- Email: [email protected]
Transmission dynamics of Plasmodium falciparum
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12 Schad, G.A. et ai. (1973) Arrested development in human hookworm infections: an adaptation to a seasonally unfavorable external environment. Science 180, 502-504 13 Schad, G.A. (19o0) in Hoot~Tt,orm Disease: Current Status and New Directions (Schad, G.A. and Warren, K.S., eds), pp 71-88, Taylor & Francis 14 Schad, G.A. (~991) Hooked on hookworm: twenty-five years of attachment. ]. Panlsitol. 77, 177-186 15 Yu, S-H. and Shen, W-X. (1990) in HooKer,arm Disease. Current Status and New Directions (Schad, G.A. and Warren, K.S., eds), pp 44-54, Taylor & Francis 16 Scott, J.A. (1028) An experimental study of the development of Ancylostoma canhumz in normal and abnormal hosts. Am. ].
Hyg. 8, 158-204 17 Schad, G.A. (1982) in Aspects Of Parasitoh)gy: A Festschrift Dedi-
cated to the Fiftieth Anniversary Of the Institute of Parasitoh~gy of McGill University (Meerovitch, E., ed.), pp 361-391, McGill University 18 Rogers, R.A., Rogers, L.A. and Martin, L.D. (1992) How the door opened: the peopling of the New World. Hum. Biol. 64, 281-302 19 Kliks, M.M. (1990~, Helminths as heirlooms and souvenirs: a review of New World paleoparasitology. Parasitoh~gy Today 6, 93-100 20 Kliks, M.M. (1982) Parasites in archaeological material from Brazil. Trans. R. Sac. Trap. Med. Hyg. 76, 701
Transmission Dynamics of Plasrnodiurn falciparum A, Saul
Allan Saul is in the Australian Centre for International and Tropical Health and Nutrition, Queensland Institute of Medical Research, PO Royal Brisbane Hospital, Brisbane, Queensland 4029, Australia. Tel: +61 7 3362 0402, Fax: +61 7 3362 0104, e-mail: [email protected]
values of each strain circulating ill the community (eg. Eqn 4 in Ref. 2). Third, in the original description of the model, they generally assume that, once infected, a person develops lifelong strain-specific immunity that prevents infection. More recently, Gupta and Day have explored variations of this model where several infections are required to give lifelong immunity ",7. Through this assumption of lifelong immunity, when the process comes to equilibrium, only a small proportion (1/R 0) of the population remains susceptible to infection to a particular strain. This set of assumptions is unconventional. Data from infections ill relatively nonimmune people would put tile period of infectiousness much longer, of the order of at least 100 days ~. Strains are commonly thought to interact so that force of infection for the population is determined from the average of the individual strain R0 values, giving estimates for Rt) closer to estimates based on measurements of entomological parameters. Data from experimental infections also suggest that, in relatively nonimmtme subjects, people can be repeatedly infected with the same isolate with evidence of immunity lasting for a maximuna of a few months '). How can we decide which of these views of malaria is mar ~. likely to be correct? One way is to check the starting asstimptions. The data supporting the conventional view have come mainly from work on experimental or therapeutic infections, largely in people with little previous exposure to malaria. The actual situatior under field conditions is much harder to determine. A second way is to examine predictions of a model based on these assumptions to see how they compare with our experiences of malaria. This paper examines the predictions of the Gupta model. This requires that the qualitative description of the model is converted into a quantitative framework. Gupta et al. have followed the general methods pioneered by Ross and Macdonald"), and extensively developed by Anderson and May ~. In this approach, the probabilities that individual mosquitoes and humans will move from one state to another (eg. noninfected to infected) are converted into rate constants, for the population, and the infection modelled as a series of differential equations. This approach generates
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Recent models of malaria have been developed by Gupta and her co-worker~. A frequent assumption used to illustrate these models is that levels of malaria are controlled by lifelong strain-specific immunity. In this article, Allan Saul examines the predictions this model makes about the equilibrium values of parasite prevalence and the dynamics of an epidemic following the introduction of a new strain. He reaches the conclusion that the stability of malaria makes long-term strain-specific immunity highly improbable, thus rendering models requiring lifelong strain-specific immunity unlikely to be of relevance in most epMemioh~ical contexts. A central problem for modelling malaria transmission is how to reconcile the high potential multiplication rates [as usually defined in models by the concept of the basic reproduction ratio (R,): the potential number of secondary cases resulting from a primary case in a completely susceptible populationl with the actual reproduction ratio (R,.: the actual number of secondary cases resulting from a primary case in a real population where immunity and other feedback mechanisms operate). In endemic areas, as the number of cases stays constant, R~, is one. Recent articles by Gupta and her co-workers present a novel view of the way in which this may happen t-.~.This paper extends comments made by others 4 to examine this model critically. In tile original formulation of the model, an R~. of one is achieved by several mechanisms. First, Gupta et al. suggest that the ratio for an individual strain is much less than that generally believed ~.2 arguing that tile period of infectiousness in the field is very short, ie. of the order of a few weeks I. Second, they assume that the multiple strains present in endemic areas 5 are independent fie. one strain does not affect transmission or growth of other strains even in mixed infections). In this case, the rate at which people become infected with any strain of malaria (called tile force of infection) can be calculated from the sum of the R0
© 1996. ElsevierSoence Lid
Parasitology Today, vol.
12. no. 2. 1996
FOCUS Box 1. P~oI~abiiistic M o d e l
Susceptible(x.) ~ 1
Infectious(Xs)i~_~ i (t~days)
Death
Death
,
,I I
"
Death
Incubating(Xc) H I (t~days)
|
Immune(xm) ~_ (tindays) ....
|
~
Death
Ii
I
I I
!
Incubating (Edays)
Infectious mosquitoes
mosquitoes
Death
Death
4
Uninfected i mosquitoes
' ~ ' ~
Birth
Death
As shown on the flow diagram above, humans and mosquitoes occupy a series of compartments. For humans, these are defined as the susceptible, incubating, infectious and immune, with x., x¢, x~ and x m proportion of the population, respectively, in each cowpartment. In practice, immune individuals may still be parasitaemic, but since such people will no longer contribute to transmission, they will not be considered in this model. People are lost from each compartment through death, with a daily probability of dying of (1 - Ph), where Ph is the daily survival probability and is assumed to be independent of age or infection status. Transition from the susceptible to the incubating compartment is determined by the daily probability of being bitten by an infectious mosquito. All other transitions occur at fixed time intervals t~, t~ and t m days for the transition times through the incubating, infectious and immune compartments, respectively. Similar states and transitions occur for the mosquito population and derivations of equations for R0 based on the probability of traversing these compartments have been described"L Using a similar approach, a series of recursion equations can be obtained that describes the proportion of the human population in each compartment on any day (i) in terms of the proportion of people in the susceptible and infectious compartments at earlier times, ie. on Day j. Assuming the numbers of infectious humans do not change appreciably over the average life of a mosquito, and that there is a low probability of a human being bitten by more than one infectious bite in a day, then these equations are: i
x,, =
~ x.ix.( ~ r)Ph, iRo/t~ j=i--t,+l
(1)
j --t L da-i
j=i--I,.-I,+
"£ n l 'l ", l l
t ~Ph, q'I,/ t.
