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Deterministic models for the eradication of poliomyelitis: Vaccination with the inactivated (IPV) and attenuated (OPV) polio virus vaccine
ELSEVIER
Deterministic Models for the Eradication of Poliomyelitis: Vaccination with the Inactivated (IPV) and Attenuated (OPV) Polio Virus Vaccine M. EICHNER Institut far Medizinische Biometrie, Universitiit Tiibingen, Germany AND
K. P. HADELER Lehrstuhl far Biomathematik, Universitiit Tiibingen, Germany Received 23 December 1993; revised 11 July 1994
ABSTRACT Currently two polio vaccines, IPV and OPV, are in use which differ markedly in their epidemiological parameters. A simple epidemiological model in terms of ordinary differential equations is proposed to study the effects of vaccination campaigns using these vaccines. The numbers of interest are the reproduction number of the disease in the presence of vaccination and the critical vaccination coverage necessary to prevent an outbreak. For these numbers explicit representations are determined which can be used in comparing different vaccination strategies.
INTRODUCTION Widespread vaccination led to polio elimination in many countries of the northern hemisphere; for all continents the number of poliomyelitis cases reported has been decreasing for the last 15 years [40]. In May 1988 the World Health Assembly accepted a resolution to eradicate poliomyelitis worldwide by the year 2000 [39, 40]. At present there are two poliomyelitis vaccines in use that have completely different epidemiological features. The inactivated polio virus vaccine (IPV, Salk vaccine) consists of "dead" virus particles (inactivated by formaldehyde treatment) and is applied by injection [30]. IPV vaccination protects most of the vaccinated individuals against infantile paralysis [45, 48], but does not readily protect all of them against infection with the wild virus [5, 9, 15, 18, 19, 35, 37]. If vaccinated children are infected they can become a source of infection with wild virus to their environment [14, MATHEMATICAL BIOSCIENCES 127:149-166 (1995)
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M. EICHNER AND K. P. HADELER
29, 33]. The oral polio virus vaccine (OPV, Sabin vaccine) consists of attenuated ("live") viruses. It is given orally (usually on a cube of sugar). Vaccination with OPV leads to infection (with vaccine virus) of the pharynx and intestine cells where the attenuated virus replicates; large numbers of virus particles are excreted for a long period of time [16, 18, 19, 51]. Susceptible individuals can become infected with the vaccine virus if they come into close contact with a vaccinated person [7, 15, 17, 22, 36, 43]. Successful vaccination or infection with OPV virus is nearly as effective as wild virus vaccination in providing immunity and preventing further infections [18]. Unfortunately the oral vaccine often fails in developing countries [2, 6, 10, 20] and in rare cases the vaccine virus itself leads to paralysis [3, 7, 26, 34]. These facts have caused extended discussions about the appropriate vaccination strategy [2, 6, 21, 23, 31, 32, 41, 44, 46, 49]. Models in the form of ordinary differential equations provide an insight into the mechanism and perhaps crude parameter estimates. More realistic models must take into account age structure and possibly social structure, mixing pattern, and stochastic effects. Also there are many features of poliomyelitis that are important but not sufficiently well documented to be incorporated into a simple model. In fact there are three types of virus which come with about the same frequency, although most cases of paralysis are caused by type 1. There are indications that these types compete and that infection with one type need not lead to protection against infections with other types. There are several modeling approaches towards poliomyelitis specifically. Elveback et al. (see [13] and the work cited there) have simulated the spread of polio and the effect of OPV vaccination in small highly structured populations. Cvjetanovic, Grab, and Dixon [8] simulate vaccination policies for an age structured population of children and young adults. They assume a specific distribution for the loss of immunity. Uehleke [50] studies models for the OPV vaccination; in his IPV model he assumes complete protection. In [12] the problem of minimum population size for polio persistence has been approached by simulation. The eradication problem has been studied in detail in [11]. Several authors [47, 52] have studied the seasonality (dependence on temperature and humidity) of polio transmission. Here a transmission model for polio infection and vaccination is studied. The model is deterministic and assumes a simple population structure. The aim of the modeling approach is to determine the necessary minimum vaccination coverage that must be reached for polio elimination. Two models are studied which take the special features of the two currently used vaccines into account.
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T H E OPV M O D E L The population is homogeneously mixing and of constant size (birth rate = death rate =/z). It is subdivided into the susceptible fraction s, the wild virus and vaccine virus infectious fractions (called w and v, respectively), and the immune fraction r. Each newborn individual is regarded to be susceptible. Immediately after birth, a fraction p is vaccinated. The vaccinated infants become infectious with the vaccine virus and can spread it to contacts. Each individual has /3w contacts per unit of time that are sufficiently close for wild virus transmission (similarly fl~ for the vaccine virus). Every infected individual immediately becomes infectious without an incubation period. Wild virus and vaccine virus infectivity is lost at a constant rate (Yw and y~, respectively) and thereafter the individuals are fully and permanently protected against further infection. It is assumed that the vaccine virus is, in some sense, less effective than the wild virus, e.g., that it is not as easily transmitted (/3~ Yw). In the present context these single effects are not really important. What matters is the basic reproduction number of the vaccine virus as compared to the wild virus. The model equations read = (1-
p)lx-
flwSW-
b = P i t + ~ v s v - yv v = flw s w - yw TM =
yw w
+
yv v
-
~vSV tzv,
ItS,
(1)
tzw,
tzr.
The basic reproduction number with respect to an infection with wild virus is Rw
~W
yw+/.
(2)
We assume that the wild virus can establish itself in the population, i.e., we assume R w > 1. Similarly, the basic reproduction number for an infection with vaccine virus is
Rv
~V vv+
(3)
We assume that the vaccine virus is unable to compete with the wild virus unless it is constantly reintroduced by vaccination. Thus, in the
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M. EICHNER AND K. P. HADELER
presence of the wild virus the vaccine cannot establish itself in the population, i.e., Rv < R w.
(4)
At this moment it is not necessary to distinguish the case where the vaccine virus can survive in the absence of the wild virus or not, i.e., the cases R~ > 1 and R v ~<1. The system (1) is essentially three-dimensional since the limit sets are contained in the hyperplane s+w+v+r=-l.
Since there are no general tools for three-dimensional systems we will not be able to analyze the qualitative behavior. We shall restrict ourselves to a discussion of stationary points and their stability. Since the vaccination virus is constantly introduced into the population there is no virus-free stationary solution. The state where the wild virus is absent is obtained from the nonlinear equations ~vSV + / z s = ( 1 - p ) ~ ,
(Sa)
yvv +/~v - / 3 vsv = p t z .
(5b)
Adding both equations gives a linear equation for v and s, yvv +/.,v + #s = #.
