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The population dynamics of two vertically transmitted infections
THEORETICAL
POPULATION
33, 181-198 (1988)
BIOLOGY
The Population Dynamics of Two Vertically Transmitted Infections STAVROS BUSENBERG* Department of Mathematics,
Harvey Mudd College, Claremont, California
92711
AND KENNETH Department of Mathematics,
L. COOKE+
Pomona College, Claremont, California
91711
Received September 16, 1986
The transmission of Keystone virus in the mosquito Aedes atlanticus and of rickettsii in the tick Dermacentor andersoni is modeled and analyzed. Both of these infections can be transmitted vertically from an infective parent to newborn offspring as well as horizontally via direct or indirect contacts with infected individuals. The vertical transmission mechanism plays a major role in the maintenance of these infections and its effects are analyzed in detail. This same mechanism can act as a means for controlling the size of the infected host population and an analysis of this effect is also provided. The sensitivity of the threshold parameters and the endemic prevalence rates of the disease to variations in the basic infection transmission components are investigated, The transmission components that are considered include the ability to transmit the pathogen vertically as well as horizontally, the size of the host population, and survival probabilities of the hosts. 0 1988 Academic press, hc. Rickettsia
1.
INTRODUCTION
Rickettsia rickettsii causes Rocky Mountain spotted fever and is transmitted to humans and other mammals by a vector population which acts as the host for this pathogen. This pathogen is vertically transmitted in the host population; that is, it is passed on from infective parents to their newborn offspring, and this mode of transmission plays a major role in the maintenance of the disease it causes.Keystone virus is an infection that is also vertically transmitted in its host. The mechanism of vertical trans* Partially supported by NSF Grant DMS 8507193. 7 Partially supported by NSF Grant DMS 8603450.
181 0040-5809/88$3.00 Copyright 0 1988 by Academic Press. Inc. All rights of reproduction III any form reserved.
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mission occurs commonly and is discussed in Fine (1975) where many examples of vertically transmitted diseasesare given. Mathematical models of such diseaseshave been formulated and analyzed by Fine and Le Due (1978) Busenberg and Cooke (1982), Cooke and Busenberg (1982), Busenberg et al. (1983), Regniere (1984), and El Doma (1987). Anderson and May (1981) give an extensive discussion of the role that vertical transmission can play in the transmission of infections within invertebrate populations. We present and analyze a model for the transmission of Keystone virus and one for the transmission of R. rickettsii. Our aim is to emphasize the role of vertical transmission in maintaining these infections and its effect on the size of the vector population. In the case of Keystone virus we show that the diseasedisappears in the absence of a high enough rate of vertical transmission. We also give a quantitative analysis of the effect of changes in the number of gonotrophic cycles of the mosquito hosts and in the horizontal transmission rate of the infection. In the case of R. rickettsii we also study the effect that vertical transmission, through several generations, can have on the population size of the ticks. This effect occurs becauseeach successivegeneration in a vertically infected lineage of ticks may start life with a larger load of pathogen than did the parent generation (Burgdorfer and Brinton, 1975). After a certain number of such successive vertically infected generations, the pathogen load is large enough to prevent successful oviposition and essentially causes infertility in such ticks. Finally, the age-structure of the host population plays a significant role in the population dynamics of vertically transmitted diseasesas has been already shown by us (Busenberg and Cooke, 1982; Busenberg et al., 1983; Cooke and Busenberg, 1982) and by El Doma (1987). This fact will again appear in both of the models that we discuss here even though we use only discrete dynamic equations in these models. 2. A MODEL FOR THE TRANSMISSION OF KEYSTONE VIRUS A model for the transmission of Keystone virus in the mosquito Aedes atlanticus was suggested by Fine and Le Due (1978) on the basis of field observations and was reformulated as a dynamic system and partially analyzed by Cooke and Busenberg (1982). Here we present a special case of this dynamic model which we have been able to analyze completely. Recall that mosquitoes need a blood meal from a vertebrate host prior to oviposition and that there are several such gonotrophic cycles (blood meals followed by oviposition) in a given season. The mosquitoes overwinter in the egg stage, so vertical transmission is a primary mechanism for the perpetuation of the infection. Since emergence and oviposition by the
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mosquitoes are roughly synchronized, a discrete time formulation of the dynamics of this infection is possible. We use the following notation which is in agreement with our previous work (Cooke and Busenberg, 1982): B, = prevalence rate of the virus among female mosquitoes that oviposit in the kth year, 0 < Bk < 1. (Thus, Bk is the fraction
of eggs laid in the kth year that carries the virus.) d = maternal vertical transmission rate, 0 < d c 1.
