The impact of media coverage on the transmission dynamics ... · Hypodermic theory • The original...

Robert Smith?

Department of Mathematics and Faculty of MedicineThe University of Ottawa

The impact of media coverage on the transmission dynamics of

human influenza

Outline

• Effects of media

Outline

• Effects of media• The model

Outline

• Effects of media• The model• Analysis

Outline

• Effects of media• The model• Analysis• Optimal controls

Outline

• Effects of media• The model• Analysis• Optimal controls• Adverse outcome

Outline

• Effects of media• The model• Analysis• Optimal controls• Adverse outcome• Implications.

The media

The media influences:

The media

The media influences:• individual behaviour

The media


(eg gift-chasing)

The media


(eg gift-chasing)• formation and

implementation of public policy

The media



implementation of public policy(eg biometrics)

The media




• perception of risk

The media




• perception of risk(eg SARS in Chinatown).

During a pandemic

• Government information released is often restricted to only the number of infections and deaths

During a pandemic


• Mass media are key tools in risk communication

During a pandemic


• Mass media are key tools in risk communication

• However, they have been criticised for making risk a spectacle.

Hypodermic theory

• The original interpretation of media effects in communication theory was the “hypodermic needle”

Hypodermic theory


• It was thought that a particular media message would be directly injected into the minds of media spectators

Hypodermic theory



• This suggests that media have a direct and rapid influence on everyday understanding

Hypodermic theory



• This suggests that media have a direct and rapid influence on everyday understanding

• However, this has been revised in recent years.

Contemporary media theories• Media is shaped by the dominant cultural

norms


norms• It is impossible to separate the message

from the society from which it originates



from the society from which it originates(eg WNV vs Chagas’ Disease)




• Consumers might only partially accept a particular media message




• Consumers might only partially accept a particular media message

• Or they may resist the dominant media messages altogether.

Implications for a pandemic

• Media effects may sway people into a panic


• Media effects may sway people into a panic• Especially for a disease where scientific

evidence is thin or nonexistent



evidence is thin or nonexistent(eg swine flu and pig-burning)




• Conversely, media may have little effect on more familiar diseases




• Conversely, media may have little effect on more familiar diseases(eg seasonal influenza).

Media in a crisis

• Media reporting play a key role in

Media in a crisis

• Media reporting play a key role in– perception

Media in a crisis

• Media reporting play a key role in– perception– management

Media in a crisis

• Media reporting play a key role in– perception– management– and even creation of a crisis

Media in a crisis


• Non-state-controlled media thrive in a crisis

Media in a crisis


• Non-state-controlled media thrive in a crisis(eg Wikileaks)

Media in a crisis



• However, state-controlled media are rewarded for creating an illusion of normalcy

Media in a crisis



• However, state-controlled media are rewarded for creating an illusion of normalcy(eg embedded journalists).

An intersubjective anchorage

• Media messages are widely distributed


• Media messages are widely distributed• Reports are retrievable


• Media messages are widely distributed• Reports are retrievable• Thus, they gain authority as an intersubjective

anchorage for personal recollection



anchorage for personal recollection• This may make information

appear “more true” the more exposure it gets fromthe media, regardless of the evidence



anchorage for personal recollection• This may make information

appear “more true” the more exposure it gets fromthe media, regardless of the evidence(eg climate change).

Media and risk protection

• The evaluation of epidemics may be driven by the complex interplay between information and action



• Individuals may overprotect, which may have additional consequences for the disease




• eg, after an announcement of the 1994 outbreak of plague in Surat, India, many people fled to escape the disease, thus carrying it to other parts of the country




• eg, after an announcement of the 1994 outbreak of plague in Surat, India, many people fled to escape the disease, thus carrying it to other parts of the country

• Media influences behaviour, which in turn influences media.

Vaccination

• One of the most effective tools for reducing the burden of infectious diseases

Vaccination


• However, individuals often refuse or avoid vaccinations they perceive to be risky

Vaccination



• eg, rumours that the polio vaccine could cause sterility and spread HIV hampered polio eradication efforts in Nigeria

Vaccination



• eg, rumours that the polio vaccine could cause sterility and spread HIV hampered polio eradication efforts in Nigeria

• Misplaced fears of autism in the developed world have stoked fears of vaccinations against childhood diseases.