(2)
I
i -t,.-t, x,,,, =
~ j=i-t~-t~-t,,
x,,i"~',~i r)Ph ' rR,/t~
(3)
+l
where E is tile length of tile extrinsic incubation period. Each of these equations contains the term Ro/G which is the potential number of new cases arising each day a primary case remains infectious. These equations are in a form that allows direct translation into computer programs for numerical solutions for any time following introduction of parasites. Provided the human population is large enough, these equations generally produce a series of damped oscillations in the proportion of incubating, infectious and immune humans with time, eventually giving stable equilibrium values. The following general solutions can be obtained for equilibrium values:
x,.* = (1 - Pht')lPh q - Pht'' t._ (1 - Ph)lJRol/I(Ph t':- Pht' ' t')(l - Pht'' t ,ft.)] X,* = [ PhI' -- Ph '' ~ t _ ( 1 -- P,)t~
/ R0l / ( 1 - Ph"' t~, t,,, )
X,,,* = (Ph"' " - Ph~' ' "' ":')!Pht' - Pht'' " - (1 -- Ph)tJ R.I/l(Ph
(4) (5)
~'- Pl," +t.)(l - Pl,q. L' fi,,)]
(6)
These equations can be simplified for two conditions: (1) where the period of immunity, t m is short compared with the human life expectancy:
x,.*=ltJ(t~.+t~+t.,)l(l-1/R
o) (7)
.r*=ltJ(t,.+t~+t,,)](1-1/R
o)
(8)
x,,*=It,,,/(t~+t~+t.,)](1-1/R.)
(11 )
x,.* = 1 - 1/R o
(9)
and (2). if imlnunity is lifelong, then: x¢*= to(1 - Ph)(1 -- 11R.)
Parasitology
(10)
Today,vol. 12, no. 2, 1996
x~*= t~(1 - Ph)(l - 1/R o)
(12}
75
Focus .
.
.
.
sets of equations which give a general picture of the process, but it is relatively difficult to accommodate processes which have constant transition times. An alternative way of formulating the model is to use the transition probabilities directly. These lead to a series of recursion equations where the number of individuals passing through each state can be expressed as a series of sums of the product of individuals in the previous state at earlier times and the probability of the transition 12. This approach readily accommodates transiE, ons through fixed times, such as the extrinsic and intrinsic incubation periods of malaria in the mosquito and human hosts, respectively. These recursion equations are directly translatable to computer-simulation programs for both deterministic and stochastic modelling. Although the intermediate equations may be complex, for steady-state values, they can be solved to give forms which are as simple as the solutions to sets of differential equations. A general set of equations for malaria tran,~mission is given in Box 1. These include not on!y '&:e situation modelled by Gupta et al., but also the m~,re general situation where immunity to malaria can have any specified duration. Both quantitative formulations of the Gupta model are used below for analysing the predicted equilibrium values of malaria. Although Gupta et al. have modelled the dynamics using the formulation based on their differential equations, this method does not consider delays in the transmission such as the extrinsic incubation period, so only the probabilistic model has been used for this purpose in this paper.
Model predictions: equilibrium values Equilibrium values using the Gupta et al. differenth~l equations. From the differential equations presented by Gupta el al. I, the proportion of people w h , are immune (x*) or infectious (xt') and the proportion of infected mosquitoes (tt*) when the system comes to equilibriuna is approximately: X* = 1 ~ 1/R,
(13)
xl* = ~(! -11R.)IS
(14)
y* = t.t/abm
(15)
where/.t is the death rate of the human population, S is the rate at which people cease to be infectious, a is the rate at which mosquitoes bite, b is the proportion of bites by infected mosquitoes which cause an infection in humans and m is the ratio of mosquitoes to humans. Using values similar to those employed by Gupta et al. (R,=3.34, a = 100 per year, b =0.1, m = 1.5,/z = 0.02 per year, S = 20 per year); this gives values of xl* =0.0007, or one person infectious in 1400, and y* of 0.0013, of which only about 10% will survive to be infectious. Equilibrium values using the probabilistic approach. The probabilistic modelling (Box 1), using the same underlying assumptions and values for parameters but with the addition of an estimate of the pre-infectious period in humans (k-), predicts that the proportion of the population acting as a reser,voir is 0.0015, or about twice as large as that predicted by the Gupta formulation. This results from the introduction of a further 16
compartment to the size of the infected human reservoir (ie. people incubating malaria but not yet infectious). This increase in the human reservoir is about twofold because the pre-incubation period (assumed to be 20 days), is close to the length of the infectious period chosen by Gupta et al. of 18 days. If the infectious period is really much longer, then the two formulations of the model would predict similar proportions infected and consequently a similar critical community size based on the equilibrium values. Equilibrimn values: sensitivity oll underlying assumptions. With either formulation, increasing R0 only marginally increases the size of the human reservoir of infection. However, as the size of the human reservoir is directly proportional to the period from infection to becoming noninfectious, changing this period from 18 days (assumed by Gupta et al.) to 100 days increases the proportion of the population acting as a reservoir to 0.0038 for the Gupta formulation and to 0.0046 for the probabilistic formulation. The low frequency of infectious people poses two problems. First, the strain will become extinct unless the population in the village being modelled is large enough to have at least a single person or a single mosquito either infectious or incubating the disease. For an 18 day period of infectiousness and for the Gupta formulation, only one person in 1400 is a reservoir. In practice, the population would have to be greater than 1400 to avoid extinction: allowing for stochastic fluctuations, critical community size needs to be greater than 10000 to avoid a high probability that random fluctuations in numbers of people infected would not at some stage result in no people infected. How big is the effective human popu!ation? For example, since the population of the endemic north coast of l'apua New Guinea is greater than one million people, why is this limit on the low frequency of a particul~r strain a problem? The difficulty is that both fort~v~lations of the model are valid only for populations over which transmission can occur freely, if there is no movement of people or mosquitoes, this limits the population under consideration to roughly the number living within the flight range of a mosquito. As a result, an endemic area can be thought of as a mosaic of small transmission areas and a particular strain could persist only if the population of each is larger than the critical community size. Alternative models could be ~.onstructed in which maintenance of the strain cart be achieved by frequent introduction of infectious or nonimmune people from one area into another which will increase the critical community size for maintaining a particular strain. However, in this scenario, interactions between parasites and levels of immunity in the human population will not be in equilibrium and it would be dangerous to draw conclusions on such parameters as the development of immunity and selection of virulence, on the basis of strain prevalence values calculated fi'om equilibrium levels predicted by the Gupta et al. model. The second problem is the difficulty in reconciling the high parasite prevalence in highly endemic areas" with the equilibrium prevalence levels predicted by either formulation of the model. Adults in such areas typically have a 20% prevalence of patent infections. Because many infections will be subpatent, realistic estimates of infection prevalence are probably greater Parasitology Today, vol. 12, no. 2, 1996
FOCUS imm
than 50% and may approach 100%. Even if people remain infected for six months, both the Gupta and the probabilistic formulations require at least 1000 different strains to be present in an area to account for the parasite prevalence. While multiple strains do occur, isolates of the same strain (as defined by Gupta et al. 2) can be detected. Therefore the level of diversity must be considerably less than this figure. It would appear that the only way in which both the parasite prevalence and the critical community size can both be satisfied is through a long period of infectiousness and short°term immunity to infection. For example, an alternative model of no long-term immunity where the preinfectious period (t~) is 20 days; the infectious period (t~) is 100 days; the period of immunity to infection (t m) is 100 days; R, >5 gives an equilibrium level of 43-54% of the population infectious with each isolate. These starting parameters are in accord with the periods of infectiousness noted by Jeffrey and Eyless, the period of immunity noted by Powell et al.", estimates of R, from entomological rates in the Madang region of Papua New Guinea 13. Model predictions: dynamic values following introduction of a new
isolate
Neither Gupta's differential equations nor the equations in Box 1 readily lead to analytical solutions for the proportion of people infected during the rapid rise then fall of parasites following the introduction of a new isolate. However, numerical solutions can be obtained for both approaches. Gupta et al. illustrate the damped oscillations in numbers of immune people under these conditions ~. Unfortunately they do not give values which illustrate the size of the infectious reservoir, but state:
100.0000 10.0000
100 -
-
u
OJ ¢-,
o.100o 0.0100
E E
-
604020-
0.0010 0.0001 0
i
!
