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Introduction to ODE Modeling

Plan. . We will learn in this tutorial: translating biological knowledge to differential equations modelsIn the afternoon Lab we use matlab to simulate models generate graphs, make predictions, ...In particular we will modelReactions, Trafficking, Simple infectionsParticipants w

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Introduction to ODE Modeling

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    1. Introduction to ODE Modeling Shlomo Ta’asan Carnegie Mellon University

    2. Plan

    3. Modeling

    4. Ordinary Differential Equations (ODE)

    5. Basic Examples da/dt = 0 - This means da = 0 * dt = 0 ? change in a is 0, ? a does not change da/dt = 1 - This means da = dt ? a changes by dt 3. da/dt = - a da = - a * dt ? a changes by – a * dt - This means that a decreases, and the reduction is large when a is large and getting small when a is getting smaller. Lets look at the results of simulation of these equations.

    6. Want to model - interactions in the immune system cell-cell, virus/bacterium – cell, molecule (cytokine/chemokine)-cell, etc - trafficking - Natural killer trafficking between organs in the body - Dendritic cell migration from tissue to lymph-node Spread of disease in a population (in a given location) HIV, Influenza A ?? Pandemic – worldwide spread of an infection focus on the spatial aspect – spread between countries, continents

    7. Preparation for Modeling Some Syntax

    8. Modeling Reactions - The Law of Mass Action The rate of change of products is proportional to the product of reactants concentration A ? 0 The only reactant (left side) is a : ? rate of change is proportional to a, ODE da/dt = -k*a (minus sign since we loose a) A ? B : Similar to the previous case but here one B is created per each A that disappear ODE da/dt = -k*a as before but we also have db/dt = k*a; here the sign is +

    9. Modeling Reactions – cont. A + B ? C; Here the reactants (left side) is A and B, the product (right side) is C. dc/dt = k *a*b; C is created at a rate proportional to the product of the concentration of A and B da/dt = - k*a*b; The rate of change of A is tha same as the rate of change of C – per each C that is created one A is lost db/dt = - k*a*b, similar to A.

    10. Modeling Trafficking

    11. Trafficking – cont. We use our syntax: L ? LN and LN ? L written also as L ?? LN The rate at which cells arrive to the lymph node from the lung is proportional to the number of cells in the lung. Similarly, rate at which cells arrive to the lung from the lymph node is proportional to the number of cells in the lymph node. The ODE: dL/dt = -k1*L + k2 * LN loss + gain dLN/dt = – k2*LN + k1*L loss + gain

    12. Modeling Infection -The SIR model

    13.

    14. SIR model

    15. An HIV model

    16. HIV model cont. Model assumptions: -> T ; (lambda) % target cells production T -> 0 ; (d) % target cells natural death T + V -> I + V ; (k) % target cell becomes infected by virus I -> 0; (delta) % infected cells death I -> I + V; (p) % virus replication in infected cells V -> 0; (c) % virus clearance We construct the equations similar to the SIR model. Each reaction contribute to changes in several of the variables. We add all the changes together for each variable separately

    18. HIV model cont. The ODE: dT/dt = lambda – d * T – k * V * T dI/dt = k * V * T – delta * I dV/dt = p* I – c * V This model has 6 parameters that may affect the behavior. We will study this in the lab

    20. Now You are Ready to Do Your Own ODE Models

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