Demographic interruption

• Media exposure and attention partially mediate the effects of demographics and personal experience on risk judgements


• Media exposure and attention partially mediate the effects of demographics and personal experience on risk judgements(eg anti-smoking campaigns)



• However, this may be especially problematic for vaccines



• However, this may be especially problematic for vaccines(eg HPV vaccine).

The model

• We model the dynamics of influenza based on a single strain without effective cross-immunity

The model


• We include a vaccine that confers temporary immunity

The model



• Vaccinated individuals may still become infected but at a lower rate than susceptibles

The model



• Vaccinated individuals may still become infected but at a lower rate than susceptibles

• Media converage is included via a saturated incidence function.

S R

V I

μ μ+α

μμ

Λ

θ ω λ

σ

βSI

β(1-γ)SV

S R

V I

μ μ+α

μμ

Λ

θ ω λ

σ

βSI

β(1-γ)SV

Susceptible

S R

V I

μ μ+α

μμ

Λ

θ ω λ

σ

βSI

β(1-γ)SV

Susceptible

Vaccinated

S R

V I

μ μ+α

μμ

Λ

θ ω λ

σ

βSI

β(1-γ)SV

Susceptible

Vaccinated

Infected

S R

V I

μ μ+α

μμ

Λ

θ ω λ

σ

βSI

β(1-γ)SV

Susceptible

Vaccinated

Recovered

Infected

S R

V I

μ μ+α

μμ

Λ

θ ω λ

σ

βSI

β(1-γ)SV

Susceptible

Vaccinated

Recovered

Infected

birth rate

S R

V I

μ μ+α

μμ

Λ

θ ω λ

σ

βSI

β(1-γ)SV

Susceptible

Vaccinated

Recovered

Infected

birth rate

background death rate

S R

V I

μ μ+α

μμ

Λ

θ ω λ

σ

βSI

β(1-γ)SV

Susceptible

Vaccinated

Recovered

Infected

birth rate


vaccination rate

S R

V I

μ μ+α

μμ

Λ

θ ω λ

σ

βSI

β(1-γ)SV

Susceptible

Vaccinated

Recovered

Infected

infection rate

birth rate


vaccination rate

S R

V I

μ μ+α

μμ

Λ

θ ω λ

σ

βSI

β(1-γ)SV

Susceptible

Vaccinated

Recovered

Infected

infection rate

vaccine efficacy

birth rate


vaccination rate

S R

V I

μ μ+α

μμ

Λ

θ ω λ

σ

βSI

β(1-γ)SV

Susceptible

Vaccinated

Recovered

Infected

infection rate

vaccine efficacy

birth rate


vaccination rate

waning rate

S R

V I

μ μ+α

μμ

Λ

θ ω λ

σ

βSI

β(1-γ)SV

Susceptible

Vaccinated

Recovered

Infected

infection rate

vaccine efficacy

birth rate


vaccination rate

waning rate

recovery rate

S R

V I

μ μ+α

μμ

Λ

θ ω λ

σ

βSI

β(1-γ)SV

Susceptible

Vaccinated

Recovered

Infected

infection rate

vaccine efficacy

birth rate


vaccination rate

waning rate

loss of immunity

recovery rate

S R

V I

μ μ+α

μμ

Λ

θ ω λ

σ

βSI

β(1-γ)SV

Susceptible

Vaccinated

Recovered

Infected

infection rate

vaccine efficacy

birth rate


vaccination rate

waning rate

disease death rate.

loss of immunity

recovery rate

The model equationsdS

dt= � + �V � (⌅ + µ)S �

�⇥1 � ⇥2

I

mI + I

⇥SI + ⌥R

dI

dt=

�⇥1 � ⇥2

I

mI + I

⇥SI +

�⇥1 � ⇥3

I

mI + I

⇥(1� ⇤)V I � (� + µ + ⇧)I

dV

dt= ⌅S � (µ + �)V �

�⇥1 � ⇥3

I

mI + I

⇥(1� ⇤)V I

dR

dt= ⇧I � (µ + ⌥)R

Λ=birth rate µ=background death rate θ=vaccination rate α=disease death rate ω=waning rate σ=loss of immunity γ=vaccine efficacy λ=recovery rate