(6)
We solve (5b), (6) for v and equate, p. . ( 1 - s) 7v + / z _ / 3 v S = / z + 7 v
(7)
This is a nonlinear equation for the quantity s. Since the left-hand side is increasing in s and the right-hand side is decreasing, and the left-hand side is smaller than the right-hand side for s = 0 but larger for s = 1, there is exactly one positive root g. It can be obtained as the positive root of the quadratic equation 1 p S 2 - (1 + ~__~v)S+ 1-- = 0, g= R m v+l 2R,
1 f 2R~I(Rv-1)2+4pRvl
(8) ,,~/2 "
(9a)
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The corresponding value of v is = /x(1 - g) /x+yv
(9b)
Thus the three-dimensional system for the variables (s,v,w) has the stationary point (g,~,0)which we call the uninfected stationary state. For a solution with infected individuals ("infected solution") we have > 0, and thus we get 1 = Rw"
(10)
Then b is obtained from the equation (% + / x - ~vs)v = ptz,
pl~
= plx. RwRv /3v R w - R v"
yv+~-/3v~
(11)
Then = (1-p)/z-/3vgb -/~g
/3w~ ~=N
Rw-1
pROw)
R~--?~v •
(12)
The condition that ff be positive, is equivalent to Rw
l>PRw_Rv
1
+ Rw,
(13)
Rw-.v( ,)
(14)
or to P<
"Rw-
1-~--~w •
Thus there is a critical coverage of vaccination 1
1
It is evident that 0 < p * < 1. If p > p* then the wild virus cannot persist.
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M. EICHNER AND K. P. HADELER
If the spread of the vaccine virus in the population is weak, i.e., if R~ = 0, then the critical vaccination coverage is p * = 1 - 1 / R w. If, however, the basic reproduction number for the vaccine virus approaches that of the wild virus, then the reduction of the critical coverage becomes significant. PROPOSITION 1 In the O P V model there is a critical vaccination coverage p*. I f p >1p* then there is only the uninfected stationary state whereas if p < p* then there is an additional infected stationary state which represents an endemic situation.
We now look at the problem from a different point of view. The Jacobian at any point (s, v, w) is
J(s,v,w)
- # v V - # w W - JZ
- #vS
- ~wS
/3vV /3wW
/3vS-ev- ~ 0
0 /3wS - Yw - / ~
=
(16)
Thus the Jacobian at (g,~,0) splits into a matrix of order 2 with trace T=~vg-ev-~-#vV-U
(17a)
= -plz/-~-flv~-lz
and determinant D = - ( flv-~ + IX)( flv g - Yv - tz) + f l v 2 ~ = ( ~vr' + U ) p ~ / - ~ + ~ v ~ > 0
(17b)
and the eigenvalue A3 = [3wg - Yw - Iz.
(18)
Since T < 0 and D > 0, the eigenvalues of the matrix of order 2 have negative real parts. Hence the stability condition is the inequality A3<0, i.e., g <(Yw + l.t)/ fl w. The only information about g we have is contained in (7). We write that equation in the form ~O(s) = 0, where
q,(s)=
p.
~,v + . -
~vS
. ( 1 - s) ~,~ + ~
(19)
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155
Notice that ~(s) is an increasing function. Hence the inequality g < (Tw + / z ) / / 3 w is equivalent to qJ((Tw +/z)//3w) > 0, which is pp~
> / z ( 1 - (Tw +/z)//3w) 7v+/Z
(20)
One finds by elementary manipulations that this inequality is equivalent to
P>
RwRv( 1)w ' "R'-~ 1 - R-~-
(21,
which is the converse of (14). Hence we have the following:
PROPOSITION 2 If in the OPV model the vaccination coverage p satisfies the inequality p > p* then the uninfected stationary state is locally stable. If the converse inequality p < p* holds then the uninfected state is unstable and there is an infected stationary state. T H E IPV M O D E L In the case of the IPV vaccination we have original susceptibles s o and individuals with reduced susceptibility sl (as a result of vaccination). There are also infectives who had previously been fully or partially susceptible (w 0 and wl, respectively). Finally, there are those who are permanently immune; these are called r. The population again is homogeneously mixing and of constant size (birth rate = death rate =/z). The transmission rate is /3 =/3w and the recovery rate is 3' = Yw. Each newborn individual is regarded susceptible and a fraction p becomes vaccinated immediately after birth. A fraction a of the vaccinated infants becomes completely immune, while the susceptibility of the others is only reduced. The rate /3 encompasses the product of the susceptibility of a nonvaccinated susceptible and the infectivity of a nonvaccinated virus carrier. The infectivity of a vaccinated virus carrier is reduced by a factor g, 0 ~
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M. EICHNER AND K. P. HADELER
The system has the following form: 0 + g % ) s 0 - I~So,
s0 = / z ( 1 - p ) - / 3 ( w
$1 = /x(1 - a)p -/3(w o + gw1)fs 1 - - ~.LS1, ~0 = /3(W0 + gW~)S0 --(~ + i')W0,
(22)
~1 = /3(W0 + gw,)fS, -- ( ~ + ~')Wl,
i'=y(Wo +-~-~-)-- tzr + txap. This system is essentially four-dimensional and the statements about systems of higher dimension apply again. A simple way to obtain a system of dimension four is just to drop the last equation and consider a system for So, sa ,w o, w 1. Again we introduce the basic reproduction number
Rw= +----~ ~ /3
(23)
and the reproduction number for a population with reduced susceptibility
/3fg RI= y / h + tz"
(24)
In the uninfected situation we have w 0 = 0,w I = 0, hence s0 = (1 - p ) t z - Izs0, $1 = ~ ( 1
- -
a)p
(25)
-- I~S 1.
Thus the solutions approach a stationary point ( S 0 , g l ) w i t h go = ( 1 P), gl = ( 1 - a)p, and the full system has a stationary point (go, ~1,0,0). Next we observe that whenever w o > 0 or w a > 0 is introduced into the system then the variables w 0 and w I both become positive. Hence we look for a stationary point with all components positive. Assume that w = w 0 + gw 1 is known. Then the four variables So, sl, Wo,W1 satisfy a linear system
( /3w+ l z ) S o = ( 1 - p ) t z , ( ~Sfw + p.)s 1 = (1 - a)pl*, /3WSo = (~ + g)Wo, /3fwsl = ( ~,/ h + ~ ) w l ,
(26)
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157
from which s°=
(1- p)/x /3w+~ ~
'
/3w ×
w0= 3,+/,
Wl
=
/3fw
3"/h + t*
S1 ~---
( 1 - a)pl*
(27a)
/3fw + t.
