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On Crop Vector borne diseasesOn Vector/Pest control

On the usefulness of Mathematical Modelingin Epidemiology and Pest/Vector control

Yves Dumont

CIRAD, Umr AMAP, Montpellier, FranceUniversity of Pretoria, department of Mathematics and Applied Mathematics, South Africa

SarChi Chair M3B2: Mathematical Models and Methods in Bio-engineering and Biosciences,University of Pretoria, South Africa

17th of October 2019

Yves Dumont #DigitAg Workshop


An example of crop vector-borne diseases modelingJoined work with Michael Chapwanya (UP, South Africa)[Chapwanya-Dumont, 2018].

-αv Sv

?µ1 + µ2V

-�

δIv

?µ1 + µ2V

@@

φbIvHpK

6Hp

- Lp-k1 Ip

��

φaSvIpK

?

-k2

?γ

Rp

Main variables

Hp : Healthy Plants

Lp : Latent Plants : infectedbut not yet infective

Ip : Infective Plants

Rp : Recovered/Removed/Resistant Plants

Sv : Susceptible Vectors

Iv : Infective vectors

V = Sv + Iv

Parameters of interestδ : recovery rate (persistent vs non-persistent),γ, φ : sanitary harvest, contact rate.



Vector-borne disease Model with Diffusion

Submodel for the Crop

∂Hp

∂t = −φbIvHp

K,

∂Lp

∂t = φbIvHp

K− k1Lp,

∂Ip

∂t = k1Lp − k2Ip − γIp,∂Rp

∂t = k2Ip.

(1)

Pest are moving while plants not !

Submodel for the Vector population∂Sv∂t = D∂2Sv

∂x2 + αvV − (µ1 + µ2V) Sv − φaSvIp

K + δIv,

∂Iv∂t = D∂2Iv

∂x2 + φaSvIp

K− δIv − (µ1 + µ2V) Iv

(2)

Mathematical Analysis - Existence of Solution - Existence ofTraveling-wave solutions



Simulations : spreading of Infective vectors

δ = 0 (Persistent case : an infective vector stay infectious until itsdeath)

Both vectors and infectious vectors travel at the same speed.



Simulations : spreading of Infective vectors

δ = 5 (non-persistent case)

Vectors and infectious vectors are NOT traveling at the same speed :the velocity of the infective front is low thanks to the virus death rate(on the stylet).Impact in terms of monitoring (monitor what?), and controlintervention (when?)



2D simulations - δ = 1

(a) Vector population (b) Infective Vectors

Once the whole area has almost been invaded by (susceptible)vectors, then the infective front appears, with a delay of almost 2weeks.



The Sterile Insect Technique (SIT)

SIT control generally consists of massive releases of sterileinsects in the targeted area.Sterilization can be done either by irradiation (IAEA) or by usinga natural parasite, Wolbachia.Several questions : When? How many? Where? How long? etcHowever, practically, massive releases of sterile males are onlypossible for a short period of time (e.g. production constraints).

Mathematical ModelsMosquito control (Aedes spp)Fruit fly control (c. capitata, b. dorsalis)

Based on submitted and ongoing work with Prof. Roumen Anguelovand Dr. Valaire Yatat (UP)



The minimalistic entomological modelMinimalistic in term of state variables and parameters...

Wild vector minimalistic model

(S1)

dAdt

= φF − (γ + µA,1 + µA,2A)A, (A Immature stage )

dMdt

= (1− r)γA− µMM, (M : Male stage)

dFdt

= rγA− µFF, (F : Mature female)

with

non negative initial data.

System (S1) is a monotone cooperative system. We setR = rγφ

µF(γ+µA,1), the so-called Basic Offspring Number.



The SIT model : constant and continuous releases

Minimalistic cooperative SIT model

(S2)

dAdt

= φF − (γ + µA,1 + µA,2A)A,

dMdt

= (1− r)γA− µMM,

dFdt

=M

M + M∗TrγA− µFF,

Setting MT1 = (√R−1)2

Q ,and Q =

µA,2µM

(γ+µA,1)(1−r)γ ,we derive the dynamics of(S2) according to the sizeof the releases, M∗T , that is

Proposition1 When M∗T > MT1 then system (S2) has no Positive equilibrium.2 When M∗T = MT1 then system (S2) has two equilibria 0 and E†

with 0 < E†.3 When 0 < M∗T < MT1 then system (S2) has three equilibria 0, E1

and E2 with 0 < E1 < E2.