dt= � + �V � (⌅ + µ)S �

�⇥1 � ⇥2

I

mI + I

⇥SI + ⌥R

dI

dt=

�⇥1 � ⇥2

I

mI + I

⇥SI +

�⇥1 � ⇥3

I

mI + I

⇥(1� ⇤)V I � (� + µ + ⇧)I

dV

dt= ⌅S � (µ + �)V �

�⇥1 � ⇥3

I

mI + I

⇥(1� ⇤)V I

dR

dt= ⇧I � (µ + ⌥)R

Media affects mixing rates



dt= � + �V � (⌅ + µ)S �

�⇥1 � ⇥2

I

mI + I

⇥SI + ⌥R

dI

dt=

�⇥1 � ⇥2

I

mI + I

⇥SI +

�⇥1 � ⇥3

I

mI + I

⇥(1� ⇤)V I � (� + µ + ⇧)I

dV

dt= ⌅S � (µ + �)V �

�⇥1 � ⇥3

I

mI + I

⇥(1� ⇤)V I

dR

dt= ⇧I � (µ + ⌥)R


• mI is the media half-saturation constant



dt= � + �V � (⌅ + µ)S �

�⇥1 � ⇥2

I

mI + I

⇥SI + ⌥R

dI

dt=

�⇥1 � ⇥2

I

mI + I

⇥SI +

�⇥1 � ⇥3

I

mI + I

⇥(1� ⇤)V I � (� + µ + ⇧)I

dV

dt= ⌅S � (µ + �)V �

�⇥1 � ⇥3

I

mI + I

⇥(1� ⇤)V I

dR

dt= ⇧I � (µ + ⌥)R


• mI is the media half-saturation constant• βi are the relative transmissibilities.


S R

V I

μ μ+α

μμ

Λ

θ ω λ

σ

βSI

β(1-γ)SV

S R

V I

μ μ+α

μμ

Λ

θ ω λ

σ

βSI

β(1-γ)SV


S R

V I

μ μ+α

μμ

Λ

θ ω λ

σ

βSI

β(1-γ)SV



S R

V I

μ μ+α

μμ

Λ

θ ω λ

σ

βSI

β(1-γ)SV



Media may also affect vaccination rates (in either

direction).

Media effects

• Susceptible and vaccinated people mix less with infecteds due to media

Media effects


• As many people become infected, effects of media are reduced

Media effects



• ie message reaches a maximum number of people due to information saturation

Media effects



• ie message reaches a maximum number of people due to information saturation

• This also reflects the fact that the media are less interested in a story once it’s established in society.

Equilibria

The model has two equilibria:

Equilibria

The model has two equilibria:• the disease-free equilibrium

Equilibria


(S, I, V , R) =�

�(µ + ⇤)µ(� + µ + ⇤)

, 0,��

µ(� + µ + ⇤), 0

⇥

S=susceptible I=infected V=vaccinated R=recovered Λ=birth rate µ=background death rate θ=vaccination rate ω=waning rate

Equilibria


• and an endemic equilibrium

(S, I, V , R) =�

�(µ + ⇤)µ(� + µ + ⇤)

, 0,��

µ(� + µ + ⇤), 0

⇥


Equilibria



(S, I, V , R) =�

�(µ + ⇤)µ(� + µ + ⇤)

, 0,��

µ(� + µ + ⇤), 0

⇥

(S, I, V , R)


Equilibria



which only exists for some parameter values.

(S, I, V , R) =�

�(µ + ⇤)µ(� + µ + ⇤)

, 0,��

µ(� + µ + ⇤), 0

⇥

(S, I, V , R)


Stability

• Using the next-generation method, we can calculate

Stability


R0 =⇥1�(µ + ⌥) + ⇥1(1� ⇤)⌅�µ(� + ⇧ + µ)(⌅ + µ + ⌥)

Λ=birth rate µ=background death rate θ=vaccination rate α=disease death rate ω=waning rate γ=vaccine efficacy λ=recovery rate β1=infection rate (susceptibles)

Stability


• We can prove:

R0 =⇥1�(µ + ⌥) + ⇥1(1� ⇤)⌅�µ(� + ⇧ + µ)(⌅ + µ + ⌥)


Stability


• We can prove:– If R0<1, the disease-free

equilibrium is globally stable

R0 =⇥1�(µ + ⌥) + ⇥1(1� ⇤)⌅�µ(� + ⇧ + µ)(⌅ + µ + ⌥)


Stability


• We can prove:– If R0<1, the disease-free

equilibrium is globally stable – If R0>1 the DFE is unstable.