i = proportion of vertically infected mosquitoes that are infectious to vertebrates, 0 < i < 1. n = average number of mosquitoes per vertebrate host at the time of the first blood meal, n > 1. p = proportion of female mosquitoes surviving through one gonotrophic cycle, 0 < p < 1. r = number of gonotrophic cycles in 1 year, r > 1. f= probability that a susceptible mosquito which feeds upon a viremic (infected) vertebrate will become infective, 0
(2.1)
where R =fs(p + .** +p’-I)/(1
+p+
... +pr-‘)=fps(l-p’-‘)/(1-p’),
(2.2)
with r > 1. R is the net horizontal transmission rate in the mosquito population mediated by the vertebrate hosts on which they feed. For this model we have the following conclusions based on Theorem 4.1 which is stated and proved in the Appendix. Define the threshold parameter T = d(inR + 1), with R defined in (2.2). Then, if T-C 1, as time increases, the prevalence rate B, tends to zero and the proportion of infective mosquitoes tends to zero. If T> 1, the prevalence rate
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B, tends to the endemic level Bz which can be computed as shown in Theorem 4.1.
First, note that in the absence of vertical transmission (d= 0), we always have T= 0, hence, the disease dies out in this case. Second, in the absence of horizontal transmission (f= 0) the rate R is equal to 0, and T= d< 1, and again the disease dies out. So, the maintenance of this infection requires a coupling of the horizontal and vertical transmission mechanisms. Finally, T can be easily shown to be an increasing function of r, the number of gonotrophic cycles, and varies from the value T=d(infip/(l+p)+ 1) when r=2 to T=d(infsp+ 1) as r+oo. Consequently, the likelihood of the disease reaching an endemic level increases with the number of gonotrophic cycles. This is illustrated in the following graph of T versus r and d where we have taken f = p = s = 0.5, i = 0.8, and n=lOoO. The above mathematical conclusions based on the dynamic model (2.1) can be heuristically explained as follows. Because of vertical transmission, the mosquito population starts each season with a positive proportion of infectives. Since only infective mosquitoes can have infective offspring, and since the probability d of vertical transmission is less than one, the disease would eventuallly die out if this were the sole mechanism for its transmission. However, because the population goes through several gonotrophic cycles (r 2 2), it initiates the mechanism of horizontal transmission by infecting vertebrate hosts who, in turn, become the source 1.6-
d = 0.004
0.
2
I
4 Number
of
gonotropbic
6 cyclea
FIG. 2.1. Variation of the threshold parameter T with the number of gonotrophic cycles r and the vertical transmission rate d.
VERTICALLY TRANSMITTED INFECTIONS
185
of new infections in the mosquito population. The process of horizontal transmission is enhanced by an increase in the number of gonotrophic cycles, hence, by any prolongation of the period of fertility of the mosquitoes. It is the close coupling of all these mechanisms that makes an endemic level of the disease possible. The threshold parameter T = d[ in&( ( 1 -p’ - ’ )/( 1 -p’) + 1] is easily seen to be an increasing function of each of the basic epidemiological parameters. Its variation with respect to all of these parameters except p (the survival probability) and r (the number of gonotrophic cycles) is linear and is largest with respect to d, the maternal vertical transmission rate. T varies in a sublinear way with respect to p from the value d at p = 0 to the value d[infi(r - 1)/r + 1] when p = 1. With respect to r, T again increases in a sublinear way with value d[infp/( 1 +p) + 1] at r = 2 and reaching the value d[infsp + l] as r -+ co. The sensitivity of the prevalence rate B*, when T> 1, to variations in the epidemiological parameters is more dihicult to see analytically since Eq. (4.1) canot be explicitly solved for B* in the most interesting cases
FIG. 2.2. (a) Variation of the prevalence rate with time when T-c 1. the prevalence rate with time when Ts 1.
