1. Mathematical Modeling of the Life Cycle of Toxoplasma gondii A Sullivan, W Jiang, F Agusto, S Bewick, C Su, M Gilchrist, M Turner, and X Zhao 1

2. Agent-Based Model for Transmission Dynamics Compartment Model for Stage Conversion Future Work 2 Outline

3. A Prototype Agent-Based Model for the Transmission Dynamics of Toxoplasma gondii 3

4. Life cycle of T. gondii. Sibley and Ajioka, Annu. Rev. Microbiol. 2008;62:329-351 What is Toxoplasma gondii ? Cause life-threatening disease in AIDS and cancer patients, recipients of organ transplants and fetus Cause infection in all warm-blooded vertebrates

5. Can Toxoplasma gondii change the world? Change mice behavior  Imprudent attraction to cats ( Torrey et al., 2006; Flegr et al., 2003; Webster et al., 2006 )  Ensuring the completion of the life cycle of T. gondii Cause long-term personality change in humans  Higher guilt proneness, more self-doubting ( Webster, 2001 ) Is variation in culture ultimately be related to how climate affects the distribution of T. gondii? ( Lafferty, 2006 )

6. Models of T. gondii Transmission Differential/ difference equation models  Mateus-Pinilla et al., 2002 ;  Trejos and Duarte, 2005 ; Aranda et al., 2008; Gonzalez-Parra et al., 2009; Arenas et al., 2010;  Lelu et al. 2010 Agent-based Model on a farm Small population sizes Inherent stochasticity Emergent properties

7. Problem Description Schematic of the transmission routes of T. gondii; figure edited from Jone et al., Am. Fam. Physician. 2003;67:2131-2138. 7

8. ABM of Toxoplasma in a Farm catmouse oocyst clean cell contaminated cell Sketch of ABM of Toxoplasma in a cat-mouse-environment system Agents cat (susceptible, infected or immuned) mouse (susceptible, infected or immuned) Environment cell (contaminated or clean) 8

9. Agents Cats ( Griffin, 2001 ) Mice Cells  Contaminated or clean Contain detectable oocysts or not 9 weaningmature 050240 2 × 365 Age (days) weaningmature Age (days)021500.4 × 365

10. Birth and Death Birth rate  Breeding female cats gave birth to an average of 7.1 kittens per year ( Warner, 1995 )  Annual rhythms Natural death rate  Age ( Warner, 1995 )  Carrying capacity 10 090270365 b1b1 b2b2 b2b2 Cat: b 1 = 5.6/365, b 2 = 1.4/365; Mouse: b 1 = 40/365, b 2 = 10/365.

11. Predator Prey Rule Random walk rule  Post-weaning cats or mice  Max_step_cat = 5 and max_step_mouse = 1 Predator prey rule 11 1 0.5 1 0.70.5 0.7

12. Population Dynamics 12

13. Oocyst Shedding & Decay Rule Latent: 3 days for primary and 7 days for secondary Recovery: 17 days Oocyst spread time: 2 weeks for primary infection; 10 days for secondary infection Amount: 20×10 6 units of oocysts are excreted per day during primary infection and less during secondary infection (1×10 6 units) Decay: oocyst can survive 26 or 52 weeks in outdoor environment detection threshold 2000 units, time constant 20 or 40 days 13

14. Infection Rule (I) Cats Mice 14 latentrecovery (chronic infection) 0317 Infected Days recovery(chronic infection) 01428 Infected Days recovery(chronic infection) 0710 Infected Days infection latent infection

15. Infection Rule(II) Infection by Oocyst  Contact risk  A f =2×10 6.  Infection probability when contacted: Cats (p 0 =2.5%) and mice (p 0 =25%)  Infection risk  Average infection risk of the farm 15

16. Infection Rule(III) Infection by tissue cysts Cat gets infected from eating mouse (Dubey)  after the latent period of mouse: 100%  before latent: certain probability  t: how long the mouse has been infected 16

17. Infection Rule(IV) Secondary infection (Dubey)  After the initial infection: very low before 6 years and 50% chance after 6 years Vertical transmission  Mice (75%); none in cats Maternal immunity  Cats (weaning period) 17

18. Virulence Rule Strain type  Type I (high virulent)  Type II (intermediate virulent) Produce 10 to 20 times more tissue cysts than type I and III (Suzuki and Joh)  Type III (non virulent) More tissue cysts -> higher infection risk Relations between lethal rate (v) and transmission 18

19. Pseudo Code 19

20. Pseudo Code 20

21. Pseudo Code 21

22. Pseudo Code 22

23. Pseudo Code 23

24. Results under Nominal Parameters 24

25. Stochasticity

26. Transmission Routes 26

27. Influence of Vertical Transmission

28. Influence of Latent Period 28

29. Influence of Prey Probability 29

30. Influence of Virulence and # of Mice 30

31. Possible prevention strategies Reduce the survival time of oocysts Mice elimination  Role of mice in T. gondii transmission Pass disease to cats  95% of cats are infected through predation on infected mice Pass disease to the next generation of mice  80% of mice are infected through vertical transmission 31

32. Future Work Decision based on internal states and local interactions  Cats and mice may adjust their activities according to their experience and sense of the environment Include human activities  Vaccination of cats  Mice elimination Pattern-oriented modeling  Demographics of cats and mice 32

33. Future Work 33 Stochastic Dynamics Model

34. A Mathematical Model for Stage Conversion of Toxoplasma gondii 34

36. Scheme 36

37. Model 37

38. Simplification 38

39. Stability 39 Disease-free Equilibrium Endemic Equilibrium

40. Numerical Results 40

41. Numerical Results

43. Host-pathogen Interaction Compartment Model PDE model Individual-base Model

44. Host-pathogen Interaction

45. Future Work 45 More accurate description of within-host life cycle More detailed and accurate immune response Whole-body kinetics

46. Future Work 46

47. Thank you!