Controlling malaria with indoor residual spraying in spatially heterogeneous environments

(1)

Controlling malaria with indoor residual spraying in

spatially heterogeneous environments

Robert Smith?

Department of Mathematics and Faculty of Medicine

The University of Ottawa

(2)

• Epidemiology of malaria

Outline

(3)

• Indoor Residual Spraying

Outline

(4)

• Research questions

Outline

(5)

• The mathematical model

Outline

(6)

• Spraying in an interior disc

Outline

(7)

• Fixed vs nonfixed spraying

Outline

(8)

• Fixed vs nonfixed spraying

• The effects of wind.

Outline

(9)

Malaria

• One of the most important human diseases throughout the tropical and sub-tropical

regions of the world

Source: NMCC Central Board of Health, 2000

(10)

Malaria

• More than 300 million acute illnesses each year

(11)

Malaria

• More than 300 million acute illnesses each year

• 1,000,000 deaths

annually.

(12)

Symptoms

• Repeated episodes of fever

(13)

Symptoms

• Pregnancy complications

(14)

Symptoms

• Impairs development

(15)

Symptoms

• Weakness

(16)

Symptoms

• Weakness

• Anemia

(17)

Symptoms

• Weakness

• Anemia

• Death.

(18)

• 90% of malaria deaths in sub-Saharan Africa

Endemic areas

(19)

• Mostly among young children

Endemic areas

Admissions to St. Kitzo-Matany hospital, Uganda

(20)

• Even when it

doesn’t kill, acute illness can

devastate

economies in the developing world

Endemic areas

(21)

• Even when it

doesn’t kill, acute illness can

devastate

economies in the developing world

• Impact of malaria has been estimated to cost Africa $US12 billion every year.

Endemic areas

(22)

(23)

(24)

Control

Malaria control primarily consists of

(25)

Control

• chemoprophylaxis

(26)

Control

– drugs, vaccines, etc

(27)

Control

• vector control

(28)

Control

• vector control

– insecticides, larvacides, etc

(29)

Control

• vector control

– insecticides, larvacides, etc – aim is to reduce vector

population density and survival.

(30)

Indoor Residual Spraying

• Malaria vectors are endophilic, resting inside houses after feeding

(31)

Indoor Residual Spraying

• Indoor Residual Spraying (IRS) involves spraying houses or dwellings on

the inside and under eaves on the outside

(32)

Indoor Residual Spraying

• Kills mosquitos after they’ve fed

(33)

Indoor Residual Spraying

• Kills mosquitos after they’ve fed

• Duration of effective action is 2-6 months.

(34)

Effectiveness of IRS

• When implemented well, it can be effective

(35)

Effectiveness of IRS

• IRS has been responsible for suppression of at least one vector of malaria transmission, An. funestus

(36)

Effectiveness of IRS

• IRS has been responsible for suppression of at least one vector of malaria transmission, An. funestus

• Indoor residual spraying is a

powerful method of

malaria control, but is limited to the physical location of structures.

(37)

• Careful delineation of spray areas and

populations is necessary to determine the scale of impact for each intervention

Limitations of a spraying program

(38)

• IRS cannot be used in areas devoid of structures

Limitations of a spraying program

(39)

– eg forests, swamps

Limitations of a spraying program

(40)

• Spatial heterogeneity is thus important

Limitations of a spraying program

(41)

– eg landspace

Limitations of a spraying program

(42)

– eg landspace

– urban/rural population densities

Limitations of a spraying program

(43)

– eg landspace

– urban/rural population densities – distribution of structures.

Limitations of a spraying program

(44)

Crucial questions

Prevention of malaria

• Indoor residual spraying is used without any plan

• Cost of Spray

5 / 24

(45)

Crucial questions

What are the effects of spraying in

different geographic

areas?

• Cost of Spray

5 / 24

(46)

Crucial questions

areas?

How do the results depend

on the regularity of

spraying?

• Cost of Spray

5 / 24

(47)

Crucial questions

areas?

How do the results depend

on the regularity of

spraying?

Can we alter our control strategies to

account for asymmetric phenomena such as wind?

