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Plan de travailIntroductionConclusion

The generalized eco-epidemic model

DJEDDI Kamel

Universite des sciences et de la technologieHouari Boumediene

Algiers, 10-13 June 2013

DJEDDI Kamel (Universite de Oum El Bouaghi) The generalized eco-epidemic model


Introduction

Mathematical models are essential tools in order to understand themechanisms responsible for persistence or extinction of species in naturalsystems. In ecological models persistence is generally desired. Bycontrast, investigations in epidemic models usually aim at findingmechanisms that lead to the extinction of the parasites or infectionsHowever, it is known that diseases can not only greatly affect their hostpopulations, but can also affect other. In this paper we discuss thedemographic as well as the epidemic model and introduce theeco-epidemic model. Furthermore, we normalize the model and discussthe possible emergence of bifurcations for the steady state.

We begin our analysis we will briefly outline the construction of one ofthe simplest predator-prey models and one of the most elementaryepidemic models, which together form the building blocks for the moregeneral eco-epidemic model we would like to consider. The basicdemographic model in general accounts for two interacting species.



Chapitre 1: Generalites

The nature of interactions can be of competing, predator-prey, orsymbiotic nature. A typical formulation for a predator-prey model is givenby {

X = SX −MXX2 −G (X) Y

Y = EG (X)Y −MY Y(1)

where S is the specific growth rate of the prey X. Apart from predationthe growth of the prey is limited by intraspecific competition assumed toincrease quadratically in X with the coefficient MX thereby giving alogistic evolution for the prey dynamics. The predation is expressed bythe so-called per capita functional response G(X). The efficiency ofbiomass conversion is given by the yield constant E. MY is the mortalityrate of the predator population.



Chapitre 1: Generalites

This simple model structure has been analyzed for a variety of differentfunctional responses G(X) describing the prey-dependent predation rate.However, in our case the function G(X) is not specified to keep themodel more general.

A predator-prey problem (Volterra’s model)In a lake there are two species of fish: A, which lives on plants of whichthere is a plentiful supply, and B (the predator) which subsists by eatingA (the prey). We shall construct a crude model for the interaction of Aand B.Let X(t) be the population of A and Y (t) that of B. We assume that Ais relatively long-lived and rapidly breeding if left alone. Then in time atthere is a population increase given by

aX.4 t, a > 0


Example.1.


due to births and natural deaths, and negative increase

−cX.Y.4 t, c > 0

owing to As being eaten by B (the number being eaten in this time beingassumed proportional to the number of encounters between A and B).The net population increase of A, 4x, is given by

4x = aX.4 t− cX.Y.4 t

so that in the limit 4t → 0

X = aX − cXY,


(2)


Assume that, in the absence of prey, the starvation rate of Bpredominates over the birth rate, but that the compensating growth of Bis again proportional to the number of encounters with A. This gives

Y = −bY + dXY

with b > 0, d > 0We now plot the phase diagram in the X, Y plane. Only the quadrant

X ≥ ≥ 0

is of interest. The equilibrium points are where

F (X, Y ) = aX − cXY = 0, G(X, Y ) = −bY + dXY = 0

that is at (0, 0) and (b/d, a/c). The phase paths are given bydY/dX = G/F , or

dY

dX=

(−b + Xd)Y(a− cY )X


, Y0

( )3


∫(a− cY )

YdY =

∫(−b + Xd)

XdX

ora lnY + b lnX − cY −Xd = C

where Cin the form a lnY − cY + b lnX −Xd = C,

point (b/d, a/c).

Figure.1. shows the phase paths for a particular case. The direction onthe paths can be obtained from the sign of X at a single point, even onY = 0. This determines the directions at all points by continuity. From

Y = 0, that is, onthe lines Y = 0 and Y = Xd/b, and those of infinite slope on X = 0,that is, on the lines X = 0 and Y = aX/c.


is an arbitrary constant, the parameter of the family. Writing

( )

( )4

4

and the isoclines of zero slope occur on(2) ( )3

the equilibrium


Since the paths are closed, the fluctuations of X(t) and Y (t), startingfrom any initial population, are periodic, the maximum population of Abeing about a quarter of a period behind the maximum population of B.As A gets eaten, causing B to thrive, the population X of A is reduced,causing eventually a drop in that of B. The shortage of predators thenleads to a resurgence of A and the cycle starts again. A sudden change instate.



Figure.1. Typical phase diagram for the predatorprey model.