(1- p)tt /3w + / x × (1 - a)pl*
-/3fiv'-~7 "
(2719)
We insert these expressions into w = w 0 + gw x, divide by w and obtain the condition
,I,(w)
( 1 - p)/z + g /3f ( 1 - a)p/z /3 × -flw+ ~ 3"/h+ ~ × /3fw + =1. (28) 3' +---~
This is a scalar equation for the variable w. The quantity ~ ( w ) is positive for w >/0 and it decreases to zero for increasing w, and
,~(o)
/3 ( l - p ) + 3" + ~
g/3f
3"/h +
×(1-a)p.
(29)
Hence a solution w > 0 of the equation ~ ( w ) --- 1 exists if and only if • (0) > 1, i.e., if
[3 g/3f × ( 1 - a ) p > l , 3' + ~ ( 1 - P ) + . 3 " / h + Ix or, equivalently Rw(1 - p ) + RI(1 - a)p > 1.
(30)
Suppose the inequality (30) is satisfied and w is the unique solution of ~ ( w ) = 1. Then s 0, s 1, w 0, and w I can be obtained from (27). These quantities satisfy the linear system (26) and thus are the components of the stationary solution. Hence we have shown the following:
PROPOSITION 3 Using IPV vaccination, the reproduction number under vaccination coverage p is given by R(p) = R w - ( R w - ( 1 There are several cases to be examined.
a)R1)p.
(31)
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M. EICHNER AND K. P. HADELER
In this case the disease cannot spread in a totally susceptible (nonvaccinated) population. Vaccination is essentially unnecessary. C a s e 1: R w < 1.
a)R r In this case the disease can spread in the susceptible population but it cannot spread in a completely vaccinated population. There is a critical vaccination level p* such that R ( p * ) = 1, C a s e 2: R w > 1 ( 1 -
p* =
R w -1
Rw-(1-a)R1
,
(32)
so the disease cannot spread for p > p*. C a s e 3 : ( 1 - a ) g I > 1. T h e infection can spread in a totally vaccinated population. Vaccination cannot cause elimination in this case (although it reduces prevalence of the infection). The Jacobian of the system (22) is (recall w = w 0 + gw 1) -
~w
-
o
j=
tx
0
-
- t3wf-
~
¢lSo
-
- 13fsl
flw
0
flSo - 3' - tx
0
flwf
firs I
¢igSo
- t3~s1 flgSo flgfs 1 - y/h
J "
- I~
(33)
For w 0 = wa = 0 the lower left 2 × 2 block vanishes and the upper left block is - / x I . Hence stability of the uninfected solution is governed by the lower right block Bso - 3' - #
¢~gso
~fsl
t
~gfs~ - ~, / h - ~z ]"
(34)
The trace is 7" = ( 13So - , / -
~ ) + (/3gfs~ - ~ , / h - t~).
(35)
In view of (31), the determinant is, up to a positive factor, equal to the quantity D = 1 - soR,~ - s i r ~
= 1-(1-p)R = 1 -
R(p).
w -(1-a)pR
1
(36)
MODELS FOR THE ERADICATION OF POLIOMYELITIS
159
The determinant is positive if and only if R(p) < 1. It remains to check the trace. We have to show that D > 0 implies T < 0. But this is obvious: If the sum of the two quotients in (36) is less than 1, then each of them is less than 1, and thus the trace is negative. Thus we have the following:
PROPOSITION4 If the reproduction number R(p) (as given by (31)) is less than 1 then the uninfected state is stable. DISCUSSION According to (15) and (31) the critical vaccination coverage for OPV and IPV, respectively, is
1
1
Rw
(37)
Both expressions contain the factor ( 1 - 1 / R w ) . The expression for OPV contains the factor (1 - R v / R w) which is less than 1, whereas the expression for IPV contains the factor R w / ( R w - ( 1 - a ) R 1) which is always greater than 1. Hence the inequality P~Pv ~
160
M. EICHNER AND K. P. HADELER
with other enteroviruses [20, 24, 43], or maternal antibodies that are transmitted by breast feeding [10, 25, 27, 28, 38]. Because of the lower take rate of OPV in tropical countries, vaccinated individuals are more likely to be protected against paralytic polyomyelitis after IPV vaccination. For individuals who live in a developing country, IPV is therefore the better choice. The community, on the other hand, will often decide to use OPV to reach a high level of herd immunity and to minimize the total number of paralytic cases. The decision of the community may also be influenced by the price of the vaccine and the ease of its application. IPV is more expensive and must be injected, whereas OPV is cheaper and easier to administer [30]. The oral application of OPV on sugar cubes might additionally increase the general acceptance by the population. For tropical countries we have to take the different take rates t o w , ttp v of the vaccines into account, in particular the low take rate of OPV. Hence the expressions (37) should be replaced by
1
) Rw
(38)
For small take rates the critical vaccination coverages can become greater than 1. The WHO advocates vaccination of all infants at the ages 6, 10, and 14 weeks. In countries with high transmission an additional vaccination shortly after birth is recommended [40]. The consequences of multiple vaccination are different for OPV and IPV. Successful OPV vaccination leads to full immunity and multiple vaccination increases the fraction of protected individuals. If each vaccination has a 60% chance being successful, only 6.4% remain unprotected after three vaccinations. Successful IPV vaccination, however, leads to a varying degree of protection. Multiple vaccinations reduce the number of nonresponders and improve the level of protection of individuals who have been "successfully" vaccinated previously. However, it is difficult to determine the extent to which each booster vaccination reduces the susceptibility and (after infection) the infectivity. The following parameter values are chosen for a developing country: The mean life expectancy is 45 years ( / z - - 1 / 4 5 per year). The basic reproduction number of the wild polio virus infection is estimated as R w = 12 and the average duration of infectivity is one month (Yw = 12 per year) which leads to an estimate flw = 144. The basic reproduction
MODELS FOR THE ERADICATION OF POLIOMYELITIS
161
number of the OPV infection is assumed to be only one quarter of that of the wild virus infection (Rv = 3). It is further assumed that 30% become completely immune (a = 0.3) after IPV vaccination while the others are only partly protected. The probability that a contact leads to infection of a partially protected individual is reduced to 50% ( f = 0.5). All infected individuals become fully infective (g = 1). For partially protected individuals the duration of the infectious period is reduced to 20% (h = 0.2). These assumptions lead to an estimate R 1 = 1.2. For the parameter set chosen above we get 1 × 0.69, P~Pv = -toPv
1 - × 0.98. P~'PV= -qPv
Empirical estimates for the take rate of IPV range between 0.9 and 1.0. Thus our estimate for P]~PV indicates that IPV vaccination, if it is successful at all in eliminating the disease in a developing country, requires a coverage close to 100%. On the other hand OPV, which looks more promising on the northern hemisphere, is competitive only if the take rate is not below 0.7. Otherwise there is a definite advantage of IPV. It m u s t be underlined, however, that the models are rather crude and the quantitative assumptions are rough estimates. We add a remark on vaccination in a growing population. The results have been obtained for a demographically stable population. In general, one cannot extend results directly to populations that are subject to demographic change, e.g., growing populations. One has to set up a demographic transmission model, e.g., in the form of a homogeneous system, and study in detail its qualitative behavior. In the present situation, in the absence of differential mortality and fertility, there is a direct approach. If we introduce a birth rate b into the models (in general one will assume b >/~), and the total population size is P, then the models (1) and (22) assume the form
= b(1-p)P-
~wsw/P-
b = bpP + ~ v S V / P -
Us,
Tvv - / z v ,
= [3~,sw/P - Tww - / z w ,
= T~w + Tvv - / ~ r .
