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

1 Math 211 Math 211 A Model of the Human Immune System

2 Model of the Developement of Immunity Model of the Developement of Immunity to Desease 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, E 1 ( t ) & E 2 ( t ) . Return

3 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 E 2 E ′ 1 = Λ 1 − µ 1 E 1 + a 1 1 + b 1 E 1 E 2 E 1 E 2 E ′ 2 = Λ 2 − µ 2 E 2 + a 2 1 + b 2 E 1 E 2 � In pplane5 use Λ 1 = Λ 1 = 1 , µ 1 = µ 1 = 1 . 25 , a 1 = a 2 = 0 . 252 , and b 1 = b 2 = 0 . 008 . Return

4 Dynamics of the Lymphocytes Dynamics of the Lymphocytes E 1 ’ = 1 − 1.25 E 1 + 0.252 E 1 E 2 /(1 + 0.008 E 1 E 2 ) E 2 ’ = 1 − 1.25 E 2 + 0.252 E 1 E 2 /(1 + 0.008 E 1 E 2 ) 30 25 Immune State 20 E 2 15 10 5 Virgin State 0 0 5 10 15 20 25 30 E 1 Return

5 Interactions with the Virus Interactions with the Virus • Virus cells have an intrinsic growth rate r. • Lymphocytes of type E 1 : � kill virus because of contacts with them � proliferate because of contacts with virus • Lymphocytes of type E 2 : � do not directly interact with the virus � regulate cells of type E 1 Return

6 The Model With Virus Present The Model With Virus Present E 1 E 2 E ′ 1 = Λ 1 − µ 1 E 1 + a 1 + KV E 1 1 + b 1 E 1 E 2 E 1 E 2 E ′ 2 = Λ 2 − µ 2 E 2 + a 2 1 + b 2 E 1 E 2 V ′ = rV − kV E 1 • For ode45 use K = 0 . 5 , k = 0 . 01 and r = 0 . 1 . Return No virus Interactions

7 Equilibrium Points Equilibrium Points • There are three realistic equilibrium points 1 5 20 E 1          =  ,  , 1 5 & 20 E 2      0 0 0 V • 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 . System Dynamics