2-Classical epidemic models partition the population into severalepidemiological classes. The population, in our case the predatorpopulation Y , is usually split into susceptibles YS , infected YI , andrecovered YR. The latter may be thought of as being immunized, at leastfor a period of time, after which they return into the class of susceptibles.Following this population division, in absence of vital dynamics, i.e.,demographic terms to account for births and natural deaths, a simpleSIRS (susceptibles-infected-recovered-suspectibles) model would bewritten as

YS = −λ (YS , YI) + δYR

YI = λ (YS , YI)− γYI

YR = γYI − δYR


due to external causes, such as a bad season for the plants, puts thestate on to another closed curve, but no tendency to an equilibriumpopulation, nor for the population to disappear, is predicted. If we expectsuch a tendency, then we must construct a different model

( )5


The recovered become susceptible again with a fixed rate δ.In general, pathogen transmission is expressed by interactions amongindividuals. The latter are modeled by the incidence function (YS , YI), forwhich the most common approaches are the mass action

(YS , YI) = bYSYI and the socalled standard incidence function orfrequency-dependent transmission (YS , YI) = bYSYI/(YS + YI). In bothcases susceptibles YS and infected YI are assumed to be well-mixed andhence, to interact randomly. However, it is not clear if the assumption ofrandom interactions and an equal distribution of infected and uninfectedis appropriate to describe pathogen transmission in wild populations.


assuming linear transition rate from the infected to the recovered class.γ

λ

λλ


Both, smallscale experiments as well as observed disease dynamics giveevidence that simple mass action is not an adequate model in manysituations. The simplest argument for an asymmetry of the incidencefunction is that due to a patchiness in the disease, on average eachinfected individual is more likely to have an infected neighbor. The morebiological details are taken into account. Proposed a more generalincidence function of the form (YS , YI) = kYSY p

I /(1 + mY pI ).

Additionally a variety of other incidence functions have been investigatedby various authors. A universal approach has not been found yet.However, using the generalized approach we avoid specifying theincidence function but study more generic properties of the model classunder consideration.


λ


A general epidemic model Consider the spread of a non-fataldisease in a population which is assumed to have constant size over theperiod of the epidemic. At time t suppose the population consists ofX(t) susceptibles: those so far uninfected and therefore liable toinfection ;Y (t) infectives: those who have the disease and are still at large ;(t) who are isolated, or who have recovered and are therefore immune.

Assume there is a steady contact rate between susceptibles and infectivesand that a constant proportion of these contacts result in transmission.Then in time ∆t, ∆ of the susceptibles become infective, where

∆ = −βXY ∆t,

and β is a positive constant.


X

X

Z

.2.Example


If γ > 0 is the rate at which current infectives become isolated, then

∆Y = βXY ∆t− γY ∆t

The number of new isolates ∆ is given by

∆ = γY ∆t,

Now let ∆t → 0. Then the systemX = −βXY

Y = βXY − γY

Z = γY


( )6

Z

Z


with suitable initial conditions, determines the progress of the disease.Note that the result of adding the equations is

d

dX(X + Y + Z) = 0

that is to say, the assumption of a constant population is built into the

restriction X ≥ 0, Y ≥ 0, equilibrium occurs for Y = 0 (all X ≥ 0)


model. and are defined by the first two equations in ( ). With theX , Y 6


To analyze the effect of a disease in the predator population Y on thedynamics of interaction with the prey X we combine the demographic

X = SX −G (X) (YS + YR + αYI)−MXX2

YS = EG (X) (YS + YR + αYI)−MY YS + δYR − λ (YS , YI)YI = λ (YS , YI)− (MY + µ) YI − γYI

YR = γYI − δYR −MY YR

Here we assume neither vertical transmission, nor vertical immunity, i.e.,that the infected predators as well as the recovered predators reproduceonly susceptibles.