(b) Variation of
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when n is large. However, it is quite easy to determine numerically and in the previous two graphs we show the variation of the prevalence rate with time for two different values of r(r = 2 and Y= 10) in two cases,one with T< 1 (Bk -+ 0) and one with T> 1 (B, + B* > 0). 3. A MODEL FOR THE TRANSMISSION OF Rickettsia rickettsii Rickettsia rickettsii causes Rocky Mountain spotted fever (RMSF) in human and other vertebrates and ticks are the primary vectors that transmit this disease. There is a significant mortality in humans infected with RMSF and consequently there have been extensive field and laboratory studies (Burgdorfer and Brinton, 1975; Burgdorfer, 1975; Garvie et al., 1978) of the transmission of this disease in the tick population. Elsewhere (Busenberg and Cooke, 1982) we have modeled and analyzed the coupling between the vertical and horizontal modes of transmission of this disease. Here we present a model of one of the most striking characteristics of this type of infection, a characteristic shared by a number of other vertically transmitted diseases,namely, the cumulative effect of the pathogen as it is vertically passeddown through several generations. This effect occurs when the pathogen load that is vertically transmitted by an infected parent increases as the pathogen load of the parent increases.Infected offspring of a parent with a large pathogen load commence life with a relatively large load and may reach their reproductive age with a pathogen load that is larger than that of their infected parent. Consequently, the pathogen load may increase monotonically in each successive generation of an infective lineage. In the case of R. rickettsii in the tick Dermacentor andersoni, it has been found (Burgdorfer and Brinton, 1975) that, in certain circumstances, after several generations in an infective lineage, the pathogen load at birth may reach a high enough level to impede successfuloviposition. As a result, vertical transmission can act as a means for controlling population size. This is the aspect of the transmission of this disease that we model here. The following assumptions are based on field and laboratory data (Burgdorfer and Brinton, 1975): (i) The Rickettsia are vertically transmitted in the tick population with transmission rate of 10 to 100%. (ii) The level of the Rickettsia load increases with each successive generation of vertically acquired infectivity. (iii) Female ticks of the fifth generation in a lineage of vertically acquired infection carry a large enough load of Rickettsia to be unable to oviposit successfully and, hence, are sterile. We use two independent discrete variables to model this situation. The
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first is the cohort number starting at some arbitrary time origin, and the second is an internal variable indicating the generation number in an infected lineage. The second variable takes on the values (0, 1, .... S}, where 0 denotes ticks that acquired the disease via horizontal transmission, 1 denotes those that acquired the diseasevertically from a parent of infective lineage 0, and so on, with 5 denoting ticks that are vertically infected by a parent of lineage 4. The notation used in this model is n = cohort number, n > 0. o!= the generation number in an infective lineage of vertical transmission, 0 < a < 5. i,(a) = the number of infective ticks of the nth cohort that are of the a th infective lineage. S, = the number of susceptible ticks in the n th cohort. b = the birth rate of female ticks adjusted for the proportion of female to male ticks in the population. q = the probability of an infected female tick of lineage a = 0, 1, ...) 4, to vertically transmit the infection to its offspring, O
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and the mathematics. We have chosen to present this simpler case since it illustrates the same basic phenomena. In a discrete model, horizontal transmission which occurs over an extended time interval needs to be idealized as occurring at a synchronized specific instant of time. Here we take this instant to be the end of the active blood meal-seeking period of the season. Since we are following cohort groups which appear at the beginning of a particular season,we distinguish between those ticks that are born with the infection and those that acquire it via horizontal transmission. We shall assume that the horizontally infective ticks in the n th cohort come from susceptible ticks of the n th cohort that are infected via contact with infective ticks of the (n - 1)th cohort, and not from the new infectives of their own cohort. This assumption allows for the maturation period of new born infectives before they take their first blood meal, and also takes into account the time lapse that is necessary before the vertically acquired pathogen in newborn infectives is large enough to be effectively transmitted via horizontal contacts. The force of horizontal infection on the n th cohort is given by a funtion
f(s,J-,)=f(s.~ i L(a))
(3.1)
LX=0
which can take a variety of forms that are mathematically described by hypothesis (H) in the Appendix. Here we consider two forms off that we will discuss in some detail and which are representative of the general forms of the force of infection that are analyzed in the Appendix. These special forms are the “mass-action” form f(xvZn-,)=QJn-1
(3.2a)
and the “saturated mass-action” form
f(s”,r,~,)=kS,z”-,/(k’+z,-,).
