EPITAG - EPIdemiological modelling and control - for Tropical AGriculture Inria associate team & LIRIMA team
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EPITAG – EPIdemiological modelling and control
for Tropical AGriculture
Inria associate team & LIRIMA team
2017–2019
Suzanne TOUZEAU Samuel B OWONG
BIOCORE, Inria Sophia Antipolis University of Douala, Cameroon
ISA, INRA Sophia Antipolis
January 2020
1 / 46
LIRIMA
= International Laboratory for Computer Sciences and Applied Mathematics
I Before: CARI (1992–), Sarima (2004–2008)
I 2009–2014: 12 project-teams
I 2015–2019: Inria International Lab “LIRIMA”
via Inria associate team programme
More: https://lirima.inria.fr/
Inria associate team
I Aim: “fostering collaborations between Inria project teams and
top-level research teams across the world”
I Lifespan: 3 to 6 years
I Exchanges of researchers and students (6 13 ke/year)
More: https://www.inria.fr/...
2 / 46
Where we are
3 / 46
Who we are
BIOCORE Cameroon
• Suzanne TOUZEAU (INRA) • Samuel B OWONG (Douala)
• Jean-Jules T EWA (Yaoundé 1)
• Jean-Luc G OUZÉ
• Berge T SANOU (Dschang)
• Frédéric G ROGNARD
• Émile M INYAKA (Douala)
• Ludovic M AILLERET (INRA)
• Myriam D JOUKWE TAPI
• Samuel N ILUSMAS (Douala, PhD 2015–)
(INRA, PhD 2016–) • Israël TANKAM C HEDJOU
• Clotilde D JUIKEM (Yaoundé I, PhD 2016–)
(Inria, PhD 2019–) • Yves F OTSO F OTSO
(Dschang, PhD 2017–)
• Janvier P ESSER N TAHOMVUKIYE
Other participants (Douala, PhD 2017–)
• Clotilde D JUIKEM
I Yves D UMONT (CIRAD)
(Douala, Master April 2018)
I Gauthier S ALLET (emeritus) • Michel M OUGANG
(Douala, Master Oct. 2019)
4 / 46
Context
• Crop pests, diseases and weeds destroy up to 40% of global
crop yields every year ⇒ threat to food security
• Cameroon: agriculture is a major sector for employment (62%)
and revenues (30% of exports, 15% of GDP)
å Controlling crop pests is a major issue
Control methods
Pesticides: high financial and environmental cost, health issues
Alternatives: cropping practices, biological control, plant resistance
Why models
I complement field studies (costly and time-consuming)
I formalise and integrate knowledge
I help design efficient strategies for integrated pest management
5 / 46
What we do
Aims
Study the epidemiology and management of tropical crop diseases,
mathematically and numerically, with a focus on Cameroon
Approach
I Tools from dynamical systems and control theory
I Development and analysis of models to:
1. understand the plant–parasite interactions
2. identify relevant parameters
3. predict the evolution of damages
4. provide efficient and sustainable control strategies to limit damages
Challenges
I Relevance of our models
⇒ collaboration with field experts and involvement of stakeholders
I “Small data” in epidemiology (scarce and often qualitatively known)
6 / 46
How we collaborate
Joint PhD supervision on specific topics