• Cost of Spray

5 / 24

(48)

Impulsive differential equations

• Assume spraying is instantaneous

Impulsive Differential Equations

• Assume drug effects

• That is, the time-to- peak is assumed to be negligible

• This results in a

system of impulsive differential equations.

are instantaneoustime (months)

(49)

Impulsive differential equations

• That is, the delay in mosquito reduction is assumed to be

negligible

time (months)

total mosquito population

Impulsive Differential Equations

are instantaneous

Impulsive Differential Equations

are instantaneous

time (months)

(50)

Impulsive differential equations

• That is, the delay in mosquito reduction is assumed to be

negligible

time (months)

total mosquito population

Impulsive Differential Equations

are instantaneous

Impulsive Differential Equations

are instantaneous

time (months)

(51)

Impulsive effect

• According to impulsive theory, we can

describe the nature of the impulse at time r_k via the difference equation

y (r

_k⁺

) y (r

_k

) = f (r

_k

, y (r

_k

))

(52)

Impulsive effect

Difference equation

y (r

_k⁺

) y (r

_k

) = f (r

_k

, y (r

_k

))

(53)

Impulsive effect

Depends on the time of impulse

and the state immediately beforehand.

Difference equation

y (r

_k⁺

) y (r

_k

) = f (r

_k

, y (r

_k

))

(54)

Impulsive DEs

• Solutions are continuous for t ≠ r_k

r_k=impulse time

(55)

Impulsive DEs

• Solutions are continuous for t ≠ r_k

• Solutions undergo an instantaneous change in state when t = r_k.

r_k=impulse time

(56)

Putting it together

• The model thus consists of a system of ODEs (humans), together with PDEs and difference equations (mosquitos).

(57)

The mathematical model

S I R

M N

β

α

β β

h

h m

m

h

δ

D D

π

µ µ

µm

h µ+µ_h _d h

Λ

Parameter Definition Parameter Definition

S susceptible humans M susceptible mosquitoes

I infected humans N Infected mosquitoes

R recovered humans β_m rate of infecting a mosquito

µ_h human death rate D Diffusion coefficient

β_h rate of infecting a human Λ mosquito birth rate

h recovery rate without immunity µ_q mosquito death rate

α immunity rate r effectiveness of spraying

δ recovery rate after temporarily immune

π humans birth rate

6 / 24

The model

(58)

The mathematical model

S I R

M N

β

α

β β

h

h m

m

h

δ

D D

π

µ µ

µm

h µ+µ_h _d h

Λ

6 / 24

The model

susceptible

(59)

The mathematical model

S I R

M N

β

α

β β

h

h m

m

h

δ

D D

π

µ µ

µm

h µ+µ_h _d h

Λ

6 / 24

The model

susceptible

infected

(60)

The mathematical model

S I R

M N

β

α

β β

h

h m

m

h

δ

D D

π

µ µ

µm

h µ+µ_h _d h

Λ

6 / 24

The model

susceptible

immune infected

(61)

The mathematical model

S I R

M N

β

α

β β

h

h m

m

h

δ

D D

π

µ µ

µm

h µ+µ_h _d h

Λ

6 / 24

The model

susceptible

immune infected

susceptible mosquitos

(62)

The mathematical model

S I R

M N

β

α

β β

h

h m

m

h

δ

D D

π

µ µ

µm

h µ+µ_h _d h

Λ

6 / 24

The model

susceptible

immune infected

infected mosquitos susceptible mosquitos

(63)

The mathematical model

S I R

M N

β

α

β β

h

h m

m

h

δ

D D

π

µ µ

µm

h µ+µ_h _d h

Λ

6 / 24

The model

susceptible

immune infected

human birth

(64)

The mathematical model

S I R

M N

β

α

β β

h

h m

m

h

δ

D D

π

µ µ

µm

h µ+µ_h _d h

Λ

6 / 24

The model

susceptible

immune infected

human birth

background death

(65)

The mathematical model

S I R

M N

β

α

β β

h

h m

m

h

δ

D D

π

µ µ

µm

h µ+µ_h _d h

Λ

6 / 24

The model

susceptible

immune infected

infection rate of humans

human birth

background death

(66)

The mathematical model

S I R

M N

β

α

β β

h

h m

m

h

δ

D D

π

µ µ

µm

h µ+µ_h _d h

Λ

6 / 24

The model

susceptible

immune infected

human birth

disease death

background death

(67)

The mathematical model

S I R

M N

β

α

β β

h

h m

m

h

δ

D D

π

µ µ

µm

h µ+µ_h _d h

Λ

6 / 24

The model

susceptible

immune

recovery infected

human birth

disease death

background death

(68)

The mathematical model

S I R

M N

β

α

β β

h

h m

m

h

δ

D D

π

µ µ

µm

h µ+µ_h _d h

Λ

6 / 24

The model

susceptible

immune acquired immunity

recovery infected

human birth

disease death

background death

(69)

