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Structured and unstructured continuous modelsfor Wolbachia infections

Peter Hinow

Department of Mathematical Sciences, University of Wisconsin - Milwaukee,P.O. Box 413, Milwaukee, WI 53201-0413, USA; email: [email protected]

8th European Conference on Mathematical and TheoreticalBiology, Cracow, Poland, June 28th - July 2nd 2011

Peter Hinow Wolbachia infections

Collaborator

Jozsef Z. Farkas (University of Stirling, United Kingdom)


Overview of the talk

I introduction of the biological background

I ordinary differential equation models

I introduction of age structure

I outlook, conclusion


Biological motivation

Wolbachia is a maternally transmitted bacterium that lives insymbiosis with many arthropod species. It inhabits testes andovaries of its hosts and can cause

I cytoplasmic incompatibility,

I induction of parthenogenesis, and

I feminization of genetic males,

depending on the Wolbachia strain and the host species.



Wolbachia bacteria inside insect cells.



Potential use as biological control (McMeniman et al. 2009):infection with Wolbachia shortens the lifespan of the mosquitoAedes aegypti, a vector for the Dengue fever virus (only oldermosquitoes are carriers).


Cytoplasmic incompatibility

Infected males mated with uninfected females results in partial orcomplete embryonic lethality, all other crosses are viable.


Mathematical modeling attempts

Models exist using

I discrete dynamical systems (Caspari & Watson 1959,Engelstadter et al. 2004)

I continuous time ODEs (Keeling et al. 2003)

We work with continuous time and focus on the effects of

I cytoplasmic incompatibility,

I partial transmission to offspring, and

I fitness cost of the infection

on diplodiploid organisms (sex is determined by sex chromosomes).


Single-sex ODE model for singular strain infection

Let I and U denote the number of infected, respectivelyuninfected, individuals in the population and

I τ ∈ [0, 1] the fraction of infected offspring from infectedparents,

I D ≥ 0 additional mortality of infected individuals, and

I q ∈ [0, 1] the probability that the offspring of an infected maleand an uninfected female is nonviable.

Then, if b > 0 is the birth rate, we have the logistic model

dI

dt= (τb − (d + D)(I + U)) I ,

dU

dt= (1− τ)bI +

(b

(1− q

I

I + U

)− d(I + U)

)U.


Rescaled model

With t → bt and setting η = d+Dd , we obtain the reduced system

for the quantities i = dI/b, u = dU/b

di

dt= (τ − η(i + u))i ,

du

dt= (1− τ)i +

(1− q

i

i + u− (i + u)

)u.

Then η−1 =: ξ ∈ (0, 1] and 1− ξ is the fitness cost associated withWolbachia infection.

Equilibrium solutions of this system are easily found (but hard toanalyze); there is always a stable disease-free steady state(i0, u0) = (0, 1).


Examples of phase portraits

Hi0,u0L

Hi1,u1L

Hi2,u2L0.5 1.0

i

0.2

0.4

0.6

0.8

1.0

1.2u

Hi0,u0L

Hi1,u1L

Hi2,u2L

0.5 1.0i

0.2

0.4

0.6

0.8

1.0

1.2

u

(Left) (ξ, τ, q) = (0.9, 1, 1) (complete transmission), stable statesare pure, low threshold for establishment of an infection;(Right) (ξ, τ, q) = (1, 0.76, 1) (partial transmission), admitscoexistence, higher threshold for establishment of an infection.


Parameter space decomposition

Region where only the disease-free equilibrium exists (A) and wherecoexistence equilibrium solutions are possible (B and above C).


Infections with multiple strains

If Wolbachia strains A and B are present, then there are singly andmultiply infected individuals iA, iB , and iAB .

0

q0,A q0,B

q0,AB

qB,A

qA,B

qA,AB qB,AB

A B

AB

Arrows X → Y indicate that a X♂× Y ♀ cross is incompatible,with incompatibility level qY ,X (=0 if mutually compatible).


Full equation set

Let p = iAB + iA + iB + u be the total population, and

diABdt

= τAτB iAB − ηABpiAB ,

diAdt

= τA(1− τB)iAB + τA

(1− qA,B

iBp− qA,AB

iABp

)iA − ηApiA,

diBdt

= (1− τA)τB iAB + τB

(1− qB,A

iAp− qB,AB

iABp

)iB − ηBpiB ,

du

dt= (1− τA)(1− τB)iAB + (1− τA)

(1− qA,B

iBp− qA,AB

iABp

)iA

+ (1− τB)

(1− qB,A

iAp− qB,AB

iABp

)iB

+

(1− q0,A

iAp− q0,B

iBp− q0,AB

iABp

)u − pu.


(One possible) Simplification

Assume the

I absence of doubly infected individuals, iAB ≡ 0,

I mutual compatibility of infected individuals, qA,B = qB,A = 0,

I equal transmission efficacy τA = τB =: τ and infection costsηA = ηB =: η

Broken symmetry by levels of incompatibility q0,A 6= q0,B , then

diAdt

= (τ − ηp)iA,diBdt

= (τ − ηp)iB ,

du

dt= (1− τ)(iA + iB) +

(1− q0,A

iAp− q0,B

iBp

)u − pu.


A conserved quantity

Planes orthogonal to the (iA, iB)-plane

Rα ={

(iA, iB , u) ∈ R3≥0 : iA − αiB = 0

}

are invariant under the flow since

d

dt(iA − αiB) = (τ − ηp)(iA − αiB) = 0

on Rα. Hence the ratio iAiA+iB

remains constant.