(3.2b)
Note that the ratio f(x, y)/x has generic graphs in these two cases. Figure 3.1 is the graph of f(x, v)/x for f given by (3.2a) and (3.2b). We shall be able to derive results on general forms off which essentially behave like (3.2a) or (3.2b) and use the two definitions (H,) and (Hb), respectively, in the Appendix for these general forms. The mass-action form (3.2a) of the force of horizontal infection f is very familiar and is based on the hypothesis of uniform mixing, hence random contacts, between infectives and susceptibles. The saturation form (3.2b) has the property that f saturates at the value kS, as I,- i + co. This reflects the possibility of susceptible ticks engorging on a non-viremic vertebrate host even when the number of infective ticks is large. This is an important
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mass-action
. 0
FIG.
Y
3.1. Generic graphs of the mass-action and the saturated mass-action nonlinearities.
consideration because a substantial proportion of the offspring of infective ticks may be susceptible, and the intense viremia in the vertebrate hosts from which these susceptible ticks may be horizontally infected lasts only a few days. Hence there is a definite possibility of having susceptible ticks feed on non-viremic vertebrates and thus avoiding horizontal infection. Based on the above assumption, we have the following dynamic model. i,(O)=cf~,,Zn-1) i,,(a) = cbqi,- ,(a - l), S,=cb
p i
a = 1, 2, ...) 5
i,-,(a)+S,-,
(3.3)
- cfaz, zn- 1).
a=0
Even though we use exactly five vertically infected generations, all of the results that we shall prove and describe hold true when 5 is replaced by any natural number Na 2 in the above model equations. In particular, note that the expression for S, has no term of the form cbpi” _ 1(5) since we have assumed that the ticks of infective lineage 5 are not fertile. For this model we have two sets of conclusions, the first of which is based on Theorem 4.2 in the Appendix and gives explicit threshold conditions for the feasibility of an endemic level of the infection. Here we suppose that the force of infection f is of either of the two forms (3.2a) or (3.2b), or more generally satisfies hypothesis (H) in the appendix. Zf the net reproduction rate cb < 1 then the total population tends to zero with time and the zero population level is stable when cb < 1 and unstable when cb > 1. Moreover, a positive endemic equilibrium can exist only if 1~ cb < (1 - qbc)/p( 1 - (qbc)‘) = l/R,.
(3.4)
As we shall see below, the exact range of parameters where an endemic
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equilibrium is feasible and stable depends on the forms of the force of infection; however, in all casesthe restriction imposed by (3.4) must apply. Now note that in the absence of vertical transmission (q= 0,p = 1) condition (3.4) cannot hold if cb # 1 and a non-trivial feasible solution does not exist. In fact, in this case, i,(u) = 0 for n > 1, a = 1, 2, .... 5; and S,+I,=P,=cbP,_,,
hence, P, -+ 0 if cb < 1 and P, + co when cb > 1. So, the tick population either decays to zero or becomes unbounded. If cb = 1 and q = 0, then i(a) = 0 for cf= 1, .... 5 and i(O)= cf(S, I(O)). Also, R, = 1, and it is shown below that S=k,, t(0) = k, is a steady state solution for any constants k, and k2 for which k2 = cf(k,, k2). In the absence of horizontal transmission (f=O), i,(a)=0 for n> 1, a=O, l,..., 5, and P, = S, = cbS,- 1 for n > 2. Again we have P,+O if cbl. If cb=l and f= 0, the population remains constant, with no infectives for n > 1. Hence, a balanced population can exist only through the effects of the coupling of vertical and horizontal transmission except in case cb = 1 and either q = 0 or f - 0. This is so because this model assumesthat there are no environmental constraints to the growth of the tick population. The parameter ranges, where such a control of the population size due to disease transmission occurs, depend on the form off as seenby the following result which is based on Theorem 4.3 of the Appendix.