R0 =⇥1�(µ + ⌥) + ⇥1(1� ⇤)⌅�µ(� + ⇧ + µ)(⌅ + µ + ⌥)


Optimal control

We introduce two controls, each representing a possible method of influenza control:

Optimal control

We introduce two controls, each representing a possible method of influenza control:• uv is the control variable for vaccination

Optimal control


(affecting the vaccination uptake)

Optimal control


(affecting the vaccination uptake)• um is the control

variable for media coverage

Optimal control


(affecting the vaccination uptake)• um is the control

variable for media coverage(affecting the media half-saturation constant).

Objective functional

• A control scheme is optimal if it maximises the objective functional



J(uv(t), um(t)) =� tf

t0[S(t) + V (t)�B1I(t)�B2(u2

v(t) + u2m(t))]dt

S=susceptible I=infected V=vaccinated uv=vaccine control um=media control




t0[S(t) + V (t)�B1I(t)�B2(u2

v(t) + u2m(t))]dt

Benefit ofuninfected populations





t0[S(t) + V (t)�B1I(t)�B2(u2

v(t) + u2m(t))]dt


Weight constraint for

infected populations





t0[S(t) + V (t)�B1I(t)�B2(u2

v(t) + u2m(t))]dt




Weight constraint for control




• B1 and B2 can represent the amount of money expended over a finite period, or the perceived risk.


t0[S(t) + V (t)�B1I(t)�B2(u2

v(t) + u2m(t))]dt




Weight constraint for control


Adjoint equations

• Given optimal controls uv and um, there exist adjoint variables λi (i=1,2,3,4) satisfying

Adjoint equations

• Given optimal controls uv and um, there exist adjoint variables λi (i=1,2,3,4) satisfying

S=susceptible I=infected V=vaccinated µ=background death rate θ=vaccination rate ω=waning rate σ=loss of immunity γ=vaccine efficacy λ=recovery rate γ=vaccine efficacy mI=media half-saturation constant B1=weight constraint (infection) B2=weight constraint (controls) β2=transmissibility reduction due to media (susceptibles) β3=transmissibility reduction due to media (vaccinated)

d⇧1

dt= � 1 + (⇧1 � ⇧2)(⇥1 � ⇥2

I

(1� um)mI + I)I + (⇧1 � ⇧3)(1� uv)⌅ + ⇧1µ

d⇧2

dt= B1 + (⇧1 � ⇧2)

⇤�⇥1 � ⇥2

I

(1� um)mI + I

⇥S � ⇥2

(1� um)mI

((1� um)mI + I)2IS

⌅

+ (⇧3 � ⇧2)⇤�

⇥1 � ⇥3I

(1� um)mI + I

⇥(1� ⇤)V � ⇥3

(1� um)mI

((1� um)mI + I)2(1� ⇤)V I

⌅

+ ⇧2(� + µ + ⇧)� ⇧4⇧

d⇧3

dt= � 1 + (⇧3 � ⇧2)(⇥1 � ⇥3

I

(1� um)mI + I)(1� ⇤)I + ⇧3µ + (⇧3 � ⇧1)�

d⇧4

dt= (⇧4 � ⇧1)⌥ + ⇧4µ.

Optimal controls

• We can calculate the optimal controls explicitly:

Optimal controls


u�v(t) = min�

max�

a11,(⇥1 � ⇥3)�S

2B2

⇥, b11

⇥

S=susceptible I=infected V=vaccinated γ=vaccine efficacy mI=media half-saturation constant B1=weight constraint (infection) B2=weight constraint (controls) β2=transmissibility reduction due to media (susceptibles) β3=transmissibility reduction due to media (vaccinated) λi=adjoint variables for the controls