i
i
20
40
60
80
b
0
0
100
I
I
I
I
i
20
40
60
80
100
Time (years)
Time(years)
100.0000
100
10.0000
B0- \
1.oooo 0.1000
"6 '~ 0.0100 i
A,~
~
_
_
60-
'10
-~
i -
0.0010 0.0001 0
I
I
I
I
I
20
40
60
80
100
40 200
0
I
I
r
I
I
20
40
60
80
100
Time (years)
Time (years)
C
6O 0.20
40
20 J
0.00
~
,
0
, , , , i , ' ' T" r " ~ - T ' ~ ' ~ - 7 4 8 12 16 20 Time (years)
0
-~
0
r-~ T r
4
r ~ 1 ~ ~
r
rT~T -v-r~
1
8
12
16
20
Time (years)
Fig. I. Percentage of people infected (ie. incubating or infectious) with malaria (left) or immune (right) following the single introduction of a new strain (a), yearly reintroduction of the same strain (b} or following changes in Ro after equilibrium values have been attained (c). In (c), Ro was altered one year into the simulation to give a 50% increase (heavy continuous curve), a 10% increase (light continuous curve), a 10% decrease (light dashed curve) and a 25% decrease (heavy dashed curve) in Ro, respectively, and it was assumed that there was no reintroduction of malaria during the period modelled. Standard conditions give a theoretical equilibrium level of 70.05% of the population immune (right). Where the immune level exceeds this figure, parasite multiplication will be substantially inhibited. Standard conditions used are: Ro, 3.34; extrinsic incubation period (E), 10 days; incubation period (to), 20 days; infectious period (ts), 18 days; lifelong immunity and the average human life 50 years (probability of surviving, P, 0.999945 per day). The simulations used conditions similar to those used by Gupta et al., and are based on the probabilistic formulation of the Gupta model.
'the infectious proportion of host and vector populations are likely to be very low under the parameter combinations that generate damped oscillations.... [and atl times fluctuate under 1%'. Because of the inherent difficulties in modelling the dynamics of transmission with fixed transition times using numerical solutions of differential equations, in this paper the dynamics have only been analysed in detail using simulations of the probabilistic model. Probabilistic formulation. To model the dynamics of an outbreak using the probabilistic formulation, beParasitology Today, vol. 12, no. 2, 1996
\
80
~1.oooo -
sides the parameters used in tile Gupta formulation, estimates must be made of the extrinsic incubation period (E) and the preinfectious period, t<~.With this formulation, it becomes apparent just how low the infectious reservoir can fall (Fig. 1). Under no feasible set of input parameters can the infection survive! Using deterministic modelling and with the standard set of parameters used by Gupta et al.~, the proportion of people who constitute the infectious reservoir falls to 8.1 x 10-'L Therefore more than one hundred million people would be needed to sustain the infection 77
during the fall which follows the initial epidemic. With more realistic stochastic modelling, the required population is even higher, ie. about a billion people. A much higher R0 or a much longer infectious period fails to save the outbreak from extinction. The reason for this prompt and invariable extinction is straightforward. During the initial spread of the infection, as shown in Fig. 2 from Ref. 1, substantially more people are infected than would be at equilibrium. As a result, the proportion of people who become immune is much greater than the equilibrium value. From the data of Gupta et al. at peak infection levels, only about 0.26 new infections occur for each person cured. Since in this model immunity is lifelong, it takes a substantial part of a human lifetime before sufficient nonimmune people are recruited to allow the R,, to again climb to a value of at least one. Under these conditions, the critical community size depends on having sufficient population so that at least one person will still be infectious when the level of immunity in the population has fallen to the point where sustained transmission can once again occur. Clearly the basic model of lifelong strain-specific inamunity leads to improbable predictions, but can modifications be raade which would save it? One possibility is to allow frequent reintroduction of the strain. Figure 1 illustrates such a simulation. Under these conditions, an equilibrium value of infection can be established, but continuous transmission only occurs when the proportion of imnaune people falls to the equilibrium value. For an R, value of 3.34, this would take about 20 years. Presunaably all surrounding areas would have had similar epidemics with the same strain, and would also be incapable of supporting transmission, it is difficult !,~ see where the inoculum for reil~troducing the strain could come froln, Dynamic vahtes: stability at eqttilit~rium, lf a mecl~anism could be foulad for equilibrium to be attained, a further test of the nlodel is to exalnine the stability of infectiorl at equilibriuln. Ft)r the Gupta Inodel, all strail~s in an area are as~utned to be independent so, ,at equilibrium, stability of any individual strain should reflect the stability of the malaria endemicity as a whole. The proportion of people infected with each strain is illustrated in Fig. 1 for scenarios following a change in the transmission rates after attaining equilibrium. For example, a 50% increase in transmission or a 25% decrease in transmission for a feasible sized community causes a rapid eradication of the infection! Larger fluctuations in transmission are usual in virtually any endemic situation. The reason for the eradication following a 50'~ increase in R 0 (for example, following a 50'/,~ increase in mosquito numbers) is similar for the eradication seen following the initial hatrodt, ction of the strain: an initial overshoot in the number infected leads to a low R,., which takes a substantial oart of a human hfetime to reverse. A similar but more direct explanation applies to the eradication Mlowing a de.. crease in R,, again the best part of a human lifetime must pass before R,. again achieves a value of one. Even a small change in transmission of 10~ leads to major fluctuations in parasite prevalence. These predictions conflict with the observed stability of malaria in areas where transmission has been changed through natural causes or through control programs. A striking demonstration of the stability of malaria in endemic 18
areas was obtained through the Garki project 14, wht re an intensive control program substantially decreased transmission, but did not result in the prompt extinction predicted by this modelling. Conclusions
Regardleqs of ovr inability to measure accurately the input parameters used in tl ese models, the outcome is unequivocal: the genetic diversity and stability of malaria in highly endemic regions rule out longterm ~';'ain-specific sterile immunity with the assumptions ot short period of infectiousness and low R0. The model proposed by Gupta et al. I produces a picture similar to that seen in measlesL~: highly unstable, or even chaotic behaviour, with generally very small numbers infectious at any one time, and requirhlg access to a large human population to be maintained. Like the malaria model proposed by Gupta et al., many processes rely on negative-feedback loops to provide stable equilibrium levels. These include stable flight in aeroplanes, regulation of chemical plants and control of fission in a nuclear power plant. A fundamental rule governil~g the design of such processes is that stability requires that the negative-feedback loop can be turned on and off with time constants comparable to the growth rate of the process being controlled, ie. a nuclear power plant will not be stable unless the control rods can be withdrawn about as fast as they can be inserted. The same must be true of stable malaria. Malaria is transmitted from one person to another on a time scale of a few months. Whatever limits transmission, such as strain or panspecific immunity, these processes must be capable of being downregulated within a time period of months. For people working on vaccines, the Gupta model is athactive, because it holds the promise that R, is nauch smaller than originally thought 2. Earlier lnodelling showed that eradication could be achieved by vaccinating a traction of the population equal to 1 ~ 1 / R , . "l'lae Gupta t t al. model suggested that this could be achieved at a 70-80'~, coverage which appears feasible. Do the results presented here mean we are once again faced with having to achieve impossible targets of the order of 99~ coverage to be useful? Fortunately, the answer is no. This estimate of coverage results from a consideration of the equilibrium value of malaria in a community. By considering the dynamics of malaria transmission, strategies resulting in substantial short-term improvements in malaria cap be obtained at feasible vaccine coverage, even where the R0 values are much higher"'.