Zf the force of infection f has the mass-action form (3.2a) (or more generally, obeys hypothesis (H,) in the Appendix) then an endemic equilibrium exists when and only when the threshold conditions in (3.4) hold. However, tf f has the saturated mass-action form (3.2b) then a unique endemic equilibrium exists when and only when the threshold conditions in both (3.4) and k>
(l-cb)
c( cbR, - 1) are satisfied. A similar condition applies for the more general forms of the force of infection which satisfy condition (Hb) in the Appendix.
The determination of the stability of the endemic state when it is feasible is difficult and we have treated this aspect of the problem only numerically. To this end we have used two independent methods for numerically studying the stability of the endemic state on a computer for several parameter ranges. These results have shown that the endemic level is stable whenever it is feasible when the force of infection takes the form (3.2b); while for the form (3.2a) there is a parameter region where this equilibrium is feasible and unstable. In this case, we have performed extensive numerical
VERTICALLY
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INFECTIONS
experiments which all show that a stable oscillatory solution exists and the subpopulations undergo bounded oscillatory changes which are quite complicated even though they are periodic. Finally, when cb exceedsthe upper threshold bound in (3.4) the total population grows without bound as time increases. This is the parameter region where the net reproduction rate is large enough to dominate the population-controlling effect due to the reduced fertility caused by the vertical transmission of the disease. Figures 3.2a and b show the dependence of the stability of the zero and the non-trivial steady states on the parameters b, c, and p obtained from the theorem and by numerically computing the eigenvalues of the linearized system about the non-trivial equilibrium forfgiven by (3.2a) and (3.2b). In the region where the zero equilibrium is unstable and the non-trivial
.075 t iI
nontrivial
,025
not
equilibrium feasible
not
equilibrium feasible
..
bc
nontrivial
.75
t a
zero equilibrium stable
.5 I
f .25t
0.
.S
1.
1.1
1.2
bc
FIG. 3.2. (a) Bifurcation boundaries and stability regions for the mass-action nonlinearity (plot independent of c). (b) Bifurcation boundaries and stability regions for the saturated mass-action nonlinearity f= ksi/(k’ + i).
f=ksi
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equilibrium is not feasible the population becomes unbounded. From this numerical computation it is clear that the region in the p vs bc plane where the disease controls the size of the tick population becomes larger as c increases.The boundary of this region, for fixed c, is monotone decreasing as a function of b and tends to infinity when bc tends to one. 4. APPENDIX: MATHEMATICAL ANALYSIS In this section we give the statements and proofs of the theorems which justify the conclusions presented in the previous sections. THEOREM 4.1. Let T = d(inR + 1) where R is given by (2.2). Then $ T < 1, B, + 0 as k -+ co and the proportion of infective mosquitoes tends to zero. If T > 1, Bk + B* > 0 as k -+ 00, where B* is the unique positive B satisfying
(l-d)B=R(l-dB)[l-(l-diB)“].
(4.1)
Hence, if T> 1, the prevalence rate tends to a constant endemic level as k increases. Proof of Theorem 4.1. We first consider the steady state solutions of (2.1) which must satisfy Eq. (4.1). Clearly, B = 0 is one such steady state, while if B # 0, the left hand side of (4.1), when viewed as a function of B, is a straight line with slope 1 - d. The right hand side of (4.1) has first derivative equal to dinR at B=O, its second derivative is negative when 0 < B c 1, and its value at B = 1 is less than 1 - d. So, there is a unique solution to (4.1): B= B* E [0, l] if and only if dinR> 1 -d. That is, the non-trivial feasible equilibrium exists if and only if T> 1. Next, writing (2.1) in the form B, = F(B,- i), we first note that, if 0 < B < 1, then 0 < F(B) < 1, so the dynamic system (2.1) maps the interval [0, l] onto itself and is well-posed. Computing the derivative of F we get F’(B)=d(l-R)+Rd(l-diB)“-‘[ni(l-dB)+(l-diB)”],
(4.2)
hence, F’(B) > 0 and F”(B) < 0 for BE [0, 11. Now, F(0) = T and we have the two situations depicted in Fig. 4.1. Because of the convexity of F and the fact that it is monotone increasing on [0, 11, it can be easily seen that, if Tl, B,-+B* as k-+co. The formal proof is as follows. When T < 1, expand F(B,- i) in a Taylor series about 0 to get from (4.2) (recall F(0) = 0, F’(0) = T) B,=TBk-,+F”(x)B:-,/2,
O
Since F’(x) < 0, we get 0 < Bk < TB,- , < B,- , , so by iterating this
VERTICALLY TRANSMITTED INFECTIONS
Bk-l
193
1 B
Y
0 B
FIG. 4.1. (a) Plot of F vs B and the iteration scheme for T< 1. (b) Plot of F vs B and the iteration scheme for T> 1.
