competing for resources in the presence of infection ...bulai introduction the redatopr-prey model...

Competingfor resources

Bulai

Introduction

Thepredator-preymodel

Disease andinterferencein thepredatorpopulation

Sensitivityanalysis:comparisonbetween themodels

Disease andBeddington-deAngelisfunctionalresponse

Conclusions

Competing for resources in the presence ofinfection: Mathematical models of foraging

interference and disease transmission

Iulia Martina BulaiJoint work with Prof. Frank Hilker

Departement of Information Engineering,

University of Padova

14 February 2018

Bulai (UniPd) Competing for resources 1 / 29


Bulai

Introduction





Conclusions

Activities made withFinanziamento GNCS Giovani Ricercatori 2016

Visiting position: Institute of Environmental SystemsResearch, Osnabreuck, January 2017.Contributed talk: DSABNS 2017, Évora � PT,January-February 2017.Publication: I. M. Bulai, F. Spina, G. C. Varese, E.Venturino. Waste-water bioremediation using white rotfungi: validation of a dynamical system with real dataobtained in laboratory. To appear in MathematicalMethods in the Applied Sciences, 2018.Work in progress: I. M. Bulai, F. Hilker. Competing forresources in the presence of infection: Mathematicalmodels of foraging interference and disease transmission.Work in progress: I. M. Bulai, F. Hilker. The "e�ect" ofthe interference on the predator population in a predatorprey model.



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Summary

1 Introduction

2 The predator-prey modelDisease and interference in the predator population

3 Sensitivity analysis: comparison between the models

4 Disease and Beddington-deAngelis functional response

5 Conclusions



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Introduction to the problem (1)

A predator-prey model is introduced with a disease in thepredator population

One can consider no vertical transmission (NoVT) modelor the vertical transmission (VT) one: we will study themost general one, partial vertical transmission (PVT)(includes both NoVT and VT)

The transmission of the disease could be density dependent(DDT) or not, thus frequency dependent (FDT). Whichare the di�erences?



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Introduction to the problem (2)

On one side the interference in the predator population isconsidered and on the other side both interference andhandling term are considered in the functional response(Beddington-DeAngelis); There are di�erences?

Which direct e�ect have the disease and the interferenceon the predator population, and indirectly on the preypopulation?



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Hypothesis for the �rst two models

There is a disease in the predator population, could bedensity-dependent disease transmission (DDT) orfrequency-dependent disease transmission (FDT)

There is the interference between the predators

The predator population is divided only in two classes, thesusceptible ones, S , and the infected, I , and not treatmentis given to the infected predators

The prey population N is the only resource of food for thepredators.



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The mathematical model

The model reads:

dN

dτ= r

(1− N

K

)N − f (S , I )S − g(S , I )I (1)

dS

dτ= −mS − β(S , I ) + eS f (S , I )S + (1− ν)eIg(S , I )I

dI

dτ= −mI − µI + β(S , I ) + νeIg(S , I )I .

With

f (S , I ) =aS N

1+ aSwSSS + aSwSI I

and

g(S , I ) =aI N

1+ aIwISS + aIwII I.



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The mathematical model (2)

Whereβ(S , I ) = βSI DDT

or

β(S , I ) =βSI

S + IFDT.

In FDT, contact rate is assumed to be independent of hostdensity. In DDT transmission, contact rate is assumed tolinearly increase with host density.



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Simpli�cation of the models

We will assume

aS = aI := a, eS = eI := e and

wSS = wSI = wIS = wII := w .

We apply the coordinate transformations

P = S + I and i =I

S + I

We nondimensionalize it by applying the next substitutions

N(t) =1KN(τ), P(t) =

a

rP(τ) and t = rτ



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Conclusions

Simpli�ed models

We get

dN

dt= (1− N)N − NP

1+ wP(2)

dP

dt= a

(−mP − µPi +

NP

1+ wP

)di

dt= ia

((β − µ)(1− i)− N(1− ν)

1+ wP

)FDT

di

dt= ia

((βP − µ)(1− i)− N(1− ν)

1+ wP

)DDT.



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Conclusions

System's equilibria for FDT model and the stabilityanalysis (1)

Proposition 1.

(i) The trivial equilibrium point E0 = (0, 0, 0) and theprey-and-predator-free point E3 = (0, 0, 1) are alwaysfeasible, furthermore they are both unstable.

(ii) The predator-and-disease-free point E1 = (1, 0, 0) is alwaysfeasible, and it's stable if m − 1 > 0 and β − µ < 1− νhold.

(iii) The disease-free point

E2 = (m(1+ wP2),P2, 0)

is feasible if the predator population is non negativem − 1 < 0. Furthermore it's stable if β − µ < (1− ν)mhold.



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(iv) The predator-disease-induced extinction point

E4 =

(1, 0,

β − µ− (1− ν)

β − µ

)is feasible if β − µ > 0, and it's stable if m − 1+ µi4 > 0hold.

(v) The coexistence equilibrium is

E∗ =

(1+ (w − 1)P∗

1+ wP∗,P∗,

β − µ−m(1− ν)

β − µν

),

P∗ is the root of the second degree polynomialAP2 + BP + C = 0.



