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@MalariaMath
304 Mathematical modelling of Anopheles mosquitoes’ breeding sites
to sustain efficient vector control
Toheeb Babatunde Ibrahim∗,†, Kristan A. Schneider∗, Martin Eichner †
∗Hochschule Mittweida, †University of Tübingen
Abstract
Background: Vector control interventions must properly account for the entomological characteristics of the targeted species to be efficient.
Proper understanding of the breeding habits and sites of a targeted mosquito vector can facilitate environmentally sustainable and cost-
efficient vector control. This, however, is a factor not properly addressed yet in predictive models. An Anopheles (An.) mosquito’s breeding
site can form when rain falls and mosquitoes find water puddles to lay their eggs. Over time, other organisms cohabit the breeding site with
the mosquito larvae, e.g., fish, tadpoles, etc. They contribute to the depletion of nutrients in the breeding site and to the mortality of mosquito
larvae.
Methods: A biological concept describing the dynamics of An. breeding sites were incorporated into a system of ordinary differential
equations. The model accounts for competition for nutrients among larvae, which results in delayed maturation of larvae. Furthermore, we
included i) organisms that cohabit the breeding site, e.g., larvivorous fish, ii) the effects of larvicides, iii) larval competition, iv) the effect of
insecticide treated bed-nets (ITNs) as a control measure, and v) the effect of introducing genetically modified mosquitoes with disruption of
the Doublesex gene resulting in blocking in formation of functional female (i.e., females that could not reproduce) to the wild. Mosquitoes are
supposed to always find a breeding site to oviposit and the dynamics in finding a site to oviposit is not explicitly studied (only the average time
for mosquitoes to initiate oviposition is considered). Mating was not explicitly modelled, since mating behaviour is not well understood and only
one mating is sufficient for a female mosquito to fertilize its lifetime eggs. The systems’ equilibrium point and the mosquito populations’ basic
reproduction number, i.e., the number of female mosquitoes arising from one female mosquito without control, are calculated analytically. We
simulated the model using realistic parameter values obtained from published field and laboratory observations, to describe the population
sizes of the different mosquito stages. Finally, the effects of mosquito control interventions were studied. Extensions of our model to
incorporate feeding habits of mosquitoes and malaria transmission are planned in the future.
Results: The density dependence is an important factor that affects the aquatic stages of the mosquitoes and, therefore, also the adult
population. The predators were discovered to have a significant effect on the reduction of the adult mosquito population. The genetically
modified mosquitoes make eradication of the vector population possible.
Conclusions: Targeting the aquatic and adult stage of the mosquitoes population is the most efficient way of controlling the vector.
1. Introduction
•Malaria is a deadly mosquito-borne disease (cf. [1]).
•Mosquitoes’ life cycle: (Fig.1)
◦Aquatic phase: juvenile states comprising the egg, larva in-
stars, and the pupa.
◦Adult phase: adults are classified into host seeking, resting and
breeding site seeking mosquitoes.
Fig.1: Mosquito life cycle
•Control interventions:
◦Predator: lavivorous fish feed on the aquatic stages.
◦Insecticide Treated Nets (ITNs): ITNs prevent adult
mosquitoes from taking blood meal and cause mortality.
◦Genetically modified mosquitoes: carries a gene that
blocks in the formation of functional female offspring (fe-
males that can ultimately reproduce) and its inheritance by
the offspring is assured by CRISPR-Ca9 based gene drive
(Fig.2). Female offsprings with two copies of the edited gene
will not be able to reproduce. (cf. [2]).
Fig.2: Gene drive inheritance (CRISPR-Cas9)
2. Model description
Mosquito population dynamics model: illustrated in Fig.3
•Eis the number of mosquito eggs in the breeding site;
•mosquitoes in the four larvae instars are grouped into two: the
early, (L1)and the late, (L2) instar larvae;
•early larvae develop into late larvae at rate lambda1;
•the breeding site has a carrying capacity of Klate larvae.
Particularly, late larvae experience competition for nutrients from
early larvae developing into late larvae;
•late larvae are prey on by larvivorous fish and develop to pupa at
rate λ2
•lavivorous fish Fare predators to the late instar larvae;
•adult mosquitoes emerge from pupae P. They are either questing
(host-seeking) MQor MR;
•ITNs causes a fraction Nof the questing mosquitoes to die;
ε
λ
L
2
φ
α
E
λ
L
11
α
M
R
n
2
φ
L
2
F
M
R
L
2
F
b
λ
L
11
L
2
EL
1
M
R
M
Q
FF
PP
M
Q
M
R
P
L
11
P
P
L
2
2
(1-L /K)
2
M
Q
π
μ
M
μ
M
μ
F
β
μ
μ
μ
E
E
μ
f
π
Fig.3: Compartmental model
•the genetically modified mosquitoes (male and female) are released to the wild to mate randomly and have offspring carries the manipulated
gene either herozygous c
MQ, or homozygous
f
f
MQ, those with double copy are non functional thus they die (cf. [2]).