Long visits in France: 3 students/year, 3–5 months/visit
T1. Cocoa plant mirids
Myriam D JOUKWE TAPI, Douala–Cirad (2015–)
S. B OWONG, L. B AGNY-B EILHE, Y. D UMONT
T2. Plantain plant-parasitic nematodes
Israël TANKAM C HEDJOU, Yaoundé I–Inria (2016–)
J.-J. T EWA, L. M AILLERET, F. G ROGNARD, S. TOUZEAU
T3. Coffee berry borers
Yves F OTSO F OTSO, Dschang–Inria (2017–)
B. T SANOU, S. B OWONG, F. G ROGNARD, S. TOUZEAU
T4. Coffee leaf rust
Clotilde D JUIKEM, MSc (2018) & PhD Inria–Douala (2019–)
S. B OWONG, F. G ROGNARD, S. TOUZEAU
7 / 46
Other research topics
Cabbage diamondback moth
I Michel M OUGANG, MSc Douala (Oct. 2019)
S. B OWONG
I Aurélien Vanes K AMBEU YOUMBI, PhD Dschang (2019–)
B. T SANOU
Maize stalk borer
I Janvier Pesser N TAHOMVUKIYE, PhD Douala (2017–)
S. B OWONG
Root-knot nematodes in horticultural crops
I Samuel N ILUSMAS, PhD INRA (2016–)
S. TOUZEAU, C. C APORALINO, V. C ALCAGNO, L. M AILLERET
8 / 46
Support
Inria associate team + UMMISCO & CIRAD
Grants
Research schools
I École de Contrôle Optimal Numérique, France, 2018 [Yves]
I International Graduate School on Control, Italy, 2019 [Yves, Clotilde]
Conferences
I CARI, Stellenbosch, South Africa, 2018 [Israël, Yves]
I BIOMATH , Bȩdlewo, Poland, 2019 [Israël, Yves, Clotilde]
Mobility
I EMS-Simons for Africa PhD development grant, 2018 [Israël]
I AUF, Collège doctoral régional de l’Afrique Centrale et des Grands Lacs
« Mathématiques, Informatique, Biosciences et Géosciences de
l’Environnement », 2019 [Yves]
PhD
I SCAC grant, 2017 [Janvier]
I Inria CORDI grant, 2019 [Clotilde]
9 / 46
SWOT
Positive Negative
Strengths Weaknesses
Internal
• High socio-economic impact • Unbalanced exchanges
(academic age structure)
• Motivated students,
jointly supervised • Still limited scientific production
(young team), fast improving
• Involvement of CIRAD
Opportunities Threats
• Links with field partners • Administrative hindrances
for Cameroonian students
External
• Financial support by
UMMISCO • Difficulties to ensure (other)
long-lasting fundings
• Links with INRA
• Associate Team format for
partners such as CIRAD
10 / 46Focus
T1. Cocoa plant mirids
Myriam D JOUKWE TAPI, Douala–Cirad (2015–)
S. B OWONG, L. B AGNY-B EILHE, Y. D UMONT
T2. Plantain plant-parasitic nematodes
Israël TANKAM C HEDJOU, Yaoundé I–Inria (2016–)
J.-J. T EWA, L. M AILLERET, F. G ROGNARD, S. TOUZEAU
T3. Coffee berry borers
Yves F OTSO F OTSO, Dschang–Inria (2017–)
B. T SANOU, S. B OWONG, F. G ROGNARD, S. TOUZEAU
T4. Coffee leaf rust
Clotilde D JUIKEM, MSc (2018) & PhD Inria–Douala (2019–)
S. B OWONG, F. G ROGNARD, S. TOUZEAU
11 / 46Plantain plant-parasitic nematodes
Rosendahl
D. Coyne
A: [Jesus, Agron Sustain Dev 2014]; B: M. MacClure, Univ. Arizona; C: [Zhang, EJPP 2012]
• Major staple food – Cameroon: 2% GDP
• Nematodes (Radopholus similis): root lesions Ô great damages
Most important pest on fruit crops in the tropics