~vsv/P-
(1')
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M. EICHNER AND K. P. HADELER
and = b(1 - p ) e -
Wo + g w , ) s o / e - USo,
sl = b(1 - a) p P - ~ ( w o + gwl ) fs 1 / P - tZSl, =
(Wo + g w l ) s o / P
+ u)wo,
(22')
Both systems are of the general form Yc= b P d + F ( x ) - tzx,
(39)
where the variable x is a vector in E~ ,n = 4 or 5, respectively, and P = erx, with e T= (1 ..... 1) ~ A n, is a scalar function. The vector d is nonnegative with erd = 1; it describes the distribution of offspring and the nonlinear function F, modeling transmission and recovery, is homogeneous of degree 1, i.e., F ( a x ) = a F ( x ) for ~ > 0, and has the property eT"F(x)- O. For the relative proportions y = x / P = x/(eT"x) one can derive a similar system .~ = b d + F ( y ) - by.
(40)
(Use .~ = (YcP- I ~ ) / P 2 and replace .~ from (39), 16 = ( b - / x ) P . ) Thus the proportions satisfy the same system as in the stationary case with /z replaced by b in (1) or (22), respectively. Of course, the exponent of demographic growth is b - / z . All formulae so far derived remain valid (with /x replaced by b), in particular the formulae for the critical vaccination rate. The results have to be appropriately interpreted, though. The finding that vaccination beyond the critical level stabilizes the uninfected situation in proportions does not exclude an increase of the absolute number of infected cases. It just says that the rate of increase of the number of infected cases is less than the rate of population growth. The discussion of absolute numbers requires more subtle arguments. M. E. thanks Evangelisches Studentenwerk Haus Villigst for a grant supporting his Ph.D. work and Klaus Dietz for advice and encouragement. Both authors thank the Issac Newton Institute (Cambridge) for invitations during the Epidemiology program 1993.
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17 H.M. Gelfand, A. H. Holguin, G. E. Marchetti, P. M. Feorino, F. R. Richardson, and D. Humphrey, A continuing surveillance of enterovirus infections in healthy children in six United States cities. 1. Viruses isolated during 1960 and 1961, Amer. J. Hygiene 78:358-75 (1963). 18 Y. Z. Ghendon and I. I. Sanakoyeva, Comparison of the resistance of the intestinal tract to poliomyelitis virus (Sabin's strains) in persons after naturally and experimentally acquired immunity, Acta l/irologica 5:265-73 (1961). 19 J.L. Henry, E. S. Jaikaran, J. R. Davies, A. J. H. Tomlinson, P. J. Mason, J. M. Barnes and A. J. Beale, A study of poliovaccination in infancy: Excretion following challenge with live virus by children given killed or living poliovaccine, J. Hygiene 64:105-20 (1966). 20 D.L. Heymann, K. Murphy, M. Brigaud, M. Aymard, A. Tembon, and G. K. Maben, Oral poliovirus vaccine in tropical Africa: Greater impact on incidence of paralytic disease than expected from coverage surveys and seroconversion rates, Bull. WHO 65:495-501 (1987). 21 A.R. Hinman, J. P. Koplan, W. A. Orenstein, E. W. Brink, and B. N. Nkowane, Live or inactivated poliomyelitis vaccine: An analysis of benefits and risks, Amer. J. Pub. Health 78:291-95 (1988). 22 D.M. Horstmann, J. Emmons, L. Gimpel, T. Subrahmanyan, and J. T. Riordan, Enterovirus surveillance following a community-wide oral poliovirus vaccination program: A seven-year study, Amer. J. Epidemiol. 97:173-86 (1973). 23 IOM, An Evaluation of Poliomyelitis Vaccine Policy Options, Institute of Medicine, Division of Health Promotion and Disease Prevention, Publication No. I OM 88-04, 1988. 24 T.J. John and S. Christopher, Oral polio vaccination of children in the tropics. III. Intercurrent enterovirus infections, vaccine virus take, and antibody response, Amer. J. Epidemiol. 102:422-28 (1975). 25 M. Katz and S. A. Plotkin, Oral polio immunization of the newborn infant: A possible method for overcoming interference by ingested antibodies, Trop. Pediatrics 73:267-70 (1968). 26 R. J. Kim-Farley, K. J. Bart, L. B. Schonberger, W. A. Orenstein, B. M. Nkowane, A. R. Hinman, O. M. Kew, M. H. Hatch, and J. E. Kaplan, Poliomyelitis in the USA: Virtual elimination of disease caused by wild virus, Lancet 11:1315-57 (1984). 27 M.L. Lepow, R. J. Warren, N. Gray, V. G. Ingram, and F. C. Robbins, Effect of Sabin type 1 poliomyelitis vaccine administered by mouth to newborn infants, New England J. Medicine 264:1071-78 (1961). 28 M.L. Lepow, R. J. Warren, V. G. Ingrain, S. C. Daugherty, and F. C. Robbins, Sabin type I (LSc2ab) oral poliomyelitis vaccine. Effect of dose upon response of newborn infants, Amer. J. Diseases of Children 104:99-103 (1962). 29 W. M. Marine, T. D. Y. Chin, and C. R. Gravelle, Limitation of fecal and pharyngeal poliovirus excretion in Salk-vaccinated children: A family study during a type 1 poliomyelitis epidemic, Amer. J. Hygiene 76:173-95 (1962). 30 A.M. McBean and J. F. Modlin, Rationale for the sequential use of inactivated poliovirus vaccine and live attenuated poliovirus vaccine for routine poliomyelitis immunization in the United States, Pediatric Infectious Disease J. 6:881-7 (1987).