Acknowledgements The idea~ developed in Wis papec have been refined through discussions with many colleagues, pacticulady in Australia and in Edinbucgh and I would like to thank them for their input. The prepacation of this ,.'title h,~sbeen assisted by a study grant flom tlw British Council. References 1 Gupta, S. (,t al. (19t)4) Theoretical studies of the effects of heterogeneity in the parasite population on the transmission dynamics of malaria. Proc. R. Soc. Lmuhm Set. B 256, 231-238 2 Gupta, S. et al. (1994) Antigenic diversity and the t~!,:.ad~,~,ion dynamics of Plasmodium falciparum. Sci~mt" 263, ~fil-963 3 Gupta, S. et al. (1994) Parasite virulence and disease patterns in Plasmodium falciparum malaria. Proc. Natl Acad. ~ci. USA 91,
3715- 3719
4 Dye, C. and Targett, G. (1994) A theory of malaria vaccination.
Parasitology Today, vol, 12, no. 2, 1996
Focus Nature 370, 95-96 5 Felger, I. et al. (1994) Plasmodium falciparmn: extensive polymorphism in merozoite surface antigen 2 alleles in an area with endemic malaria in Papua New Guinea. Exp. ParasitoL 79, 106-116 6 Gupta, S. and Day, K.P. (1994) A theoretical tramework for the immunoepidemiology of Plasmodium falciparum malaria. Parasite hnmunol. 16, 361-370 7 Gupta, S. and Day, K.P. (1994) A strain theory of malaria transmission. Parasitoiogy Today 10, 476-481 8 Jeffrey, G.M. and Eyles, D.E. (1955) h~.fectivity to mosquitoes of Plasmodium falciparum as related to gametocyte density and duration of infection. Am. ]. Trop. Med. Hyg. 4, 781-789 9 Powell R.D. et al. (1972) Clinical aspects of acquisition of i m m u n i t y to falciparum malaria. Pre~. Hehninth Soc. Wash. 39 (Special issue), 51-66 10 Macdonald, G. (1952) The Epidemh~logy and Control q# Malaria, Oxford University P, ess. ll Anderson, R.M. and May, R.M. (1991) Infectious Diseases of
Humans: Dynamics and Cmztrol, Oxford University Press 12 Saul, A.J. et al. (1990) A cyclical model for pathogen transmission and its application to determining vectorial capacity from vector infection rates. ]. Appi. Ecol. 27, 123-133 13 Graves. P.M. et al. (1990) Estimation of anopheline survival, vectorial capacity and mosquito infection probability for malaria vector infection rates in villages near Maclang, Papua New Guinea. ]. Appl. Ecol. 27, 134-147 14 Molineaux, L. and Gramiccia, G. (1980) The Garki Project: Research on the Epidemiology and Control of Malaria in the Sudan Savanah of West Afi'ica, World Heaith Organization 15 Grenfell, B.T. and Bolker, B.M. (1994) in Parasitic end Infectious Diseases. Ephh,miology and Ecology (Scott, M.E., Smith, G., eds), pp 219-233, Academic Press 16 Saul, A. (1993) Minimal efficacy requirements for malarial vaccines to significantly lower transmission in epidemic or seasonal malaria. Acta Trop. 52, 283-296 Note: See Leiters, this issue.
Second-generationAntirnalarial Endoperoxides S.R. Meshnick, C.W. Jefford, G.H. Posner, M.A. Avery and W. Peters Artemisinin, derived firm a Chinese herbal remedy, is a potent peroxide-contai,ing antimalarial. New types of peroxides, derived from this structure, as well as other naturally occurring antimalarial peroxides, have been synthesized amt foumt to have potent antimalarial activities. Studies on the activities, modes of action, and toxicities of these compomtds are discussed here by Steven Meshnick amt colh'agues. Originally developed in China, artemisinin or qinghaosu (which means 'blue-green plant extract') is a tetracyclic 1,2,4-trioxane occuring in a widely growing shrub, Artemisia aroma. Artemisinin and its semisynthetic derivatives (Fig. 1) have now been used for the treatment of malaria in at least three million people t. These 'first generation' compounds (all of which are simph: esters or ethers obtained from the lactol, dihydroartemisinin) are effective agents that act rapidly against cerebral malaria. There have not yet been any reports of clinically significant resistance or toxicity ~. Nevertheless, most of these compounds suffer from poor oral bioavailability and are associated with a high incidence of recrudescence. Furthermore, their actio"t seems to be limited to specific stages of Plasmodium inside the red blood cells. Like other drugs, the artemisinin derivatives might become ineffective with time through the development of resistance. In other words, they are by no means ideal Steven Meshnick is at the Depa,-tment of Epidemiology, University of Michigan School of Public Health, Ann Arbor, MI 48109, USA. Charles Jefford is at the Department of Organic Chemistry, University of Geneva, Geneva, Switzerland. Gary Posner is at the Department of Chemistry, Johns Hopkins University, Baltimore, MD 21218, USA. Mitchell Avery is at the Depa,-tment of Medicinal Chemistry, University of Mississippi, University, MS 38677, USA. Wallace Peters is at the CAB International Institute of Parasitology, St. Albans, UK AL4 0XU. Tel: +1 313 747 2406, Fax: + I 313 764 3192, e-mail: [email protected] Parasitology Today, vol. 12, no. 2, 1996
antimalarials. However, a. cemisinin is a 'new entity' with a structure and mode of action that are unrelated to those of any other antimalarial. This means that crossresistance is unlikely, a very important feature in light of the high incidence of resistance to the quinoline-type antimalarials. So even though the first-generation endoperoxides may have limitations, it should be possible, using a systematic program of synthesis and screening, to develop a second generation of derivatives with morefavorable pharmacological properties. In other words, one can contemplate artemisinin as a prototype for a n,,,w class of drugs, in the same way that penicillin was a prototype for the myriad of more effective Wlactam antibiotics. Despite the fact that the artemisinin ring system is complex and difficult to constuct, more than 100 analogs have been synthesized. Fortunately, the essential part of the molecule appears to be the endoperoxide bridge. Consequently, simpler structures containing this bridge, such as the trioxanes and tetroxanes, are far more accessible. Indeed, more than 1000 new endoperoxides belonging to several classes have now been prepared. Not only do these new derivatives offer new therapeutic possibilities, but they have been helpful in providing an understanding of the mode of action. Moreover, the insights so gained are likely to aid the design and development of better antimalarials. Mode of action The mode of action appears to involve two distinct steps. In the first step, cleavage of the endoperoxide bridge is catalyzed by intraparasitic iron and heine to generate unstable free radical il~termediates 2. The selective toxicity of the drug against malaria parasite is probably due to this step, since the intra-erythrocytic parasite is rich in iron and heme. In the second step, the resulting free radical, or a further rearranged
© 1996.Elsev,e,-SoenceLid 0t 69- 4758/96/$15.00
79
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12 Schad, G.A. et ai. (1973) Arrested development in human hookworm infections: an adaptation to a seasonally unfavorable external environment. Science 180, 502-504 13 Schad, G.A. (19o0) in Hoot~Tt,orm Disease: Current Status and New Directions (Schad, G.A. and Warren, K.S., eds), pp 71-88, Taylor & Francis 14 Schad, G.A. (~991) Hooked on hookworm: twenty-five years of attachment. ]. Panlsitol. 77, 177-186 15 Yu, S-H. and Shen, W-X. (1990) in HooKer,arm Disease. Current Status and New Directions (Schad, G.A. and Warren, K.S., eds), pp 44-54, Taylor & Francis 16 Scott, J.A. (1028) An experimental study of the development of Ancylostoma canhumz in normal and abnormal hosts. Am. ].
