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

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

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

Life cycle of T. gondii. Sibley and Ajioka, Annu. Rev. Microbiol. 2008;62: 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

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 )

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 Agent-based Model on a farm Small population sizes Inherent stochasticity Emergent properties

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

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

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

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 b1b1 b2b2 b2b2 Cat: b 1 = 5.6/365, b 2 = 1.4/365; Mouse: b 1 = 40/365, b 2 = 10/365.

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

Population Dynamics 12

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

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

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

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

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

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

Pseudo Code 19

Pseudo Code 20

Pseudo Code 21

Pseudo Code 22

Pseudo Code 23

Results under Nominal Parameters 24

Stochasticity

Transmission Routes 26

Influence of Vertical Transmission

Influence of Latent Period 28

Influence of Prey Probability 29

Influence of Virulence and # of Mice 30

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

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

Future Work 33 Stochastic Dynamics Model

A Mathematical Model for Stage Conversion of Toxoplasma gondii 34

Scheme 36

Model 37

Simplification 38

Stability 39 Disease-free Equilibrium Endemic Equilibrium

Numerical Results 40

Numerical Results

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

Host-pathogen Interaction

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

Future Work 46

Thank you!