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Mathematical Modeling of the Life Cycle of Toxoplasma gondii

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

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Mathematical Modeling of the Life Cycle of Toxoplasma gondii

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  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

  2. Agent-Based Model for Transmission DynamicsCompartment Model for Stage ConversionFuture Work Outline

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

  4. Models of T. gondii Transmission • Differential/ difference equation models • Mateus-Pinilla et al., 2002 ; • Trejosand Duarte, 2005 ; Arandaet 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

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

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

  7. Agents Cats (Griffin, 2001) Mice Cells Contaminated or clean Contain detectable oocysts or not weaning mature Age (days) 0 50 240 2×365 weaning mature Age (days) 0 21 50 0.4×365

  8. 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 Cat: b1 = 5.6/365, b2 = 1.4/365; Mouse: b1 = 40/365, b2 = 10/365. b2 b1 b2 0 90 270 365

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

  10. Population Dynamics

  11. 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: 20×106 units of oocysts are excreted per day during primary infection and less during secondary infection (1×106 units) Decay: oocyst can survive 26 or 52 weeks in outdoor environment detection threshold 2000 units, time constant 20 or 40 days

  12. Infection Rule (I) Cats Mice latent infection recovery (chronic infection) recovery(chronic infection) recovery(chronic infection) Infected Days Infected Days Infected Days 0 0 0 3 14 7 28 17 10 latent infection latent infection

  13. Infection Rule(II) Infection by Oocyst Contact risk Af=2×106. Infection probability when contacted: Cats (p0=2.5%) and mice (p0=25%) Infection risk Average infection risk of the farm

  14. Infection Rule(III) Infection by tissue cystsCat 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

  15. 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)

  16. 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

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  22. Results under Nominal Parameters

  23. Stochasticity

  24. Transmission Routes

  25. Influence of Vertical Transmission

  26. Influence of Latent Period 28

  27. Influence of Prey Probability 29

  28. Influence of Virulence and # of Mice

  29. 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

  30. 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

  31. Future Work • Stochastic Dynamics Model 33

  32. A Mathematical Model for Stage Conversion of Toxoplasma gondii

  33. Scheme 36

  34. Model 37

  35. Simplification 38

  36. Stability • Disease-free Equilibrium • Endemic Equilibrium 39

  37. Numerical Results 40

  38. Numerical Results

  39. Numerical Results

  40. Host-pathogen Interaction • Compartment Model • PDE model • Individual-base Model

  41. Host-pathogen Interaction

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

  43. Future Work 46

  44. Thank you!

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