Hyg. 8, 158-204 17 Schad, G.A. (1982) in Aspects Of Parasitoh)gy: A Festschrift Dedi-
cated to the Fiftieth Anniversary Of the Institute of Parasitoh~gy of McGill University (Meerovitch, E., ed.), pp 361-391, McGill University 18 Rogers, R.A., Rogers, L.A. and Martin, L.D. (1992) How the door opened: the peopling of the New World. Hum. Biol. 64, 281-302 19 Kliks, M.M. (1990~, Helminths as heirlooms and souvenirs: a review of New World paleoparasitology. Parasitoh~gy Today 6, 93-100 20 Kliks, M.M. (1982) Parasites in archaeological material from Brazil. Trans. R. Sac. Trap. Med. Hyg. 76, 701
Transmission Dynamics of Plasrnodiurn falciparum A, Saul
Allan Saul is in the Australian Centre for International and Tropical Health and Nutrition, Queensland Institute of Medical Research, PO Royal Brisbane Hospital, Brisbane, Queensland 4029, Australia. Tel: +61 7 3362 0402, Fax: +61 7 3362 0104, e-mail: [email protected]
values of each strain circulating ill the community (eg. Eqn 4 in Ref. 2). Third, in the original description of the model, they generally assume that, once infected, a person develops lifelong strain-specific immunity that prevents infection. More recently, Gupta and Day have explored variations of this model where several infections are required to give lifelong immunity ",7. Through this assumption of lifelong immunity, when the process comes to equilibrium, only a small proportion (1/R 0) of the population remains susceptible to infection to a particular strain. This set of assumptions is unconventional. Data from infections ill relatively nonimmune people would put tile period of infectiousness much longer, of the order of at least 100 days ~. Strains are commonly thought to interact so that force of infection for the population is determined from the average of the individual strain R0 values, giving estimates for Rt) closer to estimates based on measurements of entomological parameters. Data from experimental infections also suggest that, in relatively nonimmtme subjects, people can be repeatedly infected with the same isolate with evidence of immunity lasting for a maximuna of a few months '). How can we decide which of these views of malaria is mar ~. likely to be correct? One way is to check the starting asstimptions. The data supporting the conventional view have come mainly from work on experimental or therapeutic infections, largely in people with little previous exposure to malaria. The actual situatior under field conditions is much harder to determine. A second way is to examine predictions of a model based on these assumptions to see how they compare with our experiences of malaria. This paper examines the predictions of the Gupta model. This requires that the qualitative description of the model is converted into a quantitative framework. Gupta et al. have followed the general methods pioneered by Ross and Macdonald"), and extensively developed by Anderson and May ~. In this approach, the probabilities that individual mosquitoes and humans will move from one state to another (eg. noninfected to infected) are converted into rate constants, for the population, and the infection modelled as a series of differential equations. This approach generates
74
0169 47581961515.00
Recent models of malaria have been developed by Gupta and her co-worker~. A frequent assumption used to illustrate these models is that levels of malaria are controlled by lifelong strain-specific immunity. In this article, Allan Saul examines the predictions this model makes about the equilibrium values of parasite prevalence and the dynamics of an epidemic following the introduction of a new strain. He reaches the conclusion that the stability of malaria makes long-term strain-specific immunity highly improbable, thus rendering models requiring lifelong strain-specific immunity unlikely to be of relevance in most epMemioh~ical contexts. A central problem for modelling malaria transmission is how to reconcile the high potential multiplication rates [as usually defined in models by the concept of the basic reproduction ratio (R,): the potential number of secondary cases resulting from a primary case in a completely susceptible populationl with the actual reproduction ratio (R,.: the actual number of secondary cases resulting from a primary case in a real population where immunity and other feedback mechanisms operate). In endemic areas, as the number of cases stays constant, R~, is one. Recent articles by Gupta and her co-workers present a novel view of the way in which this may happen t-.~.This paper extends comments made by others 4 to examine this model critically. In tile original formulation of the model, an R~. of one is achieved by several mechanisms. First, Gupta et al. suggest that the ratio for an individual strain is much less than that generally believed ~.2 arguing that tile period of infectiousness in the field is very short, ie. of the order of a few weeks I. Second, they assume that the multiple strains present in endemic areas 5 are independent fie. one strain does not affect transmission or growth of other strains even in mixed infections). In this case, the rate at which people become infected with any strain of malaria (called tile force of infection) can be calculated from the sum of the R0
© 1996. ElsevierSoence Lid
Parasitology Today, vol.
12. no. 2. 1996
FOCUS Box 1. P~oI~abiiistic M o d e l
Susceptible(x.) ~ 1
Infectious(Xs)i~_~ i (t~days)
Death
Death
,
,I I
"
Death
Incubating(Xc) H I (t~days)
|
Immune(xm) ~_ (tindays) ....
|
~
Death
Ii
I
I I
!
Incubating (Edays)
Infectious mosquitoes
mosquitoes
Death
Death
4
Uninfected i mosquitoes
' ~ ' ~
Birth
Death
As shown on the flow diagram above, humans and mosquitoes occupy a series of compartments. For humans, these are defined as the susceptible, incubating, infectious and immune, with x., x¢, x~ and x m proportion of the population, respectively, in each cowpartment. In practice, immune individuals may still be parasitaemic, but since such people will no longer contribute to transmission, they will not be considered in this model. People are lost from each compartment through death, with a daily probability of dying of (1 - Ph), where Ph is the daily survival probability and is assumed to be independent of age or infection status. Transition from the susceptible to the incubating compartment is determined by the daily probability of being bitten by an infectious mosquito. All other transitions occur at fixed time intervals t~, t~ and t m days for the transition times through the incubating, infectious and immune compartments, respectively. Similar states and transitions occur for the mosquito population and derivations of equations for R0 based on the probability of traversing these compartments have been described"L Using a similar approach, a series of recursion equations can be obtained that describes the proportion of the human population in each compartment on any day (i) in terms of the proportion of people in the susceptible and infectious compartments at earlier times, ie. on Day j. Assuming the numbers of infectious humans do not change appreciably over the average life of a mosquito, and that there is a low probability of a human being bitten by more than one infectious bite in a day, then these equations are: i
x,, =
~ x.ix.( ~ r)Ph, iRo/t~ j=i--t,+l
(1)
j --t L da-i
j=i--I,.-I,+
"£ n l 'l ", l l
t ~Ph, q'I,/ t.