⇒ Neither strain can replace the other among the infectedindividuals based on stronger cytoplasmic incompatibility (Turelli1994, Haygood & Turelli 2009).


2-strain competition

iA

iB

u

R1iA

iB

u

(Left) Only difference are q0,A 6= q0,B . (Right) Different infectioncosts ηA < ηB lead to dominance of strain A.


Introduction of age structure

Individuals of different ages are subject to different fertility andmortality rates. We are interested in changes of stability ofequilibrium solutions compared to the unstructured case.

it(a, t) + ia(a, t) = −η1(a)(I (t) + U(t))i(a, t),

ut(a, t) + ua(a, t) = −η2(a)(I (t) + U(t))u(a, t),

i(0, t) = τ

∫ m

0β1(a)i(a, t)da,

u(0, t) = (1− τ)

∫ m

0β1(a)i(a, t) da

+

(1− q

I (t)

I (t) + U(t)

)∫ m

0β2(a)u(a, t) da,

where

I (t) =

∫ m

0i(a, t) da, U(t) =

∫ m

0u(a, t)da,


Existence of equilibrium solutions

For the existence of a disease-free equilibrium (0, u∗(a)) it isnecessary that ∫ m

0β2(a)da > 1

(whereas it always existed for the unstructured ODE model).


Existence of equilibrium solutions

For positive equilibrium solutions (i∗(a), u∗(a)) we need solutions of

1 = τ

� m

0

β1(a) exp

�−(I∗ + U∗)

� a

0

η1(r) dr

�da,

U∗�1 −

�1 − q I∗

I∗+U∗

� � m

0β2(a) exp

�−(I∗ + U∗)

� a

0η2(r) dr

�da

�

� m

0exp

�−(I∗ + U∗)

� a

0η2(r) dr

�da

=I∗

�τ−1 − 1

�� m

0exp

�−(I∗ + U∗)

� a

0η1(r) dr

�da

,

1

this leads to quadratic equations for the total populations (I∗,U∗)formally similar to the ODE case, but also to integral conditionsfor β1.


Linearization around steady state (i∗(a), u∗(a))

With the perturbations p(a, t) = i(a, t)− i∗(a) ands(a, t) = u(a, t)− u∗(a) and Taylor series expansions, we obtainthe linearized system (Farkas, Farkas & Hagen 2006–08)

pt(a, t) + pa(a, t) = −η1(a)

�p(a, t)(I∗ + U∗) + i∗(a)(P (t) + S(t))

�,

st(a, t) + sa(a, t) = −η2(a)

�s(a, t)(I∗ + U∗) + u∗(a)(P (t) + S(t))

�,

p(0, t) = τ

� m

0

β1(a)p(a, t) da,

s(0, t) = (1 − τ)

� m

0

β1(a)p(a, t) da +

�1 − q

I∗I∗ + U∗

�� m

0

β2(a)s(a, t) da

− q

�U∗

(I∗ + U∗)2P (t) − I∗

(I∗ + U∗)2S(t)

�� m

0

β2(a)u∗(a) da,

where

P (t) =

� m

0

p(a, t) da, S(t) =

� m

0

s(a, t) da.

1

The linearized system is governed by a strongly continuoussemigroup of linear operators, which is eventually compact but notnecessarily positive.


Characteristic equation

Substituting (p(a, t)s(a, t)

)= exp {λt}

(v(a)w(a)

)

into the linearized equations gives a nonlocal eigenvalue problemthat leads to a characteristic equation K (λ) = det

(. . .)

= 0. Thesigns of its roots determine the stability of the correspondingsteady state.


Special cases: trivial steady state i∗ ≡ 0, u∗ ≡ 0

TheoremThe trivial steady state is locally asymptotically stable if

τ

∫ m

0β1(a)da < 1 and

∫ m

0β2(a)da < 1.

On the other hand, if either

τ

∫ m

0β1(a) da > 1 or

∫ m

0β2(a) da > 1.

holds, then the trivial steady state is unstable.


Special cases: disease-free steady state

TheoremIf

τ

∫ m

0β1(a) exp

{−U∗

∫ a

0η1(r) dr

}da > 1,

where U∗ satisfies

1 =

∫ m

0β2(a) exp

{−U∗

∫ a

0η2(r)dr

}da,

then the disease-free steady state is unstable

Remark: It was always stable in the ODE case.


Discussion

From our simple ODE model we can make the prediction thatthere are no persistent Wolbachia strains with a transmissionefficacy τ < 3

4 .

Strains with higher transmission efficacy or lower mortality due toinfection establish themselves over competitors.

For the age-structured model, existence of a disease-freeequilibrium is now subject to a condition on the integral of thebirth rate.


Possible future works

I introduction of separate sexes (book by Iannelli et al. 2005)

I study of haplodiploid organisms (bees, ants and wasps), wheresex is determined by number of chromosomes

I agent-based models, contact networks (JZF & Mirco Musolesi,Department of Computer Science, University of St Andrews)


Acknowledgments

I my collaborator, Jozsef

I Jan Engelstadter (Institute for Integrative Biology, SwissFederal Institute of Technology, Zurich, Switzerland)

I Institute for Mathematics and its Applications (IMA,University of Minnesota), and Edinburgh MathematicalSociety for financial support

J. Farkas, P. Hinow. Structured and unstructured continuousmodels for Wolbachia infections. Bull. Math. Biol. 72:2067–2088(2010); arXiv:0906.1676

Thank you for your attention


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