A model of a malaria vaccine

Transcription

1 A model of a malaria vaccine Epidemiology of malaria Details of the vaccine Research questions The mathematical model Derive analytical thresholds Recommendations.

2 Malaria One of the most important human diseases throughout the tropical and sub-tropical regions of the world More than 300 million acute illnesses each year 1,000,000 deaths annually. Source: NMCC Central Board of Health, 2000

3 Symptoms Repeated episodes of fever Anemia Death.

4 Endemic areas 90% of malaria deaths occur in sub-saharan Africa Mostly among young children Even when it doesn t kill, acute illness can devastate economies in the developing world. Admissions to St. Kitzo-Matany hospital, Uganda

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6 Malaria vaccines The search for a malaria vaccine is now over 70 years old Recently, a candidate vaccine (RTS,S/ AS02A) was tested on 2,022 children in Mozambique It cut the risk of developing severe malaria by 58%.

7 RTS,S/AS02A vaccine The time to first infection was cut by 45% Protection did not wane after 12 months Expected on the market after 2010.

8 The downside Such vaccines hold great hope for containing the spread of the disease However, they are likely to have poor efficacy, at least initially This may result in a net increase in infections.

9 Candidate vaccines Such vaccines permit infection but reduce viral load We call these disease-modifying vaccines.

10 Disease-modifying vaccines Disease-modifying vaccines may: allow you to become infected reduce your duration of infection lower your viral load.

11 Potential effects Potential effects from a malaria vaccine could include: i. increasing the recovery rate ii. increasing the acquired immunity rate iii. reducing the rate of infection.

12 Limitations Potential limitations of a vaccination program could include: i. the vaccine may only be delivered to a proportion p of the population ii. the vaccine may on take in a proportion ε of people vaccinated iii. the vaccine may wane at rate ω iv. the vaccine may have suboptimal efficacy ψ.

13 p Population π 1-p Gets the vaccine ε 1-ε Doesn t get it Vaccine takes Doesn t take wanes, ω Vaccine success (uninfected) Vaccine failure (uninfected) Infected mosquito (1-ψ)βV βu Vaccine success (infected) Vaccine failure (infected).

14 Efficacy A disease-modifying vaccine with 35% efficacy would: stop infection 35% of the time permit infection the remaining 65% of the time lower your viral load once you became infected (so you re less likely to transmit the disease).

15 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; c) those who received the vaccine, the vaccine 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.

16 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; c) those who received the vaccine, the vaccine 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.

17 Vaccinated individuals Unvaccinated individuals = groups (a)-(c) Vaccinated individuals = group (d).

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

19 Duration of infection Thus the duration of infection for vaccinated individuals may decrease (due to higher recovery rates) increase (due to fewer deaths).

20 The model population susceptible!!!#* "#$ "#$! "!! + & " '! & ) & % ' & & $ & infected immune & &!! infecton rate "#$ "#$ susceptible mosqitoes UNVACCINATED waning take $! ". /. " " " "!"#$ #*!! ""#$!!"#$!! ""#$!!"#$!! ""#$! proportion vaccinated "#$! $ (,!!%- "!! &"! infected mosqitoes ( ) ( + ( % ( ' ( ' ( recovery acquired immunity MOSQUITOS VACCINATED efficacy. loss of immunity

21 The ODEs dm dt = Ω β M Y U M β M Y V M µ M M dn dt = β MY U M + β M Y V M µ M N dx U dt = (1 ɛp)π µx U β U NX U + ωx V + h U Y U + δ U Q U dx V dt = ɛpπ µx V (1 ψ)β V NX V ωx V + h V Y V + δ V Q V dy U dt = β U NX U (µ + γ U + α U + h U )Y U + ωy V dy V dt = (1 ψ)β V X V (µ + γ V + α V + h V )Y V ωy V dq U dt = α U Y U (µ + δ U )Q U + ωq V dq V dt = α V Y V (µ + δ V )Q V ωq V.

22 Basic reproductive numbers The average number of secondary infections caused by an infected unvaccinated individual is R 0 The average number of secondary infections caused by an infected vaccinated individual is R V.

23 Population reproductive number The total number of secondary infections caused by a single individual is R p = SR V +(1-S)R 0 S = proportion successfully vaccinated. R 0 =reproductive number (unvaccinated) R V =reproductive number (vaccinated)

24 Vaccine coverage level When R p = 1, SR V + (1-S)R 0 = 1 Thus S = ɛp cµ µ + ω = 1 R 0 R V R 0 (See Malaria Vaccination Notes) p c = (µ + ω)(1 R 0) ɛµ(r V R 0 ) is the threshold vaccine coverage level. R j =reproductive numbers (pop, vacc, unvacc) ω=waning ε=take S=proportion vaccinated pc=coverage µ=background death rate

25 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

26 First, do no harm Disease-modifying vaccines run the risk of increasing the number of secondary infections This may happen due to increasing the average duration of infection This may occur if many more people survive to become infected later.

27 Increasing secondary infections The number of secondary infections will increase if Rp > R0 Thus (1 S)R 0 + SR V > R 0 β V β U > 1 1 ψ ξ V ξ U relative rate of infection R j =reproductive numbers (pop, vacc, unvacc) Ψ=efficacy S=proportion vaccinated βj=rate of infection ξj=duration vaccine efficacy relative duration of infection.

28 1 relative transmissability Increase in secondary infections Decrease in secondary infections vaccine efficacy relative duration of infection 4

29 Decreasing rate and duration If the rate and duration of infection both decrease, the number of secondary infections will always decrease (Not terribly surprising.) 1 rate and duration both decrease relative transmissability vaccine efficacy Increase in secondary infections Decrease in secondary infections 3 relative duration of infection 4

30 A duration shoulder For a given vaccine efficacy, there is a duration shoulder 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. relative transmissability duration shoulder vaccine efficacy Increase in secondary infections Decrease in secondary infections 3 relative duration of infection 4

31 Beyond the shoulder If the duration of infection is significantly increased, then it is crucial that the rate of infection be decreased accordingly Thus is crucial for low-efficacy vaccines. 1 relative transmissability beyond the shoulder vaccine efficacy Increase in secondary infections Decrease in secondary infections 3 relative duration of infection 4

32 An example A 20% efficacious vaccine could accomodate an increase in the duration of infection by as much as 1.25 times the current duration of infection Even if there is no reduction in the rate 0.6 of infection, the net result will still be a 0 60 decrease in vaccine efficacy 30 secondary infections relative transmissability 1 2 Increase in secondary infections Decrease in secondary infections 3 relative duration of infection 4

33 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 31% of the current rate of infection. relative transmissability vaccine efficacy Increase in secondary infections Decrease in secondary infections 3 relative duration of infection 4

34 Conclusions An imperfect malaria vaccine can eradicate the disease, if the coverage levels are sufficiently high Duration of infection decreases secondary infections always decrease 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.

35 A further consequence These results primarily apply to areas where malaria is endemic 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.

36 Recommendation Low-efficacy vaccines which result in high durations of infection, but which do not significantly lower the rate of infection should not be used in endemic areas.