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31 A. M. McBean, M. L. Thomas, P. Albrecht, J. C. Cuthie, R. Bernier, et al., Serologic response to oral polio vaccine and enhanced-potency inactivated polio vaccines, Amer. J. Epidemiol. 128:615-28 (1988). 32 A . M . McBean, M. L. Thorns, R. H. Johnson, B. R. Gadless, B. MacDonald, L. Nerhood, P. Cummins, J. Hughes, J. Kinnear, W. Watts, M. Kraft, P. Albrecht, E. J. Boone, M. Moore, J. A. Frank (Jr.), and R. Bernier, A comparison of the serologic responses to oral and injectable trivalent poliovirus vaccines, Rev. Infectious Diseases 6, Suppl. 2:552-5 (1984). 33 N. Nathanson and J. R. Martin, The epidemiology of poliomyelitis: Enigmas surrounding its appearance, epidemicity, and disappearance, Amer. J. Epidemiol. 110:672-92 (1979). 34 B. M. Nkowane, S. G. F. Wassilak, W. A. Orenstein, K. J. Bart, L. D. Schonberger, A. R. Hinman, and O. M. Kew, Vaccine-associated paralytic poliomyelitis. United States: 1973 through 1984, J. Amer. Med. Assoc. 257: 1335-40 (1987). 35 P.L. Ogra and D. T. Karzon, Formation and function of poliovirus antibody in different tissues, Progress Med. Virol. 13:156-93 (1971). 36 J.R. Paul, The spread of attenuated polioviruses among household contacts, in Poliomyelitis. Papers and Discussions Presented at the Fifth International Poliomyelitis Conference, Copenhagen, Denmark, 1960, pp. 359-67. 37 J . R . Paul, D. M. Horstmann, and J. C. Niederman, Immunity in poliomyelitis infection: Observations in experimental epidemiology, in Immunity and Virus Infection, V. A. Najjar, ed, New York, 1959, pp. 233-45. 38 T. Peradze, D. Montefiore, and D. Coker, Oral poliovirus vaccination and breast feeding, W. African Med. J. 17:122-4 (1968). 39 Resolution World Health Assembly 41.28, 1988. 40 S.E. Robertson, C. Chan, R. Kim-Farley, and N. Ward, Worldwide status of poliomyelitis in 1986, 1987, and 1988, and plans for its global eradication by the year 2000, World Health Statist. Quart. 43:80-90 (1990). 41 A. B. Sabin, Commentary: Is there a need for a change in poliomyelitis immunization policy? Pediatric Infectious Disease J. 6:887-89 (1987). 42 A. B. Sabin, Paralytic poliomyelitis: Old dogmas and new perspectives, Rev. Infectious Diseases 3:543-64 (1981). 43 A. B. Sabin, M. Ramos-Alvarez, J. Alvarez-Amezquita, W. Pelon, R. H. Michaels, I. Spigland, M. A. Koch, J. M. Barnes, and J. S. Rhim, Live, orally given poliovirus vaccine. Effects of rapid mass immunization on population under conditions of massive enteric infection with other viruses, J. Amer. Med. Assoc. 173:1521-6 (1960). 44 J. E. Salk, Commentary: Poliomyelitis vaccination--Choosing a wise policy, Pediatric Infectious Diseases J. 6:889-93 (1987). 45 J. E. Salk, Persistence of immunity after administration of formalin-treated poliovirus vaccine, Lancet 279(1I):715-23 (1960). 46 D. Salk, Polio immunization policy in United States: A new challenge for a new generation, Amer. J. Pub. Health 78:296-300 (1988). 47 C. C. Spicer, Influence of some meterological factors in the incidence of poliomyelitis, Brit. J. Preventive Social Med. 13:139-44 (1959). 48 G. Stickle, Observed and expected poliomyelitis in the United States, 1958-1961, Amer. J. Pub. Health 54:1222-29 (1964).
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49 T. A. Swartz, E. Ben-Porath, H. Kanaaneh, L. Leitner, and N. Goldblum, Comparison of inactivated poliovirus vaccine and oral poliovirus vaccine programs in Israel, Rev. Infectious Diseases 6, Suppl. 2:556-61 (1984). 50 H.B. Uehleke, Quantitative Modelle zur Epidemiologie der Poliomyelitis, MD Thesis, Medical School, University of Tiibingen, 1983. 51 J. D. Verlinde and J. B. Wilterdink, A small-scale trial on vaccination and revaccination with live attenuated polioviruses in the Netherlands, in First International Conference on Live Poliovirus Vaccines, Pan American Sanitary Bureau Scientific Publication 44, 1959, pp. 355-66. 52 J. A. Yorke, N. Nathanson, G. Pianigiani, and J. Martin, Seasonality in the requirements for perpetuation and eradication of viruses in populations, Amer. J. EpMemiol. 109:103-23 (1979).
Deterministic Models for the Eradication of Poliomyelitis: Vaccination with the Inactivated (IPV) and Attenuated (OPV) Polio Virus Vaccine M. EICHNER Institut far Medizinische Biometrie, Universitiit Tiibingen, Germany AND
K. P. HADELER Lehrstuhl far Biomathematik, Universitiit Tiibingen, Germany Received 23 December 1993; revised 11 July 1994
ABSTRACT Currently two polio vaccines, IPV and OPV, are in use which differ markedly in their epidemiological parameters. A simple epidemiological model in terms of ordinary differential equations is proposed to study the effects of vaccination campaigns using these vaccines. The numbers of interest are the reproduction number of the disease in the presence of vaccination and the critical vaccination coverage necessary to prevent an outbreak. For these numbers explicit representations are determined which can be used in comparing different vaccination strategies.
INTRODUCTION Widespread vaccination led to polio elimination in many countries of the northern hemisphere; for all continents the number of poliomyelitis cases reported has been decreasing for the last 15 years [40]. In May 1988 the World Health Assembly accepted a resolution to eradicate poliomyelitis worldwide by the year 2000 [39, 40]. At present there are two poliomyelitis vaccines in use that have completely different epidemiological features. The inactivated polio virus vaccine (IPV, Salk vaccine) consists of "dead" virus particles (inactivated by formaldehyde treatment) and is applied by injection [30]. IPV vaccination protects most of the vaccinated individuals against infantile paralysis [45, 48], but does not readily protect all of them against infection with the wild virus [5, 9, 15, 18, 19, 35, 37]. If vaccinated children are infected they can become a source of infection with wild virus to their environment [14, MATHEMATICAL BIOSCIENCES 127:149-166 (1995)
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29, 33]. The oral polio virus vaccine (OPV, Sabin vaccine) consists of attenuated ("live") viruses. It is given orally (usually on a cube of sugar). Vaccination with OPV leads to infection (with vaccine virus) of the pharynx and intestine cells where the attenuated virus replicates; large numbers of virus particles are excreted for a long period of time [16, 18, 19, 51]. Susceptible individuals can become infected with the vaccine virus if they come into close contact with a vaccinated person [7, 15, 17, 22, 36, 43]. Successful vaccination or infection with OPV virus is nearly as effective as wild virus vaccination in providing immunity and preventing further infections [18]. Unfortunately the oral vaccine often fails in developing countries [2, 6, 10, 20] and in rare cases the vaccine virus itself leads to paralysis [3, 7, 26, 34]. These facts have caused extended discussions about the appropriate vaccination strategy [2, 6, 21, 23, 31, 32, 41, 44, 46, 49]. Models in the form of ordinary differential equations provide an insight into the mechanism and perhaps crude parameter estimates. More realistic models must take into account age structure and possibly social structure, mixing pattern, and stochastic effects. Also there are many features of poliomyelitis that are important but not sufficiently well documented to be incorporated into a simple model. In fact there are three types of virus which come with about the same frequency, although most cases of paralysis are caused by type 1. There are indications that these types compete and that infection with one type need not lead to protection against infections with other types. There are several modeling approaches towards poliomyelitis specifically. Elveback et al. (see [13] and the work cited there) have simulated the spread of polio and the effect of OPV vaccination in small highly structured populations. Cvjetanovic, Grab, and Dixon [8] simulate vaccination policies for an age structured population of children and young adults. They assume a specific distribution for the loss of immunity. Uehleke [50] studies models for the OPV vaccination; in his IPV model he assumes complete protection. In [12] the problem of minimum population size for polio persistence has been approached by simulation. The eradication problem has been studied in detail in [11]. Several authors [47, 52] have studied the seasonality (dependence on temperature and humidity) of polio transmission. Here a transmission model for polio infection and vaccination is studied. The model is deterministic and assumes a simple population structure. The aim of the modeling approach is to determine the necessary minimum vaccination coverage that must be reached for polio elimination. Two models are studied which take the special features of the two currently used vaccines into account.