( )7

model equation (1) with the SIRS epidemic model equation as follows( )5


Furthermore, we suppose that the disease can, in principle, influence thedemographic parameters. The disease may induce a disease-related

infected YI expressed by the factors α and β respectively. All parametersand terms denoted by Greek letters are related to the disease. Note, thatin the absence of weakening effects of the disease (i.e., α = β = 1 andµ = 0

Y = YS + YI + YR.This means that the disease could have in principle no influence on the

start by computing the steady state and its stability with respect toperturbations.


mortality rate and reduce the predation and reproduction rates of theµ

( )7

( )7) the combined model equation reproduces the populationdynamics of the uninfected model equation (1) , with

ecological dynamics. To analyze the dynamics of model one would


But a local stability analysis cannot be performed since an analyticalcomputation of the steady states is impossible because G(X) and(YS , YI) are not specified but assumed to be general functions. However,this difficulty can be overcome To use this approach we assume that apositive steady state (X∗, Y ∗

S , Y ∗I , Y ∗

R) exists. We now define normalizedvariables

x :=X

X∗ , ys :=YS

Y ∗S

, yi :=YI

Y ∗I

, yr :=YR

Y ∗R

.

Further, we define a normalized functional response g (x) := G(X∗x)G(X∗) , and

a normalized incidence function l (ys, yr) := λ(Y ∗S ys,Y ∗

I yi)

λ(Y ∗S ,Y ∗

I ) .



Note, that in the space of normalized state variables the steady state isby definition

(x∗, y∗s , y∗i , y∗r ) :=(

X∗

X∗ ,Y ∗

S

Y ∗S

,Y ∗

I

Y ∗I

,Y ∗

R

Y ∗R

):= (1, 1, 1, 1)

In the same manner, we obtain l (y∗s , y∗r ) = g (x∗) = 1.Following thenormalization procedure we can rewrite equation (3) as

x = 1X∗

[SX∗x−G (X∗) g (x) (Y ∗

S ys + Y ∗Ryr + αY ∗

I yi)−MXX∗2x2]

ys = 1Y ∗

S[EG (X∗) g (x) (Y ∗

S ys + Y ∗Ryr + αβY ∗

I yi)−MY Y ∗S ys + δY ∗

Ryr

− λ (Y ∗S , Y ∗

I ) l (ys, yr)]

yi = 1Y ∗

I[λ (Y ∗

S , Y ∗I ) l (ys, yr)− (MY + µ)Y ∗

I yi − γY ∗I yi]

yr = 1Y ∗

R[γY ∗

I yi − (MY + δ)Y ∗Ryr]

.



x = ax

[x− G(X∗)(Y ∗

S +Y ∗R+αY ∗

I )axX∗ g (x)

(Y ∗

S +Y ∗R

Y ∗S +Y ∗

R+αY ∗I

(Y ∗

S

Y ∗S +Y ∗

Rys + Y ∗

R

Y ∗S +Y ∗

Ryr

)+ Y ∗

I

Y ∗S +Y ∗

Ryi

)−MX

X∗

axx2

ys = as

[EG(X∗)(Y ∗

S +Y ∗R)

asY ∗S

g (x)(

Y ∗S

Y ∗S +Y ∗

Rys + Y ∗

R

Y ∗S +Y ∗

Ryr + αβY ∗

R

Y ∗S +Y ∗

Ryr

)− MY

asys

+ δY ∗

R

asY ∗S

yr −λ(Y ∗S ,Y ∗

I )asY ∗

Sl (ys, yr)

yi = ai (l (ys, yr)− yi)yr = ar (yi − yr)



x = ax

[x− ∼

mxg (x)(∼fα

(bys +

∼byr

)+ fαyi

)−mxx2

]ys = as

[esg (x)

(bys +

∼byr + fβyi

)−myys +

∼esyr −

∼myl (ys, yr)

]yi = ai (l (ys, yr)− yi)yr = ar (yi − yr) .

where,∼

mx := G(X∗)(Y ∗S +Y ∗

R+αY ∗I )

axX∗ = 1−mx∼fα := Y ∗

S +Y ∗R

Y ∗S +Y ∗

R+αY ∗I

= 1− fα

b := Y ∗S

Y ∗S +Y ∗

R∼b := Y ∗

R

Y ∗S +Y ∗

R= 1− b

fα := Y ∗I

Y ∗S +Y ∗

R

mx := MXX∗

ax



andes := EG(X∗)(Y ∗

S +Y ∗R)

asY ∗S

fβ := αβY ∗R

Y ∗S +Y ∗

R

my := MY

as∼es := δ

Y ∗R

asY ∗S

= 1− es∼

my := λ(Y ∗S ,Y ∗

I )asY ∗

S= 1−my.

andes := EG(X∗)(Y ∗

S +Y ∗R)

asY ∗S

fβ := αβY ∗R

Y ∗S +Y ∗

R

my := MY

as∼es := δ

Y ∗R

asY ∗S

= 1− es∼

my := λ(Y ∗S ,Y ∗

I )asY ∗

S= 1−my.