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Conclusions


For the feasibility of the coexistence equilibrium we need to have

1+ (w − 1)P∗ > 0,β − µ−m(1− ν)

β − µν> 0

and one between

(a) A < 0 and one between B or C less than 0 or

(b) A > 0 and one between B or C greater than 0

must hold, A, B and C are the coe�cient of the 2nd degreepolynomial. While for it's stability the Routh-Hurwitzconditions for a 3rd degree polynomial mus hold.



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Density-dependent disease transmission

Table: Density-dependent disease transmission

E = (N,P, i). Feasibility conditions Stability conditions

E0 = (0, 0, 0) always feasible unstableE1 = (∗, 0, 0) always feasible m − 1 > 0E2 = (∗, ∗, 0) m − 1 < 0 βP2 − µ < (1− ν)mE3 = (0, 0, ∗) always feasible unstableE4 = (∗, 0, ∗) ν = 1 unstableE∗ = (∗, ∗, ∗) N∗ > 0, i∗ > 0, Routh-Hurwitz cond.



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Comparison between the system with FDT and DDT

In both system's were found �ve equilibrium points plusthe trivial equilibrium point

E4, in the DDT case is feasible if we consider the particularcase ν = 1 and unstable while in FDT case the nonnegativity of the prevalence must be required and underproper conditions could be stable

The conditions for the stability of E1, E2 and E∗ changesfrom a model to another

The coexistence equilibrium, in FDT case an analyticalexpression of it was found while for DDT it was showedthat exist only numerically



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Transcritical bifurcation diagram for FDT and DDTvarying the transmissibility

Figure: Left: the bifurcation diagram for FDT varying β, thetransmissibility; Right: the bifurcation diagram for DDT varying β,the transmissibility.



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Transcritical bifurcation diagram for FDT and DDTvarying the disease-induced death rate

Figure: Left: the bifurcation diagram for FDT varying µ, thedisease-induced death rate; Right: the bifurcation diagram for DDT

varying µ, the disease-induced death rate.



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Transcritical bifurcation diagram for FDT and DDTvarying the waste time

Figure: Left: the bifurcation diagram for FDT varying w , the waste

time; Right: the bifurcation diagram for DDT varying w , the waste

time.



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Sensitivity analysis: comparison between the models

We would like to predict how changes in the parameters willa�ect the solutions. We will make a sensitivity analysis of theparameters: β and w .



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Sensitivity analysis: FDT



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Sensitivity analysis: DDT



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Sensitivity analysis: FDT vs DDT (predator pop.)

Figure: Predator population varying both β and w . Left: FDT;

Right: DDT.



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The predator-prey model: PVT andBeddington-deAngelis response function

For simplicity we directly write the nondimensionalized modelsagain for FDT and DDT, respectively.

dN

dt= (1− N)N − NP

1+ wP + hN(3)

dP

dt= a

(−mP − µPi +

NP

1+ wP + hN

)di

dt= ia

((β − µ)(1− i)− N(1− ν)

1+ wP + hN

)FDT

di

dt= ia

((βP − µ)(1− i)− N(1− ν)

1+ wP + hN

)DDT



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Sensitivity solutions for BdA-FDT

Figure: From top to bottom: the prey population, predator population

and prevalence varying both β and w . Left: FDT; Right: DDT.



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Sensitivity solutions for BdA-DDT

Figure: From top to bottom: the prey population, predator population

and prevalence varying both β and w . Left: FDT; Right: DDT.



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Conclusions (1)

We investigated the impact of predator interference onpredator-prey systems with infectious diseases circulating inthe predator population.We have shown that interference in�uences predatorpopulation size and can thus impact disease emergenceand infection levels. Similarly, the strength of the diseasemay impact predator population size and cascade to theprey level.We have considered two types of disease transmission,namely frequency-dependent and density-dependenttransmission. Contact rates in the former are independentof host population size and proportional to host populationsize in the latter. These are two extremes in a continuumof possibilities, and the truth for many species may besomewhere in the middle.



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Conclusions (2)

We introduced predator interference using the approach of�wasting time� similar to Beddington (1975) and carefullydistinguish between di�erent wasting times of susceptibleand infected predators.

We included handling time as well, giving a functionalresponse analogous to the Beddington-DeAngelis functionalresponse (Beddington 1975, DeAngelis et al 1975).



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Bibliography

Andrew M. Bate, Frank M. Hilker, Disease in group-defendingprey can bene�t predators, Theor Ecol (2014) 7:87�100.

J. R. Beddington, Mutual Interference Between Parasites orPredators and its E�ect on Searching E�ciency, Journal ofAnimal Ecology, Vol. 44, No. 1 (Feb., 1975), pp. 331-340 .

Hilker FM, Schmitz K (2008) Disease-induced stabilization ofpredator-prey oscillations. Journal of Theoretical Biology255(3), 299-306 .



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Thanks for your attention!


competing for resources in the presence of infection ...bulai introduction the redatopr-prey model...

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