(2)
I
i -t,.-t, x,,,, =
~ j=i-t~-t~-t,,
x,,i"~',~i r)Ph ' rR,/t~
(3)
+l
where E is tile length of tile extrinsic incubation period. Each of these equations contains the term Ro/G which is the potential number of new cases arising each day a primary case remains infectious. These equations are in a form that allows direct translation into computer programs for numerical solutions for any time following introduction of parasites. Provided the human population is large enough, these equations generally produce a series of damped oscillations in the proportion of incubating, infectious and immune humans with time, eventually giving stable equilibrium values. The following general solutions can be obtained for equilibrium values:
x,.* = (1 - Pht')lPh q - Pht'' t._ (1 - Ph)lJRol/I(Ph t':- Pht' ' t')(l - Pht'' t ,ft.)] X,* = [ PhI' -- Ph '' ~ t _ ( 1 -- P,)t~
/ R0l / ( 1 - Ph"' t~, t,,, )
X,,,* = (Ph"' " - Ph~' ' "' ":')!Pht' - Pht'' " - (1 -- Ph)tJ R.I/l(Ph
(4) (5)
~'- Pl," +t.)(l - Pl,q. L' fi,,)]
(6)
These equations can be simplified for two conditions: (1) where the period of immunity, t m is short compared with the human life expectancy:
x,.*=ltJ(t~.+t~+t.,)l(l-1/R
o) (7)
.r*=ltJ(t,.+t~+t,,)](1-1/R
o)
(8)
x,,*=It,,,/(t~+t~+t.,)](1-1/R.)
(11 )
x,.* = 1 - 1/R o
(9)
and (2). if imlnunity is lifelong, then: x¢*= to(1 - Ph)(1 -- 11R.)
Parasitology
(10)
Today,vol. 12, no. 2, 1996
x~*= t~(1 - Ph)(l - 1/R o)
(12}
75
Focus .
.
.
.
sets of equations which give a general picture of the process, but it is relatively difficult to accommodate processes which have constant transition times. An alternative way of formulating the model is to use the transition probabilities directly. These lead to a series of recursion equations where the number of individuals passing through each state can be expressed as a series of sums of the product of individuals in the previous state at earlier times and the probability of the transition 12. This approach readily accommodates transiE, ons through fixed times, such as the extrinsic and intrinsic incubation periods of malaria in the mosquito and human hosts, respectively. These recursion equations are directly translatable to computer-simulation programs for both deterministic and stochastic modelling. Although the intermediate equations may be complex, for steady-state values, they can be solved to give forms which are as simple as the solutions to sets of differential equations. A general set of equations for malaria tran,~mission is given in Box 1. These include not on!y '&:e situation modelled by Gupta et al., but also the m~,re general situation where immunity to malaria can have any specified duration. Both quantitative formulations of the Gupta model are used below for analysing the predicted equilibrium values of malaria. Although Gupta et al. have modelled the dynamics using the formulation based on their differential equations, this method does not consider delays in the transmission such as the extrinsic incubation period, so only the probabilistic model has been used for this purpose in this paper.
Model predictions: equilibrium values Equilibrium values using the Gupta et al. differenth~l equations. From the differential equations presented by Gupta el al. I, the proportion of people w h , are immune (x*) or infectious (xt') and the proportion of infected mosquitoes (tt*) when the system comes to equilibriuna is approximately: X* = 1 ~ 1/R,
(13)
xl* = ~(! -11R.)IS
(14)
y* = t.t/abm
(15)
where/.t is the death rate of the human population, S is the rate at which people cease to be infectious, a is the rate at which mosquitoes bite, b is the proportion of bites by infected mosquitoes which cause an infection in humans and m is the ratio of mosquitoes to humans. Using values similar to those employed by Gupta et al. (R,=3.34, a = 100 per year, b =0.1, m = 1.5,/z = 0.02 per year, S = 20 per year); this gives values of xl* =0.0007, or one person infectious in 1400, and y* of 0.0013, of which only about 10% will survive to be infectious. Equilibrium values using the probabilistic approach. The probabilistic modelling (Box 1), using the same underlying assumptions and values for parameters but with the addition of an estimate of the pre-infectious period in humans (k-), predicts that the proportion of the population acting as a reser,voir is 0.0015, or about twice as large as that predicted by the Gupta formulation. This results from the introduction of a further 16
compartment to the size of the infected human reservoir (ie. people incubating malaria but not yet infectious). This increase in the human reservoir is about twofold because the pre-incubation period (assumed to be 20 days), is close to the length of the infectious period chosen by Gupta et al. of 18 days. If the infectious period is really much longer, then the two formulations of the model would predict similar proportions infected and consequently a similar critical community size based on the equilibrium values. Equilibrimn values: sensitivity oll underlying assumptions. With either formulation, increasing R0 only marginally increases the size of the human reservoir of infection. However, as the size of the human reservoir is directly proportional to the period from infection to becoming noninfectious, changing this period from 18 days (assumed by Gupta et al.) to 100 days increases the proportion of the population acting as a reservoir to 0.0038 for the Gupta formulation and to 0.0046 for the probabilistic formulation. The low frequency of infectious people poses two problems. First, the strain will become extinct unless the population in the village being modelled is large enough to have at least a single person or a single mosquito either infectious or incubating the disease. For an 18 day period of infectiousness and for the Gupta formulation, only one person in 1400 is a reservoir. In practice, the population would have to be greater than 1400 to avoid extinction: allowing for stochastic fluctuations, critical community size needs to be greater than 10000 to avoid a high probability that random fluctuations in numbers of people infected would not at some stage result in no people infected. How big is the effective human popu!ation? For example, since the population of the endemic north coast of l'apua New Guinea is greater than one million people, why is this limit on the low frequency of a particul~r strain a problem? The difficulty is that both fort~v~lations of the model are valid only for populations over which transmission can occur freely, if there is no movement of people or mosquitoes, this limits the population under consideration to roughly the number living within the flight range of a mosquito. As a result, an endemic area can be thought of as a mosaic of small transmission areas and a particular strain could persist only if the population of each is larger than the critical community size. Alternative models could be ~.onstructed in which maintenance of the strain cart be achieved by frequent introduction of infectious or nonimmune people from one area into another which will increase the critical community size for maintaining a particular strain. However, in this scenario, interactions between parasites and levels of immunity in the human population will not be in equilibrium and it would be dangerous to draw conclusions on such parameters as the development of immunity and selection of virulence, on the basis of strain prevalence values calculated fi'om equilibrium levels predicted by the Gupta et al. model. The second problem is the difficulty in reconciling the high parasite prevalence in highly endemic areas" with the equilibrium prevalence levels predicted by either formulation of the model. Adults in such areas typically have a 20% prevalence of patent infections. Because many infections will be subpatent, realistic estimates of infection prevalence are probably greater Parasitology Today, vol. 12, no. 2, 1996
FOCUS imm
than 50% and may approach 100%. Even if people remain infected for six months, both the Gupta and the probabilistic formulations require at least 1000 different strains to be present in an area to account for the parasite prevalence. While multiple strains do occur, isolates of the same strain (as defined by Gupta et al. 2) can be detected. Therefore the level of diversity must be considerably less than this figure. It would appear that the only way in which both the parasite prevalence and the critical community size can both be satisfied is through a long period of infectiousness and short°term immunity to infection. For example, an alternative model of no long-term immunity where the preinfectious period (t~) is 20 days; the infectious period (t~) is 100 days; the period of immunity to infection (t m) is 100 days; R, >5 gives an equilibrium level of 43-54% of the population infectious with each isolate. These starting parameters are in accord with the periods of infectiousness noted by Jeffrey and Eyless, the period of immunity noted by Powell et al.", estimates of R, from entomological rates in the Madang region of Papua New Guinea 13. Model predictions: dynamic values following introduction of a new
isolate
Neither Gupta's differential equations nor the equations in Box 1 readily lead to analytical solutions for the proportion of people infected during the rapid rise then fall of parasites following the introduction of a new isolate. However, numerical solutions can be obtained for both approaches. Gupta et al. illustrate the damped oscillations in numbers of immune people under these conditions ~. Unfortunately they do not give values which illustrate the size of the infectious reservoir, but state:
100.0000 10.0000
100 -
-
u
OJ ¢-,
o.100o 0.0100
E E
-
604020-
0.0010 0.0001 0
i
!
