Could low-efﬁcacy malaria vaccines increase secondary...

Could low-efficacy malaria vaccines increase secondary infections in

endemic areas?

Robert Smith?

The University of Ottawa

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

Malaria

• More than 300 million acute illnesses each year

Malaria

• More than 300 million acute illnesses each year

• 1,000,000deaths annually.

Symptoms

• Repeated episodes of fever

Symptoms

• Anemia

Symptoms

• Anemia

• Death.

• 90% of malaria deaths occur in sub-Saharan Africa

Endemic areas

• Mostly among young children

Endemic areas

Admissions to St. Kitzo-Matany hospital, Uganda

• Mostly among young children• Even when it

doesn’t kill, acute illness can devastate economies in the developing world.

Endemic areas

Admissions to St. Kitzo-Matany hospital, Uganda

Malaria vaccines

• The search for a malaria vaccine is now over 70 years old

Malaria vaccines

• Recently, a candidate vaccine (RTS,S/AS01) completed Phase III trials

Malaria vaccines

• It cut the risk of developing severe malaria by 26%

Malaria vaccines

• It cut the risk of developing severe malaria by 26%

• The efficacy in infants was only 31%.

RTS,S/AS01 vaccine (Mosquirix)

• The time to first infection was cut by 45%

• The time to first infection was cut by 45%• Protection did not wane after 15 months

• The time to first infection was cut by 45%• Protection did not wane after 15 months• Phase III trial completed in 2012

• The time to first infection was cut by 45%• Protection did not wane after 15 months• Phase III trial completed in 2012• Currently in development commercially

• The time to first infection was cut by 45%• Protection did not wane after 15 months• Phase III trial completed in 2012• Currently in development commercially• Not expected on the market for a few years.

The downside

• Such vaccines hold great hope for containing the spread of the disease

The downside

• However, they are likely to have poor efficacy, at least initially

The downside

• However, they are likely to have poor efficacy, at least initially

• This may result in a net increase in infections.

Candidate vaccines

• Such vaccines permit infection but reduce parasite burden

Candidate vaccines

• Such vaccines permit infection but reduce parasite burden

• We call these “disease-modifying” vaccines.

Disease-modifying vaccines

Disease-modifying vaccines may:

• allow you to become infected

• reduce your duration of infection

• lower your parasite burden.

Potential effects

Potential effects from a malaria vaccine could include:

Potential effects

i. increasing the recovery rate

Potential effects

ii. increasing the acquired immunity rate

Potential effects

ii. increasing the acquired immunity rate

iii. reducing the rate of infection.

Limitations

Potential limitations of a vaccination program could include:

Limitations

Potential limitations of a vaccination program could include:i. the vaccine may only be delivered to a

proportion p of the population

Limitations

proportion p of the populationii. the vaccine may on “take” in a proportion ε

of people vaccinated

Limitations

of people vaccinatediii. the vaccine may wane at rate ω

Limitations

of people vaccinatediii. the vaccine may wane at rate ωiv. the vaccine may have suboptimal efficacy ψ.

Efficacy

A disease-modifying vaccine with 35%efficacy would:

Efficacy

A disease-modifying vaccine with 35%efficacy would:• stop infection 35% of the time

Efficacy

A disease-modifying vaccine with 35%efficacy would:• stop infection 35% of the time• permit infection the remaining 65% of the

Efficacy

time• lower your parasite burden once you

became infected

Efficacy

time• lower your parasite burden once you

became infected(so you’re less likely to transmit the disease).

Four groups

For any vaccine, there are four groups:

Four groups

For any vaccine, there are four groups: a) those who never received the vaccine;

Four groups

For any vaccine, there are four groups: a) those who never received the vaccine; b) those who received the vaccine but the vaccine

did not take;

Four groups

did not take;c) those who received the vaccine, the vaccine

took, but the vaccine waned over time; and

Four groups

took, but the vaccine waned over time; and d) those who received the vaccine, the vaccine took

and for whom the vaccine did not wane over time.

Four groups

took, but the vaccine waned over time; and d) those who received the vaccine, the vaccine took

and for whom the vaccine did not wane over time.

‘Vaccinated’ individuals

• ‘Unvaccinated’ individuals = groups (a)-(c)

‘Vaccinated’ individuals

• ‘Unvaccinated’ individuals = groups (a)-(c)

• ‘Vaccinated’ individuals = group (d).