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T H E OPV M O D E L The population is homogeneously mixing and of constant size (birth rate = death rate =/z). It is subdivided into the susceptible fraction s, the wild virus and vaccine virus infectious fractions (called w and v, respectively), and the immune fraction r. Each newborn individual is regarded to be susceptible. Immediately after birth, a fraction p is vaccinated. The vaccinated infants become infectious with the vaccine virus and can spread it to contacts. Each individual has /3w contacts per unit of time that are sufficiently close for wild virus transmission (similarly fl~ for the vaccine virus). Every infected individual immediately becomes infectious without an incubation period. Wild virus and vaccine virus infectivity is lost at a constant rate (Yw and y~, respectively) and thereafter the individuals are fully and permanently protected against further infection. It is assumed that the vaccine virus is, in some sense, less effective than the wild virus, e.g., that it is not as easily transmitted (/3~ Yw). In the present context these single effects are not really important. What matters is the basic reproduction number of the vaccine virus as compared to the wild virus. The model equations read = (1-
p)lx-
flwSW-
b = P i t + ~ v s v - yv v = flw s w - yw TM =
yw w
+
yv v
-
~vSV tzv,
ItS,
(1)
tzw,
tzr.
The basic reproduction number with respect to an infection with wild virus is Rw
~W
yw+/.
(2)
We assume that the wild virus can establish itself in the population, i.e., we assume R w > 1. Similarly, the basic reproduction number for an infection with vaccine virus is
Rv
~V vv+
(3)
We assume that the vaccine virus is unable to compete with the wild virus unless it is constantly reintroduced by vaccination. Thus, in the
152
M. EICHNER AND K. P. HADELER
presence of the wild virus the vaccine cannot establish itself in the population, i.e., Rv < R w.
(4)
At this moment it is not necessary to distinguish the case where the vaccine virus can survive in the absence of the wild virus or not, i.e., the cases R~ > 1 and R v ~<1. The system (1) is essentially three-dimensional since the limit sets are contained in the hyperplane s+w+v+r=-l.
Since there are no general tools for three-dimensional systems we will not be able to analyze the qualitative behavior. We shall restrict ourselves to a discussion of stationary points and their stability. Since the vaccination virus is constantly introduced into the population there is no virus-free stationary solution. The state where the wild virus is absent is obtained from the nonlinear equations ~vSV + / z s = ( 1 - p ) ~ ,
(Sa)
yvv +/~v - / 3 vsv = p t z .
(5b)
Adding both equations gives a linear equation for v and s, yvv +/.,v + #s = #.
(6)
We solve (5b), (6) for v and equate, p. . ( 1 - s) 7v + / z _ / 3 v S = / z + 7 v
(7)
This is a nonlinear equation for the quantity s. Since the left-hand side is increasing in s and the right-hand side is decreasing, and the left-hand side is smaller than the right-hand side for s = 0 but larger for s = 1, there is exactly one positive root g. It can be obtained as the positive root of the quadratic equation 1 p S 2 - (1 + ~__~v)S+ 1-- = 0, g= R m v+l 2R,
1 f 2R~I(Rv-1)2+4pRvl
(8) ,,~/2 "
(9a)
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153
The corresponding value of v is = /x(1 - g) /x+yv
(9b)
Thus the three-dimensional system for the variables (s,v,w) has the stationary point (g,~,0)which we call the uninfected stationary state. For a solution with infected individuals ("infected solution") we have > 0, and thus we get 1 = Rw"
(10)
Then b is obtained from the equation (% + / x - ~vs)v = ptz,
pl~
= plx. RwRv /3v R w - R v"
yv+~-/3v~
(11)
Then = (1-p)/z-/3vgb -/~g
/3w~ ~=N
Rw-1
pROw)
R~--?~v •
(12)
The condition that ff be positive, is equivalent to Rw
l>PRw_Rv
1
+ Rw,
(13)
Rw-.v( ,)
(14)
or to P<
"Rw-
1-~--~w •
Thus there is a critical coverage of vaccination 1
1
It is evident that 0 < p * < 1. If p > p* then the wild virus cannot persist.
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M. EICHNER AND K. P. HADELER
If the spread of the vaccine virus in the population is weak, i.e., if R~ = 0, then the critical vaccination coverage is p * = 1 - 1 / R w. If, however, the basic reproduction number for the vaccine virus approaches that of the wild virus, then the reduction of the critical coverage becomes significant. PROPOSITION 1 In the O P V model there is a critical vaccination coverage p*. I f p >1p* then there is only the uninfected stationary state whereas if p < p* then there is an additional infected stationary state which represents an endemic situation.
We now look at the problem from a different point of view. The Jacobian at any point (s, v, w) is
J(s,v,w)
- # v V - # w W - JZ
- #vS
- ~wS
/3vV /3wW
/3vS-ev- ~ 0
0 /3wS - Yw - / ~
=
(16)
Thus the Jacobian at (g,~,0) splits into a matrix of order 2 with trace T=~vg-ev-~-#vV-U
(17a)
= -plz/-~-flv~-lz
and determinant D = - ( flv-~ + IX)( flv g - Yv - tz) + f l v 2 ~ = ( ~vr' + U ) p ~ / - ~ + ~ v ~ > 0
(17b)
and the eigenvalue A3 = [3wg - Yw - Iz.