process: 0 < B, < TkBo+ 0 as k + co, and the proof is complete in this case. When T> 1, we consider the two situations where 0 1, the monotonicity and convexity of F imply that B < F(B) < F(B*). Hence, since Bo E (0, B*), Bol and the sequence {Bk}pcO is monotone increasing and bounded from above by B*. So, {B,),“_,, must converge, say B, + B' < B* as k + co. Since F is continuous, we get B' = hmk _ o. B, + , = hmk _ m F( Bk) = F(B') and, hence, B’= B* because the positive steady state B* is unique. This completes the proof for the case T> 1 and of Theorem 4.1.
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Theorems 4.2 and 4.3 require certain hypotheses on the form of the force of infection f which we collect below. (H). f is a monotone increasing function of S, and Z,,~ 1, with f(x, y ) = 0 if either x=0 or y=O and f(x,y)>O if x>O and y>O. f is also continuously differentiable and
Two particular forms of f that we shall consider in Theorem 4.3 generalize the mass-action and the saturated mass-action forms and are respectively defined via the following hypotheses: (H,). f satisfies hypothesis (H) and f (x, ax)/x is a convex function of x for every constant a > 0, and (H,,). f satisfies hypothesis (H) and f(x, ax)/x function of x for every constant a > 0.
is a concaue monotone
Note that, iff (x, ax)/x = g,(x) has a second derivative g:(x) with respect to x, then condition (H,) requires g:(x) 20 for all a >O and (Hb) requires g:(x) < 0 for all a 2 0.
As they stand, Eqs. (3.3) are not an explicit dynamic system expressing the variables at time n in terms of their values at (n - 1) because of the presence of the term S, in the force of infection. However, from the last equation in (3.3) we get S,+cf(S,,I,-,)=cb
[
p i
1
i,-,(cr)+S,-,
a=0
(4.3)
which, because of hypothesis (H), has a unique non-negative solution whenever i,- ,(a) and S,_ 1 are non-negative: S,=G
(
Znel,SSn-,,
i
i,-,(a)
Cr=O
>
.
(4.4)
Using (3.3) and (4.4) we obtain the explicit dynamic system: i,(O)=cf
((
G InPl,Ll,
i
Ll(a)
Or=0
a = 1, 2, ...) 5
l),
i,(a)=cbqi,_,(cr-
L,,
>
i Cl=0
indl(a) . >
,I,-,
>
(4.5)
VERTICALLY TRANSMITTED INFECTIONS
195
The function G depends on the specific form of J however, it is always non-negative and G(0, 0,O) = 0. For the particular form off given by (3.2b) we get the explicit dynamic system (4.5) with G given by
G=Ch(k’+I,-,)Cp~~=,i”-,(cr)+S,-,l k’+(l
(4.6)
+ck)Z,-,
The following result holds for the system (4.5) under hypothesis (H): THEOREM 4.2. Suppose that f satisfies hypothesis (H). Then the system (4.5) preserves positivity in the sense that non-negative initial data &(a) > 0, So 2 0 yield non-negative solutions. Moreover, if the net reproduction rate cb < 1, zero is the only feasible (non-negative) steady state of (4.5) and it is stable and attracts all non-negative solutions tf and only tf cb < 1; and it is unstable ifcb > 1. A non-trivialfeasible equilibrium solution to (4.5) can exist only if
1 d cb d (1 - qbc)/p( 1 - (qbc)‘).
(4.7)
(This is only a necessary but not a sufficient condition.)