Vaccinated individuals

Vaccinated individuals may have

Vaccinated individuals may have• a reduced rate of infection

Vaccinated individuals may have• a reduced rate of infection• increased life expectancy

Vaccinated individuals may have• a reduced rate of infection• increased life expectancy• faster recovery.

Duration of infection

Thus the duration of infection for ‘vaccinated’ individuals may

• decrease (due to higher recovery rates)

• increase (due to fewer deaths).

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The model

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MOSQUITOS

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population

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proportion vaccinated

population

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population

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susceptible

population

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susceptibleinfected

population

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susceptibleinfected immune

population

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susceptible mosqitoes

population

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infected mosqitoes

population

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infected mosqitoes

infecton rate

population

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infected mosqitoes

infecton rate

recoverytake

population

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infected mosqitoes

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acquired immunity

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infected mosqitoes

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loss of immunity

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infected mosqitoes

infecton rate

acquired immunity

loss of immunity

recovery

efficacy.

population

MOSQUITOSwaning

1 Malaria vaccines in endemic areas 3

mosquitos are not vaccinated). The birth rate is π, the background death rateis µ and γk is the death rate due to malaria (k = U, V ). Thus, the model is

dt= Ω − βMYUM − βMYV M − µMM

dt= βMYUM + βMYV M − µMN

dt= (1 − εp)π − µXU − βUNXU + ωXV + hUYU + δUQU

dt= εpπ − µXV − (1 − ψ)βV NXV − ωXV + hV YV + δV QV

dt= βUNXU − (µ + γU + αU + hU )YU + ωYV

dt= (1 − ψ)βV XV − (µ + γV + αV + hV )YV − ωYV

dt= αUYU − (µ + δU )QU + ωQV

dt= αV YV − (µ + δV )QV − ωQV .

The model is illustrated in Figure 1.1.With the notation ξk = µ + γk + αk + hk (k = U, V ), 1

ξKis the total du-

ration of the infectious period for unprotected and “successfully vaccinated”individuals, respectively. It is expected that the recovery rates αV , hV willincrease due to the vaccine, but that the disease-induced death rate γV willdecrease. It follows that the total duration of the infectious period for vacci-nated individuals may either increase or decrease. It is also expected that therate of infection βV will not increase.

1.3 Analysis

The disease-free equilibrium satisfies M = ΩµM

XU = π(µ(1−εp)+ω)µ(µ+ω) , XV =

εpπµ+ω

and N = YU = YV = QU = QV = 0. Thus the proportion of the

population who are “successfully vaccinated”, S, satisfies S = XV

= εpµµ+ω

In particular, XU = πµ(1 − S) and XV = π

At the disease-free equilibrium, the Jacobian matrix is J =

µM 0 0 0 −βMM −βMM 0 00 −µM 0 0 βMM βMM 0 00 −βUXU −µ ω hU 0 δU 00 −(1 − ψ)βV XV 0 −µ − ω 0 hV 0 δV

0 βU XU 0 0 −ξU ω 0 00 (1 − ψ)βU XV 0 0 0 −ξV − ω 0 00 0 0 0 αU 0 −µ − δU ω0 0 0 0 0 αV 0 −µ − δV − ω

The ODEs

Basic reproductive numbers

• The average number of secondary infections caused by an infected unvaccinated individual is R0

Basic reproductive numbers

• The average number of secondary infections caused by an infected unvaccinated individual is R0

• The average number of secondary infections caused by an infected vaccinated individual is RV.

Population reproductive number

• The total number of secondary infections caused by a single individual is

Rp = SRV+(1-S)R0

R0=reproductive number (unvaccinated)RV=reproductive number (vaccinated)

Rp = SRV+(1-S)R0

• S = proportion “successfully” vaccinated.

R0=reproductive number (unvaccinated)RV=reproductive number (vaccinated)

• When Rp = 1, SRV + (1-S)R0 = 1

Vaccine coverage level

Rj=reproductive numbers (pop, vacc, unvacc) ω=waning ε=takeS=proportion vaccinated pc=coverage µ=background death rate

• When Rp = 1, SRV + (1-S)R0 = 1• Thus

S =pcµ

µ + =

pc =(µ + )(1R0)µ(RV R0)

S =pcµ

µ + =

pc =(µ + )(1R0)µ(RV R0)

is the threshold vaccine coverage level.