• Control: nematicides ↔ cropping practices (soil sanitation),
tolerant or resistant banana varieties, biological control
– Complex interaction model for the West Indies [Tixier et al. 2006]
– Partners: CIRAD and CARBAP
12 / 46Burrowing nematode (Radopholus similis) disease cycle
[F.E. Brooks 2008]
I Obligate migratory endoparasite, < 1 mm
I Life cycle: 20–25 days
I Sexual reproduction & hermaphrodism (mature males not
infective)
13 / 46Plant growth
New sucker flowering harvest New sucker
root growth fruit growth fallow
I Herbaceaous flowering plant, ca. 5 m
I Single inflorescence that dies after fruiting → bunch 30–50 kg
I Cycle: ca. 11 months, 7 months until flowering
I Asexual reproduction by offshoots / new sucker
14 / 46Plant root – nematode interactions
New sucker flowering harvest New sucker
root growth fruit growth fallow
R. similis destroy roots and affect:
I uptake of water and nutrients Ô reduced growth & bunch weight
I anchorage Ô toppling
åDoubly hybrid system
15 / 46Model: initialisation
New sucker flowering harvest New sucker
root growth fruit growth fallow
0 d D τ T
S
X
P
S(0) = S0 plant root
P(0) = P0 nematodes
in soil
X (0) = 0 nematodes
in root
Hypothesis: nematode-free sucker
16 / 46Model: cropping season
New sucker flowering harvest New sucker
root growth fruit growth fallow
0 d D τ T
ρ(t)=ρ ρ(t)=0
ρ(1-X/K) S
δX/(S+Δ)
X
βS μ αδX/(S+Δ)
ω γ plant root nematodes
P 1-γ
root growth root consumption
z }| { z }| {
S SX
Ṡ = ρ(t) S 1 − −δ
K S+∆
SX
Ṗ = − βPS + δ α (1 − γ) − ωP
S+∆
SX
Ẋ = +βP S +δ αγ −µX
| {z } S+∆ | {z }
root entering
| {z } mortality
feeding & reproduction
17 / 46Model: harvest
New sucker flowering harvest New sucker
root growth fruit growth fallow
0 d D τ T
S
X
P
S(D + ) = 0 plant root
P(D + ) = P(D) + q X (D) nematodes
in soil
X (D + ) = 0 nematodes
in root
18 / 46Model: fallow
New sucker flowering harvest New sucker
root growth fruit growth fallow
0 d D τ T
S
X
ω
P
Ṡ = 0 plant root
Ṗ = −ω P nematodes
in soil
Ẋ = 0 nematodes
in root
⇒ P(T ) = P(D) + q X (D) e−ω τ
19 / 46Model: new sucker
New sucker flowering harvest New sucker
root growth fruit growth fallow
0 d D τ T
S
X
P
S(T + ) = S0 plant root
P(T + ) = P(T ) nematodes
in soil
X (T + ) = 0 nematodes
in root
Etc. for the next periods, with transition law:
P(Tk +1 ) = P(Tk + D) + q X (Tk + D) e−ω τ
20 / 46Analysis
I Model reduction and analysis (fast root infections)
I Local stability results for the pest-free equilibrium
I Exact solutions for the linearised model
I Conditions for pest extinction, depending on fallow duration (τ )
I. Tankam, S. Touzeau, F. Grognard, L. Mailleret, J.-J. Tewa, 2018.
A multi-seasonal model of the dynamics of banana plant-parasitic
nematodes.
I. Tankam, S. Touzeau, L. Mailleret, J.-J. Tewa, F. Grognard, 2019. Modelling
and control of a banana soilborne pest in a multi-seasonal framework.
Mathematical Biosciences. In revision.