i
i
20
40
60
80
b
0
0
100
I
I
I
I
i
20
40
60
80
100
Time (years)
Time(years)
100.0000
100
10.0000
B0- \
1.oooo 0.1000
"6 '~ 0.0100 i
A,~
~
_
_
60-
'10
-~
i -
0.0010 0.0001 0
I
I
I
I
I
20
40
60
80
100
40 200
0
I
I
r
I
I
20
40
60
80
100
Time (years)
Time (years)
C
6O 0.20
40
20 J
0.00
~
,
0
, , , , i , ' ' T" r " ~ - T ' ~ ' ~ - 7 4 8 12 16 20 Time (years)
0
-~
0
r-~ T r
4
r ~ 1 ~ ~
r
rT~T -v-r~
1
8
12
16
20
Time (years)
Fig. I. Percentage of people infected (ie. incubating or infectious) with malaria (left) or immune (right) following the single introduction of a new strain (a), yearly reintroduction of the same strain (b} or following changes in Ro after equilibrium values have been attained (c). In (c), Ro was altered one year into the simulation to give a 50% increase (heavy continuous curve), a 10% increase (light continuous curve), a 10% decrease (light dashed curve) and a 25% decrease (heavy dashed curve) in Ro, respectively, and it was assumed that there was no reintroduction of malaria during the period modelled. Standard conditions give a theoretical equilibrium level of 70.05% of the population immune (right). Where the immune level exceeds this figure, parasite multiplication will be substantially inhibited. Standard conditions used are: Ro, 3.34; extrinsic incubation period (E), 10 days; incubation period (to), 20 days; infectious period (ts), 18 days; lifelong immunity and the average human life 50 years (probability of surviving, P, 0.999945 per day). The simulations used conditions similar to those used by Gupta et al., and are based on the probabilistic formulation of the Gupta model.
'the infectious proportion of host and vector populations are likely to be very low under the parameter combinations that generate damped oscillations.... [and atl times fluctuate under 1%'. Because of the inherent difficulties in modelling the dynamics of transmission with fixed transition times using numerical solutions of differential equations, in this paper the dynamics have only been analysed in detail using simulations of the probabilistic model. Probabilistic formulation. To model the dynamics of an outbreak using the probabilistic formulation, beParasitology Today, vol. 12, no. 2, 1996
\
80
~1.oooo -
sides the parameters used in tile Gupta formulation, estimates must be made of the extrinsic incubation period (E) and the preinfectious period, t<~.With this formulation, it becomes apparent just how low the infectious reservoir can fall (Fig. 1). Under no feasible set of input parameters can the infection survive! Using deterministic modelling and with the standard set of parameters used by Gupta et al.~, the proportion of people who constitute the infectious reservoir falls to 8.1 x 10-'L Therefore more than one hundred million people would be needed to sustain the infection 77
during the fall which follows the initial epidemic. With more realistic stochastic modelling, the required population is even higher, ie. about a billion people. A much higher R0 or a much longer infectious period fails to save the outbreak from extinction. The reason for this prompt and invariable extinction is straightforward. During the initial spread of the infection, as shown in Fig. 2 from Ref. 1, substantially more people are infected than would be at equilibrium. As a result, the proportion of people who become immune is much greater than the equilibrium value. From the data of Gupta et al. at peak infection levels, only about 0.26 new infections occur for each person cured. Since in this model immunity is lifelong, it takes a substantial part of a human lifetime before sufficient nonimmune people are recruited to allow the R,, to again climb to a value of at least one. Under these conditions, the critical community size depends on having sufficient population so that at least one person will still be infectious when the level of immunity in the population has fallen to the point where sustained transmission can once again occur. Clearly the basic model of lifelong strain-specific inamunity leads to improbable predictions, but can modifications be raade which would save it? One possibility is to allow frequent reintroduction of the strain. Figure 1 illustrates such a simulation. Under these conditions, an equilibrium value of infection can be established, but continuous transmission only occurs when the proportion of imnaune people falls to the equilibrium value. For an R, value of 3.34, this would take about 20 years. Presunaably all surrounding areas would have had similar epidemics with the same strain, and would also be incapable of supporting transmission, it is difficult !,~ see where the inoculum for reil~troducing the strain could come froln, Dynamic vahtes: stability at eqttilit~rium, lf a mecl~anism could be foulad for equilibrium to be attained, a further test of the nlodel is to exalnine the stability of infectiorl at equilibriuln. Ft)r the Gupta Inodel, all strail~s in an area are as~utned to be independent so, ,at equilibrium, stability of any individual strain should reflect the stability of the malaria endemicity as a whole. The proportion of people infected with each strain is illustrated in Fig. 1 for scenarios following a change in the transmission rates after attaining equilibrium. For example, a 50% increase in transmission or a 25% decrease in transmission for a feasible sized community causes a rapid eradication of the infection! Larger fluctuations in transmission are usual in virtually any endemic situation. The reason for the eradication following a 50'~ increase in R 0 (for example, following a 50'/,~ increase in mosquito numbers) is similar for the eradication seen following the initial hatrodt, ction of the strain: an initial overshoot in the number infected leads to a low R,., which takes a substantial oart of a human hfetime to reverse. A similar but more direct explanation applies to the eradication Mlowing a de.. crease in R,, again the best part of a human lifetime must pass before R,. again achieves a value of one. Even a small change in transmission of 10~ leads to major fluctuations in parasite prevalence. These predictions conflict with the observed stability of malaria in areas where transmission has been changed through natural causes or through control programs. A striking demonstration of the stability of malaria in endemic 18
areas was obtained through the Garki project 14, wht re an intensive control program substantially decreased transmission, but did not result in the prompt extinction predicted by this modelling. Conclusions
Regardleqs of ovr inability to measure accurately the input parameters used in tl ese models, the outcome is unequivocal: the genetic diversity and stability of malaria in highly endemic regions rule out longterm ~';'ain-specific sterile immunity with the assumptions ot short period of infectiousness and low R0. The model proposed by Gupta et al. I produces a picture similar to that seen in measlesL~: highly unstable, or even chaotic behaviour, with generally very small numbers infectious at any one time, and requirhlg access to a large human population to be maintained. Like the malaria model proposed by Gupta et al., many processes rely on negative-feedback loops to provide stable equilibrium levels. These include stable flight in aeroplanes, regulation of chemical plants and control of fission in a nuclear power plant. A fundamental rule governil~g the design of such processes is that stability requires that the negative-feedback loop can be turned on and off with time constants comparable to the growth rate of the process being controlled, ie. a nuclear power plant will not be stable unless the control rods can be withdrawn about as fast as they can be inserted. The same must be true of stable malaria. Malaria is transmitted from one person to another on a time scale of a few months. Whatever limits transmission, such as strain or panspecific immunity, these processes must be capable of being downregulated within a time period of months. For people working on vaccines, the Gupta model is athactive, because it holds the promise that R, is nauch smaller than originally thought 2. Earlier lnodelling showed that eradication could be achieved by vaccinating a traction of the population equal to 1 ~ 1 / R , . "l'lae Gupta t t al. model suggested that this could be achieved at a 70-80'~, coverage which appears feasible. Do the results presented here mean we are once again faced with having to achieve impossible targets of the order of 99~ coverage to be useful? Fortunately, the answer is no. This estimate of coverage results from a consideration of the equilibrium value of malaria in a community. By considering the dynamics of malaria transmission, strategies resulting in substantial short-term improvements in malaria cap be obtained at feasible vaccine coverage, even where the R0 values are much higher"'.