The proof of this result is given at the end of this section. The non-trivial steady state which may exist under condition (4.7) can be easily seen to satisfy t(o)J’-W~ cbR,-
1’
a = 1, 2, .... 5,
I(a) = (cbq)’ t(O),
where
s,
1 - (cbq)6 l-cbq
1 - cb S .cbR,- 1 >
with
R =P(’ - (qbC)5) P l-qbc ’
(4.8) Note that, by hypothesis (H,), the equation for 3 in (4.8) has a unique positive solution whenever (1 - cb)/(cbR, - 1) > 0. However, the other components of the steady state given by (4.8), as we shall show below, are non-negative if and only if (4.7) holds. THEOREM 4.3. Zf f satisfies hypothesis (H,) then the conclusions of Theorem 4.2 hold and a non-trivial feasible equilibrium exists if and only tf (4.7) holds. Iff satisfies (H,,) then the conclusions of Theorem 4.2 hold and
(i) rf lim,, mf (x, Ax)/x = 00, where A = (1 - (cbq)6)/( 1 - cbq) . ( 1 - cb)/(cbR, - 1), then a unique non-trivial equilibrium exists if and only if (4.7) holds with the strict inequalities.
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(ii) rf lim,, o.f(x, 2x)/x= K< Co, then a unique non-triuial equilibrium exists if and only if (4.7) with strict inequalities and K>(l
1)
-cb)/c(cb&-
(4.9)
both hold.
Condition (4.9) can also be written in the form c’“:_“:; 1)-g (0,O) + c & (0,O) > 1. ” n 1
(4.10)
We now give the proofs of Theorems 4.2 and 4.3. Proof of Theorem 4.2. The fact that system (4.5) preserves positivity follows from the non-negativity of G in (4.4). Now, if cb < 1, upon adding the first and last equation in (3.3) we get i,(O)+S,=cb
(
p f
i,-,(a)+S,-,
)
LX=0
(4.11)
so (4.5) cannot have a steady state with either S > 0 or i(0) > 0. But, if i(0) = 0, the second equation in (4.5) implies i(u) = 0 for c1= 1,2, .... 5 and (4.5) has only the zero steady state when cb < 1. In order to prove the stability of the trivial steady-state, we note that (4.11) yields the relation i,(O) + S, < cb[i,-
l(O) + S,- ,] Q (cb)“[i,(O)
+ So],
hence, if cb < 1, i,(O) + 0 and S, --f 0 as n + co for any initial data io(0), So. Since i,(O) + 0, the second equation in (4.5) implies i,(a) + 0 as n + co for a = 1, 2, ...) 5 and the zero steady state attracts all non-negative solutions when cb< 1. Next, when cb > 1 we use hypothesis (H) to linearize (4.5) about the zero equilibrium and obtain the linear system
i,(O iA1 L(5 s,
0
0
.
.
-00
cbq 0
0 cbq
. .
. ’
. .
0
0
0
0
cbq
0
0
cbp
cbp
.
.
cbp
0
cb
0 .
0 0
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197
One of the eigenvalues of this lower triangular matrix is cb > 1, and the others are all zero, so the trivial equilibrium is unstable in this case. It remains to establish condition (4.7). From the expression for i(0) in (4.8) we easily see that a non-trivial equilibrium with all subpopulations positive can exist only if (1 - cb)/(cbR, - 1) > 0, hence, only if either (a) 1
t&J51 = bcp - bcp(qbc)5 < , bcp+ 1 -bc
1 -qbc
whenever bc< 1. So, the strict inequality in (4.7) must hold - -in this case. If cb = 1 and cbR, < 1, then i(a) = 0, a = 0, 1, .... 5, hence,f(s, I) =f(S, 0) = 0 and any constant can be an equilibrium value of s. If cb > 1 and cbR, = 1, then S must be zero, hence, i(0) = cf(0, Z) = 0, and there is no non-trivial steady state. Finally, if cb = 1 and cbR, = 1, then q = 0, p = 1, z(cx)= 0 for a = 1, 2, ...) 5, and (3.3) reduces to z(O)= cf(s, i(0)). Then S=ki, i(O)= k,, k,, k, any constants such that k, = cf(k,, k2), is a steady state solution. So, steady state solutions can exist only if (4.7) holds, and Theorem 4.2 is proved. Proof of Theorem 4.3. Since both (H,) and (H,,) include (H) we only need to prove the existence and uniqueness of the steady state. When either one of the inequalities in (4.7) is not strict, we have seen in the previous proof that the non-trivial solutions, when they exist, are not unique. Hence, we can limit our considerations to the case where the inequalities in (4.7) are strict. In this case, by (4.8) we need to consider the existence of a unique S > 0 satisfying
p/c=f(s,ns)/s,
(4.12)
where /I = (1 - cb)/(cbR, - 1) and I = (( 1 - (cbq)6)/( 1 - cbq))( 1 - cb)/ (cbR,- 1) ~0. From hypothesis (H) the right hand side of (4.12) tends to zero as s-0, and by (H,) as well as by (Hb)(i) it tends to co as S+ co, consequently, (4.12) has at least one positive solution for any A > 0. When (H,,)(ii) holds then it is clear that (4.12) can be satisfied by a finite S>O if and only if (4.9) holds. Uniqueness follows by the monotonicity of the right hand side of (4.12). Since by Theorem 4.2 we already know that (4.7) is necessaryfor a non-trivial solution to exist, it follows that the conditions in Theorem 4.3 are boh necessary and sufficient for a unique non-trivial solution to exist and the proof is finished.