21 / 46Optimisation
I Aim: maximise the cumulated yield, while minimising the costs
(new suckers), on a fixed multi-seasonal time horizon (Tmax )
I Control: fallow period durations (τk )
New sucker flowering harvest New sucker flowering harvest
root growth fruit growth fallow
0 d D τ1 T1 T1+d T1+D τ2 T2 Tmax
W(t)=0 w 0 w 0
• fixed cropping period D & fixed delay before flowering d
• fixed cost of a new pest-free sucker c
R T +D R T +D
• seasonal yield Yk = Tkk W (t) S(t) dt = w Tkk+d S(t) dt
seasonal profit Rk = Yk − c
Pn
Maximise profit: maxτ =(τ1 ,...,τn ) R(τ ) with R(τ ) = k =0 Rk
22 / 46Variable fallow durations
D τ1 D τ2 D τ3 D
1 2 3 4
0 T1 T2 T3 Tmax
Solutions
• τk ∈ [0, Tmax − 2 D] → at least 2 cropping seasons
• Pests “not too abundant”
Admissible fallows τ = (τ1 , . . . , τn ) such that last season ends at Tmax
n + 1 complete cropping seasons
→
n fallows
Procedure
1. Solve the optimisation problem for n = 1, . . . , nmax
Pn
• over the n-simplex : k =1 τk = Tmax − (n + 1)D
• Adaptive Random Search algorithm
2. Select the highest profit among the nmax optima
23 / 46Numerical application
Parameter value unit Parameter value unit
banana roots nematode
d 210 days 1 δ 2.10−4 g.day−1
D 330 days 1 ∆ 60 g
ρ 0.025 day−1 α 400 g−1
K 150 g 2 γ 0.5
S0 60 g 2 β 10−1 g−1 .day−1
banana production µ 0.045 day−1 3
w 0.3 CFA.g−1 .day−1 ω 0.0495 day−1 4
simulation P0 100
Tmax 4000 days harvest
1
[Banana cultivation guide, 2018] 2 [Serrano, 2005] q 5%
3
[Sarah et al., 1996] 4 [Chabrier et al., 2008]
24 / 46Result
å Maximum profit: 54,600 CFA for variable fallows over 11 seasons
(nmax = 12 cropping seasons)
25 / 46Regulations distance to
simplex centre
z }| {
Bounded fallow: τsup = 60 days Penalty: R̄ = R − r d(τ, τ0 )
R(τ0 )
with r = 10×dmax
(penalty < “regular” profit / 10)
å Maximum profit: 54,400 CFA å Maximum profit: 53,500 CFA
over 11 seasons over 11 seasons
Still 11 cropping seasons
Almost no profit loss with regulations
26 / 46Constant fallow duration
D τ D τ D τ D
1 2 3 4
0 T 2T 3T Tmax
Solutions
Solution(s) of the optimisation problem in discrete set:
Tmax − D
Ξ= τ >0: ∈N
D+τ
Procedure
Exhaustive exploration
27 / 46Result
12 11 10 9 8 7 6 5 4 3 2
profit (CFA)
τ (days)
å Maximum profit: 52,000 CFA over 11 cropping seasons
for τ = 37 days ∈ Ξ
28 / 46Comparisons
Similar profits
High final soil infestations
with variable fallows
29 / 46Conclusions
I Banana – nematode interaction model, with pest-free suckers
planted at each cropping season
I Optimal fallow durations: best profit with non constant fallows
(even when regulated), but minor gains and high final soil
infestations
I. Tankam, S. Touzeau, F. Grognard, L. Mailleret, J.-
J. Tewa, 2018. An agricultural control of Radopholus
similis in banana plantations
I. Tankam, S. Touzeau, F. Grognard, L. Mailleret, J.-J.
Tewa, 2019. Agricultural control of Radopholus similis
in banana and plantain plantations
I. Tankam, F. Grognard, L. Mailleret, J.-J. Tewa, S. Touzeau, 2019. Optimal and
sustainable management of a soilborne banana pest. In prep.