Optimal controls


u�v(t) = min�

max�

a11,(⇥1 � ⇥3)�S

2B2

⇥, b11

⇥

u�m(t) = min�

max�

a22,(⇤1 � ⇤2)�2mISI2

2B2((1� um)mI + I)2+

(⇤2 � ⇤3)�3mI(1� ⇥)V I2

2B2((1� um)mI + I)2

⇥, b22

⇥


Optimal controls


– a11 and b11 are lower and upper bounds for uv

u�v(t) = min�

max�

a11,(⇥1 � ⇥3)�S

2B2

⇥, b11

⇥

u�m(t) = min�

max�

a22,(⇤1 � ⇤2)�2mISI2

2B2((1� um)mI + I)2+

(⇤2 � ⇤3)�3mI(1� ⇥)V I2

2B2((1� um)mI + I)2

⇥, b22

⇥


Optimal controls



– a22 and b22 are lower and upper bounds for um

u�v(t) = min�

max�

a11,(⇥1 � ⇥3)�S

2B2

⇥, b11

⇥

u�m(t) = min�

max�

a22,(⇤1 � ⇤2)�2mISI2

2B2((1� um)mI + I)2+

(⇤2 � ⇤3)�3mI(1� ⇥)V I2

2B2((1� um)mI + I)2

⇥, b22

⇥


Optimal controls



– a22 and b22 are lower and upper bounds for um

• The optimal controls are unique if tf is small.

u�v(t) = min�

max�

a11,(⇥1 � ⇥3)�S

2B2

⇥, b11

⇥

u�m(t) = min�

max�

a22,(⇤1 � ⇤2)�2mISI2

2B2((1� um)mI + I)2+

(⇤2 � ⇤3)�3mI(1� ⇥)V I2

2B2((1� um)mI + I)2

⇥, b22

⇥


Media has beneficial effect on vaccine

Figure 1: Schematic model flow diagram

10

15

20

25

30

35

40

0 20 40 60 80 100

Infe

ctives

Time/Days

i1i2i3

4

6

8

10

12

14

16

18

0 20 40 60 80 100

Vaccin

ate

d

Time/Days

v1v2v3

(a) (b)

Figure 2: Graphs of the optimality system when media coverage has a beneficial e�ect on the vaccinationrate, when the weight constraint for the infected population varies. (a) Infected individuals. (b) Vaccinatedindividuals. Initial conditions: S(0) = 20.0, I(0) = 25.0, V (0) = 50.0, R(0) = 40.0. The value of the weightsused are (i) B1 = 0.0025 corresponds to variables with subscript 1, (ii) B1 = 25.0 corresponds to variableswith subscript 2, (iii) B1 = 250000.0 corresponds to variables with subscript 3. The value B2 = 0.0025 is keptconstant in all three cases.

18

Costs of infection

high, control

low

Media has beneficial effect on vaccine

Figure 1: Schematic model flow diagram

10

15

20

25

30

35

40

0 20 40 60 80 100

Infe

ctives

Time/Days

i1i2i3

4

6

8

10

12

14

16

18

0 20 40 60 80 100

Vaccin

ate

d

Time/Days

v1v2v3

(a) (b)

Figure 2: Graphs of the optimality system when media coverage has a beneficial e�ect on the vaccinationrate, when the weight constraint for the infected population varies. (a) Infected individuals. (b) Vaccinatedindividuals. Initial conditions: S(0) = 20.0, I(0) = 25.0, V (0) = 50.0, R(0) = 40.0. The value of the weightsused are (i) B1 = 0.0025 corresponds to variables with subscript 1, (ii) B1 = 25.0 corresponds to variableswith subscript 2, (iii) B1 = 250000.0 corresponds to variables with subscript 3. The value B2 = 0.0025 is keptconstant in all three cases.

18

10

15

20

25

30

35

40

0 20 40 60 80 100

Infe

ctives

Time/Days

i1i2i3

0

5

10

15

20

25

30

35

40

45

50

0 20 40 60 80 100

Vaccin

ate

d

Time/Days

v1v2v3

(a) (b)

Figure 3: Graphs of the optimality system when media coverage has a beneficial e�ect on the vaccination rate,when the weight constraint for the infected population varies. (a) Graph of infectives, (b) Graph of vaccinatedindividuals. Initial conditions: S(0) = 20.0, I(0) = 25.0, V (0) = 50.0, R(0) = 40.0. The value of the weightsused are (i) B2 = 25.0 corresponds to variables with subscript 1 (++), (ii) B2 = 2500.0 corresponds to variableswith subscript 2 (xx), (iii) B2 = 250000.0 corresponds to variables with subscript 3 (**). The value B1 = 0.0025is kept constant in all three cases.