The mathematical model

S I R

M N

β

α

β β

h

h m

m

h

δ

D D

π

µ µ

µm

h µ+µ_h _d h

Λ

6 / 24

The model

susceptible

recovery infected

loss of immunity

human birth

disease death

background death

(70)

The mathematical model

S I R

M N

β

α

β β

h

h m

m

h

δ

D D

π

µ µ

µm

h µ+µ_h _d h

Λ

6 / 24

The model

susceptible

recovery infected

mosquito birth loss of immunity

human birth

disease death

background death

(71)

The mathematical model

S I R

M N

β

α

β β

h

h m

m

h

δ

D D

π

µ µ

µm

h µ+µ_h _d h

Λ

6 / 24

The model

susceptible

recovery infected

mosquito death

human birth

disease death

background death

(72)

The mathematical model

S I R

M N

β

α

β β

h

h m

m

h

δ

D D

π

µ µ

µm

h µ+µ_h _d h

Λ

6 / 24

The model

susceptible

infection rate of mosquitos

recovery infected

mosquito death

human birth

disease death

background death

(73)

The mathematical model

S I R

M N

β

α

β β

h

h m

m

h

δ

D D

π

µ µ

µm

h µ+µ_h _d h

Λ

6 / 24

The model

susceptible

infection rate of mosquitos

recovery infected

diffusion of mosquitos.

mosquito death

human birth

disease death

background death

(74)

The differential equations

• At non-spraying times, the PDEs are

(75)

The differential equations

S=Susceptible humans I=Infected humans R=Recovered humans M=Susceptible mosq.

N=Infected mosq.

π,Λ=birth rates D=diffusion µm,µH=death rates

βH,βM=transmissibility µd=malaria death rate

h=recovery rate α=immunity rate δ=loss of immunity

S_t = ⇡ _hSN + hI + R µ_hS I_t = _hSN hI ↵I (µ_h + )I R_t = ↵I R µ_hR

M_t = ⇤ µ_mM _mM I + D M t 6= t_k N_t = µ_mN + _mM I + D N t 6= t_k

(76)

The differential equations

• Boundary conditions: S=Susceptible humans I=Infected humans R=Recovered humans M=Susceptible mosq.

N=Infected mosq.

(77)

The differential equations

• Boundary conditions: S=Susceptible humans I=Infected humans R=Recovered humans M=Susceptible mosq.

N=Infected mosq.

@M

@⇢ (t, ⇢₀) = @N

@⇢ (t, ⇢₀) = 0 on @B(0, ⇢₀)

(78)

The differential equations

• Boundary conditions:

(B is a disc with radius ρ0).

S=Susceptible humans I=Infected humans R=Recovered humans M=Susceptible mosq.

N=Infected mosq.

@M

@⇢ (t, ⇢₀) = @N

@⇢ (t, ⇢₀) = 0 on @B(0, ⇢₀)

(79)

Spraying impulse

• At spraying times tk, the impulsive effect is

(80)

Spraying impulse

M=Susceptible mosq.

N=Infected mosq.

M⁺ = (1 r)M t = t_k

N⁺ = (1 r)N t = t_k.

(81)

Spraying impulse

• Here, r is the effectiveness of the insecticide.

M=Susceptible mosq.

N=Infected mosq.

M⁺ = (1 r)M t = t_k

N⁺ = (1 r)N t = t_k.

(82)

• If we define the total mosquito population by

Analysis of the impulsive system

(83)

• If we define the total mosquito population by

Analysis of the impulsive system

Λ=mosq. birth rate µ=mosq. death rate

r=spraying effectiveness D=diffusion B=disc

ρ0=radius

= M + N

(84)

• If we define the total mosquito population by then we have the partial differential equation

Analysis of the impulsive system

ρ0=radius

= M + N

(85)

• If we define the total mosquito population by then we have the partial differential equation

Analysis of the impulsive system

ρ0=radius

= M + N

t = ⇤ µ_m + D in B(0, ⇢₀)

(86)

• If we define the total mosquito population by then we have the partial differential equation with boundary condition

Analysis of the impulsive system

ρ0=radius

= M + N

t = ⇤ µ_m + D in B(0, ⇢₀)

(87)

• If we define the total mosquito population by then we have the partial differential equation with boundary condition

Analysis of the impulsive system

ρ0=radius

= M + N

t = ⇤ µ_m + D in B(0, ⇢₀)

@

@⇢ (t, ⇢₀) = 0 in @B(0, ⇢₀)