30 / 46Focus
T1. Cocoa plant mirids
Myriam D JOUKWE TAPI, Douala–Cirad (2015–)
S. B OWONG, L. B AGNY-B EILHE, Y. D UMONT
T2. Plantain plant-parasitic nematodes
Israël TANKAM C HEDJOU, Yaoundé I–Inria (2016–)
J.-J. T EWA, L. M AILLERET, F. G ROGNARD, S. TOUZEAU
T3. Coffee berry borers
Yves F OTSO F OTSO, Dschang–Inria (2017–)
B. T SANOU, S. B OWONG, F. G ROGNARD, S. TOUZEAU
T4. Coffee leaf rust
Clotilde D JUIKEM, MSc (2018) & PhD Inria–Douala (2019–)
S. B OWONG, F. G ROGNARD, S. TOUZEAU
31 / 46Coffee berry borers (CBB)
[Burbano, JIS 2011]
Uccao Cameroun
I Coffee is an important cash crop in the tropics
I Borers (Hypothenemus hampei) mostly develop and feed in coffee
berries, inducing great damages
• reduction in berry quality and yield
• economic losses > 500 million $/year
I Simulation model with crop growth and pest control [Gutierrez et al.
1998; Rodríguez et al. 2011; Rodríguez et al. 2013]
32 / 46CBB life cycle
I Microscopic beetle < 1.5 mm
I Life cycle ca. 4 weeks
I 1–3 eggs/day during ca. 20 days
I Sex ratio 10 ~ / 1 |
33 / 46CBB life cycle
I ~ bore into a berry
I Mating occurs inside the berry
I Fertilised ~ fly out to infest new berries
I | stay in berry
[Bustillo et al. 1998]
34 / 46Variables
s
z i
y
Single season model
I s(t): healthy coffee berries
I i(t): infested coffee berries
I y (t): colonizing females (outside the berries)
I z(t): infesting females (inside the berries)
Hypothesis: males are not limiting
35 / 46Model
μ
Λ β μi
s y+αs
z i
ε
y μz
φ
μy
infestation
new berries
sy mortality
ṡ = Λ −β −µs
y + αs
sy
i̇ = Λ + β − µi i infected berries
y + αs
emergence
sy
ẏ = ϕz − εβ − µy y
y + αs
sy
ż = ϕz + εβ − µz z
y + αs
36 / 46Stability of the equilibria
Λ
Pest-free equilibrium (PFE) µ , 0, 0
I LAS → Basic reproduction number (next-generation matrix):
εβϕ α1
N = 1
37 / 46CBB control
I Insecticides
I Cropping practices:
removing dropped berries,
strip-picking, stump pruning
I Trapping
Brocap
R
I Biological control
• Parasitoid (Phymastichus coffea) or predator (Cathartus quadricollis)
insects
A. Castillo, F. Infante
[Follett et al., 2016]
A. Ramirez
• Entomopathogenic fungi (Beauveria bassiana)
38 / 46Model with control
sy
ṡ = Λ − (1 − σ(v )) β − µs
y + αs
sy
ẏ = ϕz− εβ − µy y
y + αs
sy
ż = +(1 − σ(v )) εβ − µz z
y + αs
v̇ = −γv + h(t)
I Fungus load v (t) reduces the infection of healthy berries
with a saturation effect:
ξv
σ(v ) = with maximum efficacy ξ 6 1
v +k
I Fungus load v (t) persists in the plantation
→ control h(t) ∈ [0, hmax ]
39 / 46Optimal control
Maximise the yield at the end of the cropping season s(tf ),
while minimising the control costs (↔ maximise profit)
& the CBB population for the next cropping season y (tf ):
Z tf
J (h) = ζs s(tf ) − C hθ (t) dt − ζy y (tf )
0
yield costs CBB pop.
with linear (θ = 1) or quadratic (θ = 2) costs
→ Pontryagin’s Maximum Principle for quadratic cost (θ = 2):
∗ p4 (t)
h (t) = max 0, min − , hmax h∗ (tf ) = 0
2C
→ PMP for linear cost (θ = 1):
bang-bang & singular control
40 / 46Table 1
Numerical application: linear cost
Biological meaning and value of parameters (with s in number of berries, y and z in number of
females and v in gram).