10

15

20

25

30

35

40

0 20 40 60 80 100

Infe

ctives

Time/Days

i1i2i3

2

2.5

3

3.5

4

4.5

5

0 20 40 60 80 100

Vaccin

ate

d

Time/Days

v1v2v3

(a) (b)

Figure 4: Graphical representation of the evolution of the various populations when media coverage has anadverse e�ect on the vaccination rate, when the weight constraint for the control population varies. (a) Graphof infectives. (b) Graph of vaccinated individuals. Initial conditions: S(0) = 20.0, I(0) = 25.0, V (0) = 50.0,R(0) = 40.0. The value of the weights used are (i) B1 = 0.0025 corresponds to variables with subscript 1, (ii)B1 = 25.0 corresponds to variables with subscript 2, (iii) B1 = 250000.0 corresponds to variables with subscript3. The value B2 = 0.0025 is kept constant in all three cases.

19

Costs of infection

high, control

low

Costs of control high,

infection low

10

15

20

25

30

35

40

0 20 40 60 80 100

Infe

ctives

Time/Days

i1i2i3

0

5

10

15

20

25

30

35

40

45

50

0 20 40 60 80 100

Vaccin

ate

d

Time/Days

v1v2v3

(a) (b)


10

15

20

25

30

35

40

0 20 40 60 80 100

Infe

ctives

Time/Days

i1i2i3

2

2.5

3

3.5

4

4.5

5

0 20 40 60 80 100

Vaccin

ate

dTime/Days

v1v2v3

(a) (b)


19

Media has negative effect on vaccine

Costs of infection

high, control

low

10

15

20

25

30

35

40

0 20 40 60 80 100

Infe

ctives

Time/Days

i1i2i3

0

5

10

15

20

25

30

35

40

45

50

0 20 40 60 80 100

Vaccin

ate

d

Time/Days

v1v2v3

(a) (b)


10

15

20

25

30

35

40

0 20 40 60 80 100

Infe

ctives

Time/Days

i1i2i3

2

2.5

3

3.5

4

4.5

5

0 20 40 60 80 100

Vaccin

ate

dTime/Days

v1v2v3

(a) (b)


19

Media has negative effect on vaccine

Costs of infection

high, control

low

Costs of control high,

infection low

10

15

20

25

30

35

40

0 20 40 60 80 100

Infe

ctives

Time/Days

i1i2i3

0

5

10

15

20

25

0 20 40 60 80 100

Vaccin

ate

d

Time/Days

v1v2v3

(a) (b)

Figure 5: Graphs of the optimality system when media coverage has an adverse e�ect on the vaccination rate.(a) Graph of infectives, (b) Graph of vaccinated individuals. Initial conditions: S(0) = 20.0, I(0) = 25.0,V (0) = 50.0, R(0) = 40.0. The value of the weights used are (i) B2 = 25.0 corresponds to variables withsubscript 1, (ii) B2 = 2500.0 corresponds to variables with subscript 2, (iii) B2 = 250000.0 corresponds tovariables with subscript 3. The value B1 = 0.0025 is kept constant in all three cases

0 5 10 15 20 250

1

2

3

4

5

6

7

8

9

10

time (days)

Infe

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d in

div

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16.5 17 17.5 18 18.5

2.6

2.8

3

3.2

Vaccination panic threshold

Without media effects

With media effects

Figure 6: The e�ect of the vaccination panic threshold using the simplified model (4.1)-(4.2). Withoutmedia triggering a vaccinating panic, the number of infected individuals remains low (solid purplecurve). However, if the media triggers a vaccinating panic, then the number of infected individuals risessharply (dashed green curve). Inset: Comparison of the two outcomes around the vaccination threshold.

19

Adverse outcome due to media?

• To illustrate a potentially adverse outcome, consider a simplified model



• Suppose, initially, the media and the general population are unaware of the disease




• Thus, nobody gets vaccinated, allowing the disease to spread initially





• New infected individuals arrive at fixed times





• New infected individuals arrive at fixed times

• We will ignore recovery in this simple model.

Media awareness threshold

• Suppose there are a critical number of infected individuals whereupon people become aware of the disease, via the media



• Above this threshold, susceptibles do not mix with infecteds




• However, vaccinated individuals mix significantly with infecteds




• However, vaccinated individuals mix significantly with infecteds

• Even though they may still potentially contract the virus.