(18)
Since T < 0 and D > 0, the eigenvalues of the matrix of order 2 have negative real parts. Hence the stability condition is the inequality A3<0, i.e., g <(Yw + l.t)/ fl w. The only information about g we have is contained in (7). We write that equation in the form ~O(s) = 0, where
q,(s)=
p.
~,v + . -
~vS
. ( 1 - s) ~,~ + ~
(19)
MODELS FOR THE ERADICATION OF POLIOMYELITIS
155
Notice that ~(s) is an increasing function. Hence the inequality g < (Tw + / z ) / / 3 w is equivalent to qJ((Tw +/z)//3w) > 0, which is pp~
> / z ( 1 - (Tw +/z)//3w) 7v+/Z
(20)
One finds by elementary manipulations that this inequality is equivalent to
P>
RwRv( 1)w ' "R'-~ 1 - R-~-
(21,
which is the converse of (14). Hence we have the following:
PROPOSITION 2 If in the OPV model the vaccination coverage p satisfies the inequality p > p* then the uninfected stationary state is locally stable. If the converse inequality p < p* holds then the uninfected state is unstable and there is an infected stationary state. T H E IPV M O D E L In the case of the IPV vaccination we have original susceptibles s o and individuals with reduced susceptibility sl (as a result of vaccination). There are also infectives who had previously been fully or partially susceptible (w 0 and wl, respectively). Finally, there are those who are permanently immune; these are called r. The population again is homogeneously mixing and of constant size (birth rate = death rate =/z). The transmission rate is /3 =/3w and the recovery rate is 3' = Yw. Each newborn individual is regarded susceptible and a fraction p becomes vaccinated immediately after birth. A fraction a of the vaccinated infants becomes completely immune, while the susceptibility of the others is only reduced. The rate /3 encompasses the product of the susceptibility of a nonvaccinated susceptible and the infectivity of a nonvaccinated virus carrier. The infectivity of a vaccinated virus carrier is reduced by a factor g, 0 ~
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M. EICHNER AND K. P. HADELER
The system has the following form: 0 + g % ) s 0 - I~So,
s0 = / z ( 1 - p ) - / 3 ( w
$1 = /x(1 - a)p -/3(w o + gw1)fs 1 - - ~.LS1, ~0 = /3(W0 + gW~)S0 --(~ + i')W0,
(22)
~1 = /3(W0 + gw,)fS, -- ( ~ + ~')Wl,
i'=y(Wo +-~-~-)-- tzr + txap. This system is essentially four-dimensional and the statements about systems of higher dimension apply again. A simple way to obtain a system of dimension four is just to drop the last equation and consider a system for So, sa ,w o, w 1. Again we introduce the basic reproduction number
Rw= +----~ ~ /3
(23)
and the reproduction number for a population with reduced susceptibility
/3fg RI= y / h + tz"
(24)
In the uninfected situation we have w 0 = 0,w I = 0, hence s0 = (1 - p ) t z - Izs0, $1 = ~ ( 1
- -
a)p
(25)
-- I~S 1.
Thus the solutions approach a stationary point ( S 0 , g l ) w i t h go = ( 1 P), gl = ( 1 - a)p, and the full system has a stationary point (go, ~1,0,0). Next we observe that whenever w o > 0 or w a > 0 is introduced into the system then the variables w 0 and w I both become positive. Hence we look for a stationary point with all components positive. Assume that w = w 0 + gw 1 is known. Then the four variables So, sl, Wo,W1 satisfy a linear system
( /3w+ l z ) S o = ( 1 - p ) t z , ( ~Sfw + p.)s 1 = (1 - a)pl*, /3WSo = (~ + g)Wo, /3fwsl = ( ~,/ h + ~ ) w l ,
(26)
MODELS FOR THE ERADICATION OF POLIOMYELITIS
157
from which s°=
(1- p)/x /3w+~ ~
'
/3w ×
w0= 3,+/,
Wl
=
/3fw
3"/h + t*
S1 ~---
( 1 - a)pl*
(27a)
/3fw + t.
(1- p)tt /3w + / x × (1 - a)pl*
-/3fiv'-~7 "
(2719)
We insert these expressions into w = w 0 + gw x, divide by w and obtain the condition
,I,(w)
( 1 - p)/z + g /3f ( 1 - a)p/z /3 × -flw+ ~ 3"/h+ ~ × /3fw + =1. (28) 3' +---~
This is a scalar equation for the variable w. The quantity ~ ( w ) is positive for w >/0 and it decreases to zero for increasing w, and
,~(o)
/3 ( l - p ) + 3" + ~
g/3f
3"/h +
×(1-a)p.
(29)
Hence a solution w > 0 of the equation ~ ( w ) --- 1 exists if and only if • (0) > 1, i.e., if
[3 g/3f × ( 1 - a ) p > l , 3' + ~ ( 1 - P ) + . 3 " / h + Ix or, equivalently Rw(1 - p ) + RI(1 - a)p > 1.
(30)
Suppose the inequality (30) is satisfied and w is the unique solution of ~ ( w ) = 1. Then s 0, s 1, w 0, and w I can be obtained from (27). These quantities satisfy the linear system (26) and thus are the components of the stationary solution. Hence we have shown the following:
PROPOSITION 3 Using IPV vaccination, the reproduction number under vaccination coverage p is given by R(p) = R w - ( R w - ( 1 There are several cases to be examined.
a)R1)p.
(31)
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M. EICHNER AND K. P. HADELER
In this case the disease cannot spread in a totally susceptible (nonvaccinated) population. Vaccination is essentially unnecessary. C a s e 1: R w < 1.
a)R r In this case the disease can spread in the susceptible population but it cannot spread in a completely vaccinated population. There is a critical vaccination level p* such that R ( p * ) = 1, C a s e 2: R w > 1 ( 1 -
p* =
R w -1
Rw-(1-a)R1
,
(32)
so the disease cannot spread for p > p*. C a s e 3 : ( 1 - a ) g I > 1. T h e infection can spread in a totally vaccinated population. Vaccination cannot cause elimination in this case (although it reduces prevalence of the infection). The Jacobian of the system (22) is (recall w = w 0 + gw 1) -
~w
-
o
j=
tx
0
-
- t3wf-
~
¢lSo
-
- 13fsl
flw
0
flSo - 3' - tx
0
flwf
firs I
¢igSo
- t3~s1 flgSo flgfs 1 - y/h
J "
- I~
(33)
For w 0 = wa = 0 the lower left 2 × 2 block vanishes and the upper left block is - / x I . Hence stability of the uninfected solution is governed by the lower right block Bso - 3' - #
¢~gso
~fsl
t
~gfs~ - ~, / h - ~z ]"
(34)
The trace is 7" = ( 13So - , / -
~ ) + (/3gfs~ - ~ , / h - t~).