Acknowledgements The idea~ developed in Wis papec have been refined through discussions with many colleagues, pacticulady in Australia and in Edinbucgh and I would like to thank them for their input. The prepacation of this ,.'title h,~sbeen assisted by a study grant flom tlw British Council. References 1 Gupta, S. (,t al. (19t)4) Theoretical studies of the effects of heterogeneity in the parasite population on the transmission dynamics of malaria. Proc. R. Soc. Lmuhm Set. B 256, 231-238 2 Gupta, S. et al. (1994) Antigenic diversity and the t~!,:.ad~,~,ion dynamics of Plasmodium falciparum. Sci~mt" 263, ~fil-963 3 Gupta, S. et al. (1994) Parasite virulence and disease patterns in Plasmodium falciparum malaria. Proc. Natl Acad. ~ci. USA 91,
3715- 3719
4 Dye, C. and Targett, G. (1994) A theory of malaria vaccination.
Parasitology Today, vol, 12, no. 2, 1996
Focus Nature 370, 95-96 5 Felger, I. et al. (1994) Plasmodium falciparmn: extensive polymorphism in merozoite surface antigen 2 alleles in an area with endemic malaria in Papua New Guinea. Exp. ParasitoL 79, 106-116 6 Gupta, S. and Day, K.P. (1994) A theoretical tramework for the immunoepidemiology of Plasmodium falciparum malaria. Parasite hnmunol. 16, 361-370 7 Gupta, S. and Day, K.P. (1994) A strain theory of malaria transmission. Parasitoiogy Today 10, 476-481 8 Jeffrey, G.M. and Eyles, D.E. (1955) h~.fectivity to mosquitoes of Plasmodium falciparum as related to gametocyte density and duration of infection. Am. ]. Trop. Med. Hyg. 4, 781-789 9 Powell R.D. et al. (1972) Clinical aspects of acquisition of i m m u n i t y to falciparum malaria. Pre~. Hehninth Soc. Wash. 39 (Special issue), 51-66 10 Macdonald, G. (1952) The Epidemh~logy and Control q# Malaria, Oxford University P, ess. ll Anderson, R.M. and May, R.M. (1991) Infectious Diseases of
Humans: Dynamics and Cmztrol, Oxford University Press 12 Saul, A.J. et al. (1990) A cyclical model for pathogen transmission and its application to determining vectorial capacity from vector infection rates. ]. Appi. Ecol. 27, 123-133 13 Graves. P.M. et al. (1990) Estimation of anopheline survival, vectorial capacity and mosquito infection probability for malaria vector infection rates in villages near Maclang, Papua New Guinea. ]. Appl. Ecol. 27, 134-147 14 Molineaux, L. and Gramiccia, G. (1980) The Garki Project: Research on the Epidemiology and Control of Malaria in the Sudan Savanah of West Afi'ica, World Heaith Organization 15 Grenfell, B.T. and Bolker, B.M. (1994) in Parasitic end Infectious Diseases. Ephh,miology and Ecology (Scott, M.E., Smith, G., eds), pp 219-233, Academic Press 16 Saul, A. (1993) Minimal efficacy requirements for malarial vaccines to significantly lower transmission in epidemic or seasonal malaria. Acta Trop. 52, 283-296 Note: See Leiters, this issue.
Second-generationAntirnalarial Endoperoxides S.R. Meshnick, C.W. Jefford, G.H. Posner, M.A. Avery and W. Peters Artemisinin, derived firm a Chinese herbal remedy, is a potent peroxide-contai,ing antimalarial. New types of peroxides, derived from this structure, as well as other naturally occurring antimalarial peroxides, have been synthesized amt foumt to have potent antimalarial activities. Studies on the activities, modes of action, and toxicities of these compomtds are discussed here by Steven Meshnick amt colh'agues. Originally developed in China, artemisinin or qinghaosu (which means 'blue-green plant extract') is a tetracyclic 1,2,4-trioxane occuring in a widely growing shrub, Artemisia aroma. Artemisinin and its semisynthetic derivatives (Fig. 1) have now been used for the treatment of malaria in at least three million people t. These 'first generation' compounds (all of which are simph: esters or ethers obtained from the lactol, dihydroartemisinin) are effective agents that act rapidly against cerebral malaria. There have not yet been any reports of clinically significant resistance or toxicity ~. Nevertheless, most of these compounds suffer from poor oral bioavailability and are associated with a high incidence of recrudescence. Furthermore, their actio"t seems to be limited to specific stages of Plasmodium inside the red blood cells. Like other drugs, the artemisinin derivatives might become ineffective with time through the development of resistance. In other words, they are by no means ideal Steven Meshnick is at the Depa,-tment of Epidemiology, University of Michigan School of Public Health, Ann Arbor, MI 48109, USA. Charles Jefford is at the Department of Organic Chemistry, University of Geneva, Geneva, Switzerland. Gary Posner is at the Department of Chemistry, Johns Hopkins University, Baltimore, MD 21218, USA. Mitchell Avery is at the Depa,-tment of Medicinal Chemistry, University of Mississippi, University, MS 38677, USA. Wallace Peters is at the CAB International Institute of Parasitology, St. Albans, UK AL4 0XU. Tel: +1 313 747 2406, Fax: + I 313 764 3192, e-mail: [email protected] Parasitology Today, vol. 12, no. 2, 1996
antimalarials. However, a. cemisinin is a 'new entity' with a structure and mode of action that are unrelated to those of any other antimalarial. This means that crossresistance is unlikely, a very important feature in light of the high incidence of resistance to the quinoline-type antimalarials. So even though the first-generation endoperoxides may have limitations, it should be possible, using a systematic program of synthesis and screening, to develop a second generation of derivatives with morefavorable pharmacological properties. In other words, one can contemplate artemisinin as a prototype for a n,,,w class of drugs, in the same way that penicillin was a prototype for the myriad of more effective Wlactam antibiotics. Despite the fact that the artemisinin ring system is complex and difficult to constuct, more than 100 analogs have been synthesized. Fortunately, the essential part of the molecule appears to be the endoperoxide bridge. Consequently, simpler structures containing this bridge, such as the trioxanes and tetroxanes, are far more accessible. Indeed, more than 1000 new endoperoxides belonging to several classes have now been prepared. Not only do these new derivatives offer new therapeutic possibilities, but they have been helpful in providing an understanding of the mode of action. Moreover, the insights so gained are likely to aid the design and development of better antimalarials. Mode of action The mode of action appears to involve two distinct steps. In the first step, cleavage of the endoperoxide bridge is catalyzed by intraparasitic iron and heine to generate unstable free radical il~termediates 2. The selective toxicity of the drug against malaria parasite is probably due to this step, since the intra-erythrocytic parasite is rich in iron and heme. In the second step, the resulting free radical, or a further rearranged
© 1996.Elsev,e,-SoenceLid 0t 69- 4758/96/$15.00
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