3. Parameter Description
Parameter Definition Value
nNo. of eggs per oviposition states 200
fProportion of female mosquitoes hatched 0.5
NProportion of death caused by ITNs range [0 to 1]
KMosquito breeding site carrying capacity states 100000
αOviposition rate 0.5/day
Egg development rate 0.5/day
λ1Early stage larvae development rate 0.2 /day
λ2Late stage larvae development rate 0.2 / day
πPupa development rate 0.5/day
βBiting/blood feeding rate 1/day
ϕLavivorous fish feeding rate 0.0001/fish·day
µEEgg mortality rate 0.1/day
µ1Early stage larva baseline mortality rate 0.1/day
µ2Late stage larvae baseline mortality rate 0.1/day
µPPupae mortality rate 0.1/day
µMAdult mosquito mortality rate 0.1/day
µFLavivorous fish mortality rate 0.0027/day
Table 1: Mosquito dynamics parameters
4. Analytical and Numerical Results
Basic and effective reproduction number
•The number of mosquitoes arising from a female mosquito without control
intervention is:
R0=1
µM
αβn
α+β+µM
+µE
λ1
λ1+µ1
λ2
λ2+µ2
fπ
π+µP
•The number of female mosquitoes arising from a mosquito when control
intervention such as ITNs is considered,
Re=(1 −N)αβn
(α+µM)(β+µM)−(1 −N)αβ
+µE
λ1
λ1+µ1
λ2
λ2+µ2
fπ
π+µP
•Lavivorous fish decimate the mosquito population over
time (Fig.4).
•The equilibrium population size of mosquitoes (MQand
MR) is decimated by ITNs use (Fig.5).
•Transgenic mosquitoes successfully eradicate the
mosquito population (Fig.6).
0
10000
20000
30000
0 500 1000 1500 2000 2500
Time (days)
No. of Mosquitoes
With Predator
No Predator
Fig.4: Effect of lavivorous fish as predator
on mosquito population
0
10000
20000
30000
0.00 0.25 0.50 0.75 1.00
ITNs
No. of Mosquitoes
MR Equilibrium
MQ Equilibrium
Fig.5: Impact of ITNs as mosquito control
0e+00
1e+05
2e+05
3e+05
0 100 200 300 400 500
Time (days)
No. of Mosquitoes
Wild
Transgenic
Fig.6: Impact of genetically modified
mosquitoes to the wild
Discussion
•Biotic factor affecting mosquito population dynamics:
◦Larvae development is density dependent, this affects the number of adult mosquitoes emerging
from a breeding site;
◦larvivorous fish with high feeding rate on mosquito larvae can be an efficient tool to reduce the
vector population.
•Control interventions:
◦With high usage of ITNs mosquitoes and therefore malaria will be brought under control;
◦introducing genetically modified mosquitoes will wipe out mosquitoes. However the implication (fit-
ness cost) including mutation in edited vector was not investigated.
•Further research:
◦Dynamics in the formation of mosquito breeding sites;
◦implication of seasonal rainfall to the dynamics in mosquito breeding site;
◦extending the base model to female adult mosquitoes feeding habit.
Acknowledgements
This work was conducted during Toheeb Ibrahim’s stay at AIMS Cameroon, from which he received a scholarship.
This research is also supported by a grant from the DAAD (“Mathematics against malaria within the AIMS net
work”, project-ID 57417782), DFG (“Ökologisch nachhaltige Wertschöpfungsketten in in der Landwirtschaft durch
Optimierung des Insektizid-Gebrauchs aufgrund von automatisiertem Schädlings-Monitoring”).
Contact
Toheeb Babatunde Ibrahim, MSc.
University of Applied Sciences Mittweida
17, Technikumplatz,
09648, Mittweida, Germany
p: +234 806 087 2377
e-mail: toheeb.ibrahim@aims-cameroon.org
Prof. Dr. Kristan A. Schneider
University of Applied Sciences Mittweida
17, Technikumplatz,
09648, Mittweida, Germany
p: +49 3727 58-1057
e-mail: kristan.schneider@hs-mittweida.de
Prof. Dr. Martin Eichner
Eberhard Karls University of Tübingen,
Silcherstraße 5,
72076, Tübingen, Germany
e-mail: martin.eichner@uni-tuebingen.de
References
[1] W. HEALTH ORGANISATION,World malaria report 2016.
[2] KYROU, KYROS A ND HAMMOND, ANDREW MAND GALIZI, RO BE RTO A ND KRANJC, NACE AND BURT, AUSTIN AND BEAGHTON, ANDREA KAND NOLAN, TONY
AND CRISANTI, ANDREA,A CRISPR–Cas9 gene drive targeting doublesex causes complete population suppression in caged Anopheles gambiae mosquitoes,
Nature biotechnology, 36 (2018), 11, pages=1062–1066.