Towards a practical strategy

System (S2) having particular properties, for a given 0 < M∗T < MT1

we can show that 0 (E2) is LAS inside Box [0,E1[ (]E1,+∞[).

0

E1

400

100

E2

200

300 400

Fe

ma

les 300

300

Males

200

400

Immature (A)

200

500

100100

0 00



Constant and continuous release

R ≈ 30.15, MT1 = 3745, MT = 5×MT1 , and M∗T = 100.Entry time : t∗ = 162 days.

(a) Const. and cont. release : 3D plot (b) Zoom in : the ’trap box’

With a ”one Week adulticide treatment”, before SIT starts, the entrytime, t∗, reduces to 113 days.



The PDE formulation : to include spreading

∂A∂t

= φF − (γ + µA,1 + µA,2A)A, (t, x) ∈ R+ × R

∂M∂t

= dM∂2M∂x2 + (1− r)γA− µMM,

∂F∂t

= dF∂2F∂x2 +

MM + MT

rγA− µFF,

(3)

where dF and dM are the female and male diffusion rates. In compactform, setting U = (A,M,F)t, model (1) reads as

∂U∂t

= D∂2U∂x2 + HMT (U), (t, x) ∈ R+ × R (4)



About PDE formulation

When MT = 0 a monostable TW solution may exists

There exists c∗ > 0, such that for each c ≥ c∗ system (1) has anondecreasing wavefront U(x + ct) connecting 0 and E∗ ; while for anyc ∈ (0, c∗), there is no wavefront U(x + ct) connecting 0 and E∗.

When 0 < MT < MT1 a bistable TW solution may exists

System (1) being cooperative with 2 stable equilibria, E0 =0 and1≡ E2, and one unstable equilibria, α ≡ E1, then following (Fang-Zhao2009), system (1) admits a monotone wavefront (U, c) withU(−∞) =0 and U(+∞) = E2.

When MT > MT1 , no TW solution exists.

According to the ODEs study, the solution converges to 0.



About PDE formulation : when 0 < MT < MT1

With moderate SIT control (black line) on the whole domain, the wild front (inred) is moving with a speed lower than without SIT (in blue). Not efficient.



About PDE formulation : Local and Permanent control

140 ≤ x ≤ 160, MT = 1.1 ∗MT1 and 0 ≤ x < 140, MT = 0.01 ∗MT1 .Yves Dumont #DigitAg Workshop


Dynamic control strategyUsing the same releases parameters, the following dynamic strategy couldbe used to change the sign of the velocity, and push back the invasion !



The ”Art of Modeling”

The aim is not to represent ALL the informations in a model.

One should not be impressed when a complex model fits a data setwell : with enough parameters, you can fit any data set.



Discussion : why Mathematical Modeling?

Formalize, synthetize, aggregate knowledge ... but especiallyTHINKING! ... Having a new ”point of view”Check (rapidly) hypothesis when some processes are not wellknown or unknownGENERICITY of Mathematical Models !Mathematical Modeling encompasses a wide diversity ofmodels : from microscopic models to macroscopic models, usingvarious approaches (deterministic, stochastic, continuous,discrete, ...)A qualitative analysis not only help us to derive varioustheoretical properties of the (sub)model(s) but also can behelpful to consider the best numerical methods to solve thetheoretical systems.A need of pluridisciplinary projects and joined collaborations....Not so easy to put into practice.



Thank you !Some recent references

Anguelov, R., Bekker, R., Dumont, Y., 2019. Bi-stable dynamicsof a host-pathogen model. Biomath, 8 (1) : p. 1-17.Strugarek, M., Bossin, H., Dumont, Y. 2019. On the use of thesterile insect technique to reduce or eliminate mosquitopopulations. Applied Math. Modelling 68 : 443-470.Bliman, P.A., Cardona-Salgado, D., Dumont, Y., Vasilieva, O.2019. Implementation of Control Strategies for Sterile InsectTechniques, Mathematical Biosciences 314, 43-60.Chapwanya M., Dumont, Y., 2018. On crop vector-bornediseases. Impact of virus lifespan and contact rate on thetraveling-wave speed of infective fronts. Ecological Complexity34 : 119-133Chapwanya, M., Dumont, Y. Application of MathematicalEpidemiology to crop vector-borne diseases. The cassavamosaic virus disease case. Chapter of Book : ”Springer Series :Mathematics of Planet Earth”. To appear.


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