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

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Agent-Based Model for Transmission Dynamics

Compartment Model for Stage Conversion

Future Work

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Outline

A Prototype Agent-Based Model for the Transmission Dynamics of

Toxoplasma gondii

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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

Problem Description

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

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ABM of Toxoplasma in a Farm

cat mouse

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)

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Agents

Cats (Griffin, 2001)

Mice

Cells Contaminated or clean

Contain detectable oocysts or not

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weaning mature

0 50 240 2×365Age (days)

weaning mature

Age (days) 0 21 50 0.4×365

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

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0 90 270 365

b1 b2b2Cat: b1 = 5.6/365, b2 = 1.4/365;Mouse: b1 = 40/365, b2 = 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

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0 ,1

Max0/

eeep

rr

1 cell, 10 r 1 cell, 20 r

1

0.5

1

0.7 0.5

0.5 0.5

0.7 0.7

0.7

Population Dynamics

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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

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/tieAA

Infection Rule (I)

Cats

Mice

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latent recovery (chronic infection)

0 3 17InfectedDays

recovery(chronic infection)

0 14 28InfectedDays

recovery(chronic infection)

0 7 10InfectedDays

infection

infection latent

latent infection

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

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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

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10000inf t

tt

eep

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)

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365*6*1inf

2inf cct

ctstnd

eeepp

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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2

2

2)(

v

s eT

Pseudo Code

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Pseudo Code

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Pseudo Code

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Pseudo Code

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Pseudo Code

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Results under Nominal Parameters

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Stochasticity

Transmission Routes

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Influence of Vertical Transmission

Influence of Latent Period

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Influence of Prey Probability

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Influence of Virulence and # of Mice

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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

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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

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Future Work

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Stochastic Dynamics Model

A Mathematical Model for Stage Conversion of Toxoplasma gondii

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Scheme

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Model

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Simplification

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Stability

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Disease-free Equilibrium

Endemic Equilibrium

Numerical Results

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Numerical Results

Numerical Results

Host-pathogen Interaction

Compartment Model PDE model Individual-base Model

Host-pathogen Interaction

Future Work

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More accurate description of within-host life cycle

More detailed and accurate immune response

Whole-body kinetics

Future Work

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Thank you!