(35)
In view of (31), the determinant is, up to a positive factor, equal to the quantity D = 1 - soR,~ - s i r ~
= 1-(1-p)R = 1 -
R(p).
w -(1-a)pR
1
(36)
MODELS FOR THE ERADICATION OF POLIOMYELITIS
159
The determinant is positive if and only if R(p) < 1. It remains to check the trace. We have to show that D > 0 implies T < 0. But this is obvious: If the sum of the two quotients in (36) is less than 1, then each of them is less than 1, and thus the trace is negative. Thus we have the following:
PROPOSITION4 If the reproduction number R(p) (as given by (31)) is less than 1 then the uninfected state is stable. DISCUSSION According to (15) and (31) the critical vaccination coverage for OPV and IPV, respectively, is
1
1
Rw
(37)
Both expressions contain the factor ( 1 - 1 / R w ) . The expression for OPV contains the factor (1 - R v / R w) which is less than 1, whereas the expression for IPV contains the factor R w / ( R w - ( 1 - a ) R 1) which is always greater than 1. Hence the inequality P~Pv ~
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M. EICHNER AND K. P. HADELER
with other enteroviruses [20, 24, 43], or maternal antibodies that are transmitted by breast feeding [10, 25, 27, 28, 38]. Because of the lower take rate of OPV in tropical countries, vaccinated individuals are more likely to be protected against paralytic polyomyelitis after IPV vaccination. For individuals who live in a developing country, IPV is therefore the better choice. The community, on the other hand, will often decide to use OPV to reach a high level of herd immunity and to minimize the total number of paralytic cases. The decision of the community may also be influenced by the price of the vaccine and the ease of its application. IPV is more expensive and must be injected, whereas OPV is cheaper and easier to administer [30]. The oral application of OPV on sugar cubes might additionally increase the general acceptance by the population. For tropical countries we have to take the different take rates t o w , ttp v of the vaccines into account, in particular the low take rate of OPV. Hence the expressions (37) should be replaced by
1
) Rw
(38)
For small take rates the critical vaccination coverages can become greater than 1. The WHO advocates vaccination of all infants at the ages 6, 10, and 14 weeks. In countries with high transmission an additional vaccination shortly after birth is recommended [40]. The consequences of multiple vaccination are different for OPV and IPV. Successful OPV vaccination leads to full immunity and multiple vaccination increases the fraction of protected individuals. If each vaccination has a 60% chance being successful, only 6.4% remain unprotected after three vaccinations. Successful IPV vaccination, however, leads to a varying degree of protection. Multiple vaccinations reduce the number of nonresponders and improve the level of protection of individuals who have been "successfully" vaccinated previously. However, it is difficult to determine the extent to which each booster vaccination reduces the susceptibility and (after infection) the infectivity. The following parameter values are chosen for a developing country: The mean life expectancy is 45 years ( / z - - 1 / 4 5 per year). The basic reproduction number of the wild polio virus infection is estimated as R w = 12 and the average duration of infectivity is one month (Yw = 12 per year) which leads to an estimate flw = 144. The basic reproduction
MODELS FOR THE ERADICATION OF POLIOMYELITIS
161
number of the OPV infection is assumed to be only one quarter of that of the wild virus infection (Rv = 3). It is further assumed that 30% become completely immune (a = 0.3) after IPV vaccination while the others are only partly protected. The probability that a contact leads to infection of a partially protected individual is reduced to 50% ( f = 0.5). All infected individuals become fully infective (g = 1). For partially protected individuals the duration of the infectious period is reduced to 20% (h = 0.2). These assumptions lead to an estimate R 1 = 1.2. For the parameter set chosen above we get 1 × 0.69, P~Pv = -toPv
1 - × 0.98. P~'PV= -qPv
Empirical estimates for the take rate of IPV range between 0.9 and 1.0. Thus our estimate for P]~PV indicates that IPV vaccination, if it is successful at all in eliminating the disease in a developing country, requires a coverage close to 100%. On the other hand OPV, which looks more promising on the northern hemisphere, is competitive only if the take rate is not below 0.7. Otherwise there is a definite advantage of IPV. It m u s t be underlined, however, that the models are rather crude and the quantitative assumptions are rough estimates. We add a remark on vaccination in a growing population. The results have been obtained for a demographically stable population. In general, one cannot extend results directly to populations that are subject to demographic change, e.g., growing populations. One has to set up a demographic transmission model, e.g., in the form of a homogeneous system, and study in detail its qualitative behavior. In the present situation, in the absence of differential mortality and fertility, there is a direct approach. If we introduce a birth rate b into the models (in general one will assume b >/~), and the total population size is P, then the models (1) and (22) assume the form
= b(1-p)P-
~wsw/P-
b = bpP + ~ v S V / P -
Us,
Tvv - / z v ,
= [3~,sw/P - Tww - / z w ,
= T~w + Tvv - / ~ r .
~vsv/P-
(1')
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M. EICHNER AND K. P. HADELER
and = b(1 - p ) e -
Wo + g w , ) s o / e - USo,
sl = b(1 - a) p P - ~ ( w o + gwl ) fs 1 / P - tZSl, =
(Wo + g w l ) s o / P
+ u)wo,
(22')
Both systems are of the general form Yc= b P d + F ( x ) - tzx,
(39)
where the variable x is a vector in E~ ,n = 4 or 5, respectively, and P = erx, with e T= (1 ..... 1) ~ A n, is a scalar function. The vector d is nonnegative with erd = 1; it describes the distribution of offspring and the nonlinear function F, modeling transmission and recovery, is homogeneous of degree 1, i.e., F ( a x ) = a F ( x ) for ~ > 0, and has the property eT"F(x)- O. For the relative proportions y = x / P = x/(eT"x) one can derive a similar system .~ = b d + F ( y ) - by.
(40)
(Use .~ = (YcP- I ~ ) / P 2 and replace .~ from (39), 16 = ( b - / x ) P . ) Thus the proportions satisfy the same system as in the stationary case with /z replaced by b in (1) or (22), respectively. Of course, the exponent of demographic growth is b - / z . All formulae so far derived remain valid (with /x replaced by b), in particular the formulae for the critical vaccination rate. The results have to be appropriately interpreted, though. The finding that vaccination beyond the critical level stabilizes the uninfected situation in proportions does not exclude an increase of the absolute number of infected cases. It just says that the rate of increase of the number of infected cases is less than the rate of population growth. The discussion of absolute numbers requires more subtle arguments. M. E. thanks Evangelisches Studentenwerk Haus Villigst for a grant supporting his Ph.D. work and Klaus Dietz for advice and encouragement. Both authors thank the Issac Newton Institute (Cambridge) for invitations during the Epidemiology program 1993.
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163
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