10.1128/9781555817329.ch16
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Image of FIGURE 12
FIGURE 12

A simple Bayesian network. An example of a Bayesian network, representing conditional dependencies of a set of random variables relating chlamydial genes to pathology. The nodes are the random variables and the lines denote the dependencies and their directions.

Citation: Craig A, Bavoil P, Rank R, Wilson D. 2012. Biomathematical Modeling of Infection and Disease, p 352-379. In Tan M, Bavoil P (ed), Intracellular Pathogens I: . ASM Press, Washington, DC. doi: 10.1128/9781555817329.ch16
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References

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Tables

Generic image for table
TABLE 1

Summary of the published mathematical models of in terms of common features

Citation: Craig A, Bavoil P, Rank R, Wilson D. 2012. Biomathematical Modeling of Infection and Disease, p 352-379. In Tan M, Bavoil P (ed), Intracellular Pathogens I: . ASM Press, Washington, DC. doi: 10.1128/9781555817329.ch16
Generic image for table
TABLE 2

Nonexhaustive lists of key parameters used in published models

Citation: Craig A, Bavoil P, Rank R, Wilson D. 2012. Biomathematical Modeling of Infection and Disease, p 352-379. In Tan M, Bavoil P (ed), Intracellular Pathogens I: . ASM Press, Washington, DC. doi: 10.1128/9781555817329.ch16

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