Math 211 Math 211 A Model of the Human Immune System 2 Model of - - PowerPoint PPT Presentation

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Math 211 Math 211

A Model of the Human Immune System

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Model of the Developement of Immunity to Desease Model of the Developement of Immunity to Desease

• Diseases such as flu, the cold, mumps, . . .
• Infectious Diseases of Humans - Roy M. Anderson &

Robert M. May, Oxford University Press 1992

• The model includes the interactions between virus cells

and lymphocytes generated by the immune system.

V (t) = number of virus cells Two types of lymphocytes, E1(t) & E2(t).

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Interaction of the Lymphocytes Interaction of the Lymphocytes

• They are recruited from bone marrow at a constant rate
• They die at a rate proportional to their numbers
• They proliferate due to contact with each other
• The model with no virus present is:

E′

1 = Λ1 − µ1E1 + a1

E1E2 1 + b1E1E2 E′

2 = Λ2 − µ2E2 + a2

E1E2 1 + b2E1E2

In pplane5 use Λ1 = Λ1 = 1, µ1 = µ1 = 1.25,

a1 = a2 = 0.252, and b1 = b2 = 0.008.

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Dynamics of the Lymphocytes Dynamics of the Lymphocytes

E1 ’ = 1 − 1.25 E1 + 0.252 E1 E2/(1 + 0.008 E1 E2) E2 ’ = 1 − 1.25 E2 + 0.252 E1 E2/(1 + 0.008 E1 E2) 5 10 15 20 25 30 5 10 15 20 25 30 E1 E2 Virgin State Immune State

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Interactions with the Virus Interactions with the Virus

• Virus cells have an intrinsic growth rate r.
• Lymphocytes of type E1:

kill virus because of contacts with them proliferate because of contacts with virus

• Lymphocytes of type E2:

do not directly interact with the virus regulate cells of type E1

Return No virus Interactions

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The Model With Virus Present The Model With Virus Present

E′

1 = Λ1 − µ1E1 + a1

E1E2 1 + b1E1E2 +KV E1 E′

2 = Λ2 − µ2E2 + a2

E1E2 1 + b2E1E2 V ′= rV − kV E1

• For ode45 use K = 0.5, k = 0.01 and r = 0.1.

System Dynamics

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Equilibrium Points Equilibrium Points

• There are three realistic equilibrium points

  E1 E2 V   =   1 1   ,   5 5   , &   20 20  

• The first two are unstable. The third is asymptotically

stable.

• What is the global behavior? The best we can do is to

check with ode45.