S =pcµ

µ + =

pc =(µ + )(1R0)µ(RV R0)

Eradication?

• Vaccination programs whose coverage levels exceed pc are likely to eradicate the disease

pc=critical coverage level

Eradication?

• Vaccination programs whose coverage levels exceed pc are likely to eradicate the disease

• However, this may not be achievable in real terms.

pc=critical coverage level

First, do no harm

• Disease-modifying vaccines run the risk of increasing the number of secondary infections

First, do no harm

• This may happen due to increasing the average duration of infection

First, do no harm

• This may happen due to increasing the average duration of infection

• This may occur if many more people survive to become infected later.

Increasing secondary infections

• The number of secondary infections will increase if Rp > R0

Rj=reproductive numbers (pop, vacc, unvacc) Ψ=efficacyS=proportion vaccinated βj=rate of infection ξj=duration

• Thus

(1 S)R0 + SRV > R0

1(1 )2

• Thus

(1 S)R0 + SRV > R0

1(1 )2

• Thus

relative rateof infection

(1 S)R0 + SRV > R0

1(1 )2

• Thus

relative rateof infection vaccine

efficacy

(1 S)R0 + SRV > R0

1(1 )2

• Thus

relative rateof infection vaccine

efficacy

(1 S)R0 + SRV > R0

1(1 )2

relative duration of infection

fectio

vaccine efficacy (%) relative duration of infection

Increase insecondaryinfections

Decrease insecondary infections

Decreasing rate and duration

• If the rate and duration of infection both decrease, the number of secondary infections will always decrease

fectio

rate and duration

both decrease

• (Not terribly surprising.)

fectio

rate and duration

both decrease

A duration “shoulder”

• For a given vaccine efficacy, there is a duration “shoulder”

fectio

duration “shoulder”

• A small increase in the duration of infection will still decrease the number of secondary infections

fectio

• A small increase in the duration of infection will still decrease the number of secondary infections

• This is true even if the rate of infection is unchanged.

fectio

Beyond the “shoulder”

• If the duration of infection is significantly increased, then it is crucial that the rate of infection be decreased accordingly

fectio

beyond the “shoulder”

• Thus is crucial for low-efficacy vaccines.

fectio

beyond the “shoulder”

An example

• A 20% efficacious vaccine could accomodate an increase in the duration of infection by as much as 1.56 times thhe current duration of infection

fectio

An example

• A 20% efficacious vaccine could accomodate an increase in the duration of infection by as much as 1.56 times thhe current duration of infection

• Even if there is no reduction in the rate of infection, the net result will still be a decrease in secondary infections.

fectio

Reducing the infection rate

• However, a 20% efficacious vaccine that increased the duration of infection by a factor of 4 would lead to an increase in secondary infections...

fectio

Reducing the infection rate

• However, a 20% efficacious vaccine that increased the duration of infection by a factor of 4 would lead to an increase in secondary infections...

• ...unless the rate of infection for the vaccinated population were reduced to 40% of the current rate of infection.

fectio

Conclusions

• An imperfect malaria vaccine can eradicate the disease, if the coverage levels are sufficiently high

Conclusions

• Duration of infection decreases ⇒ secondary infections always decrease

Conclusions

• Small increases in the duration of infection can be tolerated, but larger increases must be accompanied by a reduction in the rate of infection

Conclusions

• Small increases in the duration of infection can be tolerated, but larger increases must be accompanied by a reduction in the rate of infection

• This is critical for low-efficacy vaccines.

A further consequence

• These results primarily apply to areas where malaria is endemic

• A disease-modifying malaria vaccine with a high duration of infection...

• A disease-modifying malaria vaccine with a high duration of infection...(for example, one which reduced mortality, but had no effect on the recovery rates)

• ...might be quite desirable for the developed world, if the prospect of reinfection is negligible.

Recommendation

Low-efficacy vaccines that result in high durations of infection must significantly lower the rate of infection if they are to be used in endemic areas.

Key reference

• R.J. Smith?, Could low-efficacy malaria vaccines increase secondary infections in endemic areas? (Mathematical Modeling of Biological Systems, Volume II (2007). A. Deutsch, R. Bravo de la Parra, R. de Boer, O. Diekmann, P. Jagers, E. Kisdi, M. Kretzschmar, P. Lansky and H. Metz (eds). Birkhäuser, Boston, 3-10)

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