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# Outcome prediction in mathematical models of immune response to infection

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### Abstract

Clinicians need to predict patient outcomes with high accuracy as early as possible after disease inception. In this manuscript, we show that patient-to-patient variability sets a fundamental limit on outcome prediction accuracy for a general class of mathematical models for the immune response to infection. However, accuracy can be increased at the expense of delayed prognosis. We investigate several systems of ordinary differential equations (ODEs) that model the host immune response to a pathogen load. Advantages of systems of ODEs for investigating the immune response to infection include the ability to collect data on large numbers of `virtual patients', each with a given set of model parameters, and obtain many time points during the course of the infection. We implement patient-to-patient variability $$v$$ in the ODE models by randomly selecting the model parameters from Gaussian distributions with variance $$v$$ that are centered on physiological values. We use logistic regression with one-versus-all classification to predict the discrete steady-state outcomes of the system. We find that the prediction algorithm achieves near $$100\%$$ accuracy for $$v=0$$, and the accuracy decreases with increasing $$v$$ for all ODE models studied. The fact that multiple steady-state outcomes can be obtained for a given initial condition, i.e. the basins of attraction overlap in the space of initial conditions, limits the prediction accuracy for $$v>0$$. Increasing the elapsed time of the variables used to train and test the classifier, increases the prediction accuracy, while adding explicit external noise to the ODE models decreases the prediction accuracy. Our results quantify the competition between early prognosis and high prediction accuracy that is frequently encountered by clinicians.

### Most cited references22

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(2001)
The innate immune system is a universal and ancient form of host defense against infection. Innate immune recognition relies on a limited number of germline-encoded receptors. These receptors evolved to recognize conserved products of microbial metabolism produced by microbial pathogens, but not by the host. Recognition of these molecular structures allows the immune system to distinguish infectious nonself from noninfectious self. Toll-like receptors play a major role in pathogen recognition and initiation of inflammatory and immune responses. Stimulation of Toll-like receptors by microbial products leads to the activation of signaling pathways that result in the induction of antimicrobial genes and inflammatory cytokines. In addition, stimulation of Toll-like receptors triggers dendritic cell maturation and results in the induction of costimulatory molecules and increased antigen-presenting capacity. Thus, microbial recognition by Toll-like receptors helps to direct adaptive immune responses to antigens derived from microbial pathogens.
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### Preferential localization of effector memory cells in nonlymphoid tissue.

(2001)
Many intracellular pathogens infect a broad range of host tissues, but the importance of T cells for immunity in these sites is unclear because most of our understanding of antimicrobial T cell responses comes from analyses of lymphoid tissue. Here, we show that in response to viral or bacterial infection, antigen-specific CD8 T cells migrated to nonlymphoid tissues and were present as long-lived memory cells. Strikingly, CD8 memory T cells isolated from nonlymphoid tissues exhibited effector levels of lytic activity directly ex vivo, in contrast to their splenic counterparts. These results point to the existence of a population of extralymphoid effector memory T cells poised for immediate response to infection.
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### HIV-1 dynamics in vivo: virion clearance rate, infected cell life-span, and viral generation time.

(1996)
A new mathematical model was used to analyze a detailed set of human immunodeficiency virus-type 1 (HIV-1) viral load data collected from five infected individuals after the administration of a potent inhibitor of HIV-1 protease. Productively infected cells were estimated to have, on average, a life-span of 2.2 days (half-life t 1/2 = 1.6 days), and plasma virions were estimated to have a mean life-span of 0.3 days (t 1/2 = 0.24 days). The estimated average total HIV-1 production was 10.3 x 10(9) virions per day, which is substantially greater than previous minimum estimates. The results also suggest that the minimum duration of the HIV-1 life cycle in vivo is 1.2 days on average, and that the average HIV-1 generation time--defined as the time from release of a virion until it infects another cell and causes the release of a new generation of viral particles--is 2.6 days. These findings on viral dynamics provide not only a kinetic picture of HIV-1 pathogenesis, but also theoretical principles to guide the development of treatment strategies.
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Quantitative & Systems biology