Simplified model - lower region

• For I<Icrit, the model is

S=susceptible I=infected V=vaccinated Λ=birth rate µ=background death rate α=disease death rate ω=waning rate λ=recovery rate Icrit=vaccination panic threshold


• For I<Icrit, the model isdS

dt= ⇥ + ⌅V � µS t ⇥= tk

dI

dt= �(� + µ + ⇥)I t ⇥= tk

dV

dt= �(µ + ⌅)V t ⇥= tk

�I = Ii t = tk




• tk are (fixed) arrival times of new infecteds

dS

dt= ⇥ + ⌅V � µS t ⇥= tk

dI

dt= �(� + µ + ⇥)I t ⇥= tk

dV

dt= �(µ + ⌅)V t ⇥= tk

�I = Ii t = tk




• tk are (fixed) arrival times of new infecteds• This approximates low-level mixing

dS

dt= ⇥ + ⌅V � µS t ⇥= tk

dI

dt= �(� + µ + ⇥)I t ⇥= tk

dV

dt= �(µ + ⌅)V t ⇥= tk

�I = Ii t = tk




• tk are (fixed) arrival times of new infecteds• This approximates low-level mixing• If arrival times are not fixed, the results are

broadly unchanged.

dS

dt= ⇥ + ⌅V � µS t ⇥= tk

dI

dt= �(� + µ + ⇥)I t ⇥= tk

dV

dt= �(µ + ⌅)V t ⇥= tk

�I = Ii t = tk


Simplified model - upper region

• For I>Icrit, the model is

S=susceptible I=infected V=vaccinated Λ=birth rate µ=background death rate θ=vaccination rate α=disease death rate ω=waning rate γ=vaccine efficacy λ=recovery rate Icrit=vaccination panic threshold


• For I>Icrit, the model isdS

dt= � + ⌥V � (⌅ + µ)S

dI

dt= ⇥5(1� ⇤)V I � (� + µ + ⇧)I

dV

dt= ⌅S � (µ + ⌥)V � ⇥5(1� ⇤)V I




• No mixing of susceptibles and infecteds

dS

dt= � + ⌥V � (⌅ + µ)S

dI

dt= ⇥5(1� ⇤)V I � (� + µ + ⇧)I

dV

dt= ⌅S � (µ + ⌥)V � ⇥5(1� ⇤)V I




• No mixing of susceptibles and infecteds • The vaccinated mix with infecteds, allowing

them to be infected

dS

dt= � + ⌥V � (⌅ + µ)S

dI

dt= ⇥5(1� ⇤)V I � (� + µ + ⇧)I

dV

dt= ⌅S � (µ + ⌥)V � ⇥5(1� ⇤)V I




• No mixing of susceptibles and infecteds • The vaccinated mix with infecteds, allowing

them to be infected (at low rates).

dS

dt= � + ⌥V � (⌅ + µ)S

dI

dt= ⇥5(1� ⇤)V I � (� + µ + ⇧)I

dV

dt= ⌅S � (µ + ⌥)V � ⇥5(1� ⇤)V I


0 5 10 15 20 250

1

2

3

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5

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2.8

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3.2



With media effects

Media awareness

0 5 10 15 20 250

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time (days)

Infe

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16.5 17 17.5 18 18.5

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2.8

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3.2



With media effects

Media awareness

0 5 10 15 20 250

1

2

3

4

5

6

7

8

9

10

time (days)

Infe

cte

d in

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16.5 17 17.5 18 18.5

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2.8

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3.2



With media effects

Media awareness

0 5 10 15 20 250

1

2

3

4

5

6

7

8

9

10

time (days)

Infe

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d in

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16.5 17 17.5 18 18.5

2.6

2.8

3

3.2



With media effects

Media awareness

Lower region

• If I<Icrit,we can prove that

µ=background death rate α=disease death rate λ=recovery rate Icrit=vaccination panic threshold

Lower region



I+ ⇤ Ii

1� e�(�+µ+⇥)⌅⇥ m+

Lower region


where τ = tk+1 - tk


I+ ⇤ Ii

1� e�(�+µ+⇥)⌅⇥ m+

Lower region


where τ = tk+1 - tk• If m+>Icrit,then the system

will eventually switch from the lower region to the upper region.