As an advantage of this approach these parameters are easy to interpretin the biological context. The scale parameters ax, as, ai and ar forinstance encode the inverse timescales of the normalized state variables.They measure the relation between the lifetimes of the different species.All other scale parameters are between 0 and 1 and describe weightfactors of certain processes of the model at the steady state. The lossesdue to intraspecific competition relative to the total losses within theprey are represented by the parameter mx. To be specific, if mx is closeto 1 the losses of prey due to intraspecific competition preponderate.



The parameter∼

mx = 1−mx expresses losses caused by predation. fα

and∼fα are the fractions of prey consumed by infected predators and

healthy predators respectively. In the same way b is the fraction of

healthy predators that are susceptible and∼b = 1− b the fraction of

healthy predators that are recovered. Further, the parameteres representsthe weight factor of the natural growth terms of susceptibles due toconsumption of X. At the steady state, the fraction of gains due torecovered predators that become susceptible again is given by

∼es = 1− es



The natural mortality for the predator relative to the total losses isexpressed by my. The stability of the steady state under consideration(x∗, y∗s , y∗i , y∗r ) := (1, 1, 1, 1) depends on the eigenvalues of the Jacobian.The steady state is stable if all eigenvalues have a negative real part.Consequently only two bifurcations can separate stable from unstableparameter regions: the saddle-node type bifurcation where a realeigenvalue crosses the imaginary axis and a Hopf bifurcation where a pairof complex conjugate eigenvalues crosses the imaginary axis. Because allnormalized state variables and the normalized processes (l (ys, yr) , g (x))are equal to one at the steady state, the Jacobian of the normalizedmodel contains, in addition to the scale parameters, only the derivativesof the normalized processes in the steady state.



We definegx := ∂g(x)

∂x |x∗ls := ∂l(ys,yr)

∂ys|y∗s ,y∗i

li := ∂l(ys,yr)∂yi

|y∗s ,y∗i

as the generalized parameters. These parameters can be interpreted asnonlinearity measures of the corresponding functions with respect to thevariable of the derivative. If the function G(X) is linear in X thederivative of the normalized function gx is equal to one. It is zero for aconstant function and two for a quadratic function. To be consistent withprevious publications we let gx be the predator sensitivity to prey. In thesame sense we denote by ls and li the incidence sensitivity to susceptiblesand to infected respectively.



In summary, the Jacobian consists of ten scaling parameters and threegeneralized parameters. How to obtain the test functions for the abovementioned bifurcations from the Jacobian. These test functions enable usto draw 3D bifurcation diagrams. Since we are essentially interested inthe influence of different mathematical expressions for the functionalresponse and for the incidence function, we focus our bifurcation analysison the generalized parameters gx, ls and li. We chose the other scaleparameters according to biological reasoning. It is known that in manycases the timescale for the lifetime of species belonging to differenttrophic levels slows down with each higher trophic levels. Hence, wecould assume that the inverse timescale of the susceptible predators isless than half the timescale of the prey, i.e., as = 0.4ax. By renormalizingthe timescale we can say that ax = 1 and as = 0.4.



It is further reasonable to expect that the timescale of the infectedpredators is slightly larger than the timescale of the susceptible predatorssince we suppose that their overall lifetime is shorter. Let us assumeai = 0.5. Clearly, this intuitive way is much more appropriate thanguessing some abstract parameters. If we would analyze a specific realsystem at the steady state we could, in principle, also gain an appropriatevalue for each scale parameter by measuring the corresponding rates.Approximating all other scale parameters, we end up with threeparameters that we consider the most interesting bifurcation parameters.These are the sensitivity of the predator with respect to prey gx and thesensitivity of the incidence function with respect to susceptibles ls andinfected li. The computation of 3D bifurcation diagrams allows us todiscuss the stability properties of the eco-epidemic model depending onthe mathematical form of the functional response G(X) and theincidence function λ (YS , YI) .


IntroductionConclusion

. Bifurcation diagram of a generalized predator-prey model. Asurface of Hopf bifurcations (dark) and two surfaces of saddle-node typebifurcations (transparent bright) are shown. The bifurcation parametersare the prey sensitivity gx, the timescale of the predator ay and thecompetition mx (intraspecific competition of the prey)


Figure

sensi

tivity

topre

yg x

competition

mx

relative

predato

r timesca

le

ay



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