Parameter Biological meaning Value
Λ Production rate of new coffee berries 1200 berries.day−1
µ Natural mortality rate of healthy coffee berries 0.002 day−1
φ Emergence of new colonizing females 2 day−1
β Infestation rate 0.01 day−1
ε Conversion rate from coffee berries to CBB 1 female berry−1
µy Natural mortality rate of colonizing females 1/20 day−1
µz Natural mortality rate of infesting females 1/27 day−1
ξ Maximum effectiveness rate of fungus 0.8
α The interference amplitude 0.7
k Maximal amount the fungus 200 g.day−1
γ Rate of fungus decay 1/50 day−1
ζs Weight of coffee berries 2.10−3 $.day −1
ζy Weight of colonizing female 10−4 $.f emale−1
C Cost per unit of fungus 0.022$.g −1 .day
hM Maximal amount fungus to apply 30g.day −1
427 6. Conclusion. In this paper, we have developed a basic model to study the
Numerical
428 infestation method
dynamic of coffee berries by CBB inside the plantation. This model is
429 based on the CBB life cycle and take into account the availability of healthy coffee
BOCOP
430 berries.direct methodanalysis show that the dynamic of CBB varies according
Our theoretical
41 / 46low initial infestation y0 = 104 ~
×10 5 (a) Healthy coffee berries ×10 5 (b) Colonizing females
3 15
y [females]
y*
s [berries]
2 10
s*
1 5
0 0
0 50 100 150 200 250 0 50 100 150 200 250
Time (days) Time (days)
×10 4 (c) Infesting females (d) Fungus load
4 1500
z [females]
z* 1000
v [g]
2
500
0 0
0 50 100 150 200 250 0 50 100 150 200 250
Time (days) Time (days)
(e) Fungus application (f) Effect of fungus load
30 1
h [g/day]
20
σ(v)
0.5
10
0 0
0 50 100 150 200 250 0 50 100 150 200 250
Time (days) Time (days)
42 / 46high initial infestation y0 = 106 ~
×10 5 (a) Healthy coffee berries ×10 5 (b) Colonizing females
4 15
y [females]
s [berries]
y*
10
2
s* 5
0 0
0 50 100 150 200 250 0 50 100 150 200 250
Time (days) Time (days)
×10 4 (c) Infesting females (d) Fungus load
3 2000
z*
z [females]
2
v [g]
1000
1
0 0
0 50 100 150 200 250 0 50 100 150 200 250
Time (days) Time (days)
(e) Fungus application (f) Effect of fungus load
30 1
h [g/day]
20
σ(v)
0.5
10
0 0
0 50 100 150 200 250 0 50 100 150 200 250
Time (days) Time (days)
43 / 46Conclusions
I CBB-berry interaction model, for a single cropping season
I Entomopathogenic fungi can limit the CBB infestation
I Optimal control on a simple model gives a rough idea of how
pest control should be applied
Y. Fotso Fotso, F. Grognard, B. Tsanou,
S. Touzeau, 2018. Modelling and control
of coffee berry borer infestation
Y. Fotso, F. Grognard, S. Touzeau, B. Tsanou,
S. Bowong, 2019. Optimal control on a simple
model of coffee berry borer infestation
Y. Fotso, F. Grognard, S. Touzeau, B. Tsanou, S. Bowong, 2019. Modelling and
optimal strategy to control coffee berry borer infestation. In prep.
44 / 46What next
PhD defenses
4 students will very soon defend their thesis (early 2020):
I Myriam D JOUKWE TAPI
I Samuel N ILUSMAS
I Janvier P ESSER N TAHOMVUKIYE
I Israël TANKAM C HEDJOU
EPITAG = fairly young team, based on joint PhD supervisions
I Associate team renewed for 3 more years, with similar overall
objectives
I Pursue the research topics and associated PhD theses
(finishing, ongoing, starting)
45 / 46In short
French & Cameroonian researchers and students
with a background in dynamical systems
and control
and an interest in crop diseases
D. Coyne
More on EPITAG: https://team.inria.fr/epitag/
46 / 46You can also read