I+ ⇤ Ii

1� e�(�+µ+⇥)⌅⇥ m+

Upper region

• If I>Icrit,there is an endemic equilibrium (S*,I*,V*)

S=susceptible I=infected V=vaccinated m+=non-media equilibrium Icrit=vaccination panic threshold

Upper region


• This equilibrium is stable if I*>Icrit


Upper region



• ie once trajectories enter the upper region, they will stabilise there


Upper region




• If I*>m+, then the outcome will be worse than without media effects


Upper region




• If I*>m+, then the outcome will be worse than without media effects

• Thus, even in this extremely simplified model, the media may make things significantly worse. S=susceptible I=infected V=vaccinated m+=non-

media equilibrium Icrit=vaccination panic threshold

Low-level mixing of susceptibles

• Low-level mixing may apply to the upper region as well



• Including these will increase the long-term number of infecteds



• Including these will increase the long-term number of infecteds

• It will also increase the peak of the epidemic wave.

0 20 40 60 80 1000

20

40

60

80

100

120

time (days)

Infe

cte

d indiv

iduals 90 95 100

22

24

26

28


With post!vaccinationfluctuations

Without post!vaccinationfluctuations

High-level mixing of susceptibles

• What if susceptibles mix with infecteds in more significant numbers?



• If these effects are included in the upper region, then the wave peak occurs earlier



• If these effects are included in the upper region, then the wave peak occurs earlier

• The long-term number of infecteds will also increase.

0 20 40 60 80 1000

20

40

60

80

100

120

140

160

180

200

time (days)

Infe

cte

d indiv

iduals

Maximal post!vaccination mixing of susceptibles and infecteds

No post!vaccination mixing of susceptibles and infecteds

50% post!vaccination mixing of susceptibles and infecteds

Adverse outcome

• Thus, a small series of outbreaks that would equilibrate at some maximal level m+>Icrit may, as a result of the media, instead equilibrate at a much larger value I*>m+

m+=non-media equilibrium Icrit=vaccination panic threshold

Adverse outcome


• The driving factor here is overconfidence in an imperfect vaccine


Adverse outcome



• ie vaccinated people mix significantly more with infecteds than susceptibles do


Adverse outcome



• ie vaccinated people mix significantly more with infecteds than susceptibles do

• This may happen if people feel invulnerable, due to media simplifications around vaccines. m+=non-media equilibrium Icrit=vaccination panic threshold

Recommendations

• As scientists, we could all benefit from media training

Recommendations


• Messages need to be straightforward

Recommendations



• Plain language is crucial

Recommendations



• Plain language is crucial• Speak in quoteable

phrases, not paragraphs

Recommendations




phrases, not paragraphs• If you can’t explain it...

Recommendations




phrases, not paragraphs• If you can’t explain it...

...you didn’t do it.

Summary

• Media simplifications can lead to overconfidence in the idea of a vaccine as a cure-all

Summary


• The result is a vaccinating panic and a net increase in the number of long-term infected

Summary



• Thus, media coverage of an emerging epidemic can have dire consequences

Summary



• Thus, media coverage of an emerging epidemic can have dire consequences

• It can also implicitly reinforce an imperfect solution as the only answer.

Limitations

• More comprehensive modelling is needed to fully understand the complex interplay between media and human behaviour

Limitations


• This will require interdisciplinary research across traditional boundaries of

Limitations


• This will require interdisciplinary research across traditional boundaries of– social

Limitations


• This will require interdisciplinary research across traditional boundaries of– social– natural

Limitations


• This will require interdisciplinary research across traditional boundaries of– social– natural– medical sciences

Limitations


• This will require interdisciplinary research across traditional boundaries of– social– natural– medical sciences– mathematics

Limitations


• This will require interdisciplinary research across traditional boundaries of– social– natural– medical sciences– mathematics

• eg people may ignore the media, de-linking the vaccination rate from the control.

Conclusions

• The media are responsible for treating risk as spectacle, panic in the face of fear and oversimplifications in the absence of data

Conclusions


• While the media may encourage more people to get vaccinated, they may also trigger a vaccinating panic

Conclusions



• Or promote overconfidence in the ability of a vaccine to fully protect against the disease

Conclusions



• Or promote overconfidence in the ability of a vaccine to fully protect against the disease

• When the next pandemic arrives, the outcome is likely to be significantly worse as a result of the media.

• J.M. Tchuenche, N. Dube, C.P. Bhunu, R.J. Smith? and C.T. Bauch (2011). The impact of media coverage on the transmission dynamics of human influenza. BMC Public Health 11(Suppl 1):S5.

Key References

http://mysite.science.uottawa.ca/rsmith43

http://www.mathstat.uottawa.ca/~rsmith

http://www.mathstat.uottawa.ca/~rsmith

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