Modeling the Effects of HIV on the Evolution of Diseases

Transcription

1 1/20 the Evolution of Mentor: Nina Fefferman DIMACS REU July 14, 2011

2 2/20 Motivating Questions How does the presence of HIV in the body changes the evolution of pathogens in the human body? How do different strains emerge at different levels of AIDS prevalence in a population?

3 3/20 Viruses Viruses are small infectious particles that can replicate only inside living cells Examples of viral diseases: influenza, HIV, ebola

4 3/20 Viruses Viruses are small infectious particles that can replicate only inside living cells Examples of viral diseases: influenza, HIV, ebola

5 3/20 Viruses Viruses are small infectious particles that can replicate only inside living cells Examples of viral diseases: influenza, HIV, ebola

6 3/20 Viruses Viruses are small infectious particles that can replicate only inside living cells Examples of viral diseases: influenza, HIV, ebola

7 3/20 Viruses Viruses are small infectious particles that can replicate only inside living cells Examples of viral diseases: influenza, HIV, ebola

8 4/20 Immune Response A cell called T-cells play a big role in fighting off pathogens

9 5/20 Immune Response

10 5/20 Immune Response

11 5/20 Immune Response

12 5/20 Immune Response

13 5/20 Immune Response

14 5/20 Immune Response

15 6/20 Human Immunodeficiency Virus (HIV) Virus that causes acquired immunodeficiency syndrome (AIDS) HIV damages the human immune system by infecting and kill T cells Causes the body to lose cell-mediated immunity Death due to other opportunistic infections

16 7/20 How are HIV and other viruses related? HIV/AIDS changes the evolutionary and ecological context for other diseases Current models exist for HIV infection and HIV strain mutations Very few models explore how HIV affects the evolution and mutation of other diseases

17 8/20 Project for the Summer ing at the individual level: seeing how HIV/AIDS affects strain emergence within a person

18 8/20 Project for the Summer ing at the individual level: seeing how HIV/AIDS affects strain emergence within a person ing at the population level: seeing how HIV/AIDS changes how strains emerge in a population

19 9/20 Characteristics of the The model needs to track the evolution new strains In healthy individuals, the model should kill the virus In individuals affected with HIV/AIDS, the model should Let the virus win Let the virus grow to some intermediate level

20 10/20 System of Differential Equations System of ODEs describes the number of virus particles of N different strains ds 1 (1 dt = n I (t) ln + r ) ( ) 1 s1 s 1 ds 1 + n s Max. ds i (1 dt = n I (t) ln + r ) ( 1 s i n s Max ds N dt. = n I (t) ln ) s i ds i + ( 1 + r ) ( 1 s ) N s N ds N + n s Max n µ j1 s j j=1 n µ ji s j j=1 n µ 1j s 1 j=1 n µ ij s i j=1 n µ jn s j j=1 n µ Nj s N j=1

21 11/20 Growth and Death of the Virus ds i (1 dt = I (t) ln + r ) ( 1 s ) i s i ds i (1) n s Max

22 11/20 Growth and Death of the Virus ds i (1 dt = I (t) ln + r ) ( 1 s ) i s i ds i (1) n s Max Growth term: n number of replication cycles per day, each cycle produces r virions

23 11/20 Growth and Death of the Virus ds i (1 dt = I (t) ln + r ) ( 1 s ) i s i ds i (1) n s Max Growth term: n number of replication cycles per day, each cycle produces r virions Logistic growth term prevents unlimited growth

24 11/20 Growth and Death of the Virus ds i (1 dt = I (t) ln + r ) ( 1 s ) i s i ds i (1) n s Max Growth term: n number of replication cycles per day, each cycle produces r virions Logistic growth term prevents unlimited growth Death term: Death representing the clearance of the virus by the body (e.g. lymphatic system)

25 11/20 Growth and Death of the Virus ds i (1 dt = I (t) ln + r ) ( 1 s ) i s i ds i (1) n s Max Growth term: n number of replication cycles per day, each cycle produces r virions Logistic growth term prevents unlimited growth Death term: Death representing the clearance of the virus by the body (e.g. lymphatic system) Immune response decreases the growth term

26 12/20 Immune Response I (t) = { a ( e b(t to) + 1 ) if s i threshold 0 if s i < threshold Has a delay between the introduction of the virions and the immune response Slowly builds up to a maximal response rather than having an instantaneous maximal response

27 13/20 Mutation of Viruses ds i dt = n µ ji s j j=1 n µ ij s i j=1 The parameter µ ij represents the rate at which strain i mutates to strain j For a given strain i, it can mutate to any of the other N 1 strains, and the other N 1 strains can mutate into i 0 µ 12 µ 1n. µ µ = µ n1 µ n2 µ nn

28 14/20 Case 1: the body wins µ = r = 5, n = 4, d = 1.2, threshold = 600, max I (t) = 0.3, S 1(0) = Number of Virions Strain 1 Strain 2 Strain 3 Strain 4 Strain 5 Strain Time (days)

29 15/20 Case 2: the virus wins µ = r = 5, n = 4, d = 1.2, threshold = 600, max I (t) = 0.6, S 1(0) = Number of Virions Strain 1 Strain 2 Strain 3 Strain 4 Strain 5 Strain Time (days)

30 16/20 Case 3: nobody wins µ = r = 5, n = 4, d = 3, threshold = 600, max I (t) = 0.6, S 1(0) = Number of Virions Strain 1 Strain 2 Strain 3 Strain 4 Strain 5 Strain Time (days)

31 17/20 From the to the The model will be SIR-like model The population model will draw conclusions made in the individual model

32 18/20 To sum it up My model can demonstrates different behaviors: The immune system effectively killing the virus A compromised immune system losing to the virus These results will provide the groundwork for a population model

33 19/20 Future Work Expand the model to capture cross-immunity Work with different parameters, including realistic parameter values Develop the population model

34 20/20 Questions?