653 37 2.6
198
BUSENBERG AND COOKE ACKNOWLEDGMENTS
We thank Keith Saints and Mark Overly for producing the graphs used in this paper. This paper was completed when the first named author (S.B.) was visiting the Centre for Mathematical Biology at the University of Oxford and we are grateful for the support of the Centre during the writing of this paper.
REFERENCES
ANDERSON,R. M., AND MAY, R. M. 1981. The population dynamics of microparasites and their invertebrate hosts, Philos. Trans. R. Sot. London Ser. B. 291, 451-524. BURGDORFER, W. 1975.Rocky Mountain spotted fever, in “Diseases Transmitted from Animal to Man” (W. Hubbert er al. Eds.), 6th ed., Chap. XXVI, Thomas, Springfield, II. BURGDORFER, W., AND BRINTON,L. P. 1975. Mechanisms of transovarial infection of spotted fever rickettsiae in ticks, Ann. N.Y. Acad. Sci. 266, 61-72. BUSENBERG, S., AND COOKE,K. L. 1982. Models of vertically transmitted diseases with sequential-continuous dynamics, in “Nonlinear Phenomena in Mathematical Sciences” (V. Lakshmikantham, Ed.), pp. 179-187, Academic Press, New York. BUSENBERG, S., COOKE,K. L., AND POZIO,M. A. 1983. Analysis of a model of a vertically transmitted disease,J. Math. Biol. 17, 305-329. COOKE,K. L., AND BUSENBERG,S. 1982. Vertically transmitted diseases, in “Nonlinear Phenomena in Mathematical Sciences” (V. Lakshmikantham, Ed.) pp. 189-197, Academic Press, New York. EL DOMA, M. 1987. Analysis of nonlinear integro-differential equations arising in agedependent epidemic models, Nonlinear Analysis TMA 11, 913-937. FINE, P. E. M. 1975Vectors and vertical transmissions: An epidemiologic perspective, Ann. N.Y. Acad. Sci. 266, 173-194. FINE, P. E. M., AND LEDUC, J. W. 1978. Towards a quantitative understanding of the epidemiology of Keystone virus in the Eastern United States, Amer. J. Trop. Med. Hyg. 27, 322-338.
GARVIE, M. B., MCKIEL, J. A., SONENSHINE, D. E., AND CAMPBELL,A. 1978. Seasonal dynamics of American dog tick, Dermacentor uariabifis (say), population in South Western Nova Scotia, Canad. J. Zoology 65, 28-39. REGNIERE, J. 1984. Vertical transmission of diseasesand pop&ion dynamics of insects with discrete generations, J. Theor. Biol. 107, 287-301.
POPULATION
33, 181-198 (1988)
BIOLOGY
The Population Dynamics of Two Vertically Transmitted Infections STAVROS BUSENBERG* Department of Mathematics,
Harvey Mudd College, Claremont, California
92711
AND KENNETH Department of Mathematics,
L. COOKE+
Pomona College, Claremont, California
91711
Received September 16, 1986
The transmission of Keystone virus in the mosquito Aedes atlanticus and of rickettsii in the tick Dermacentor andersoni is modeled and analyzed. Both of these infections can be transmitted vertically from an infective parent to newborn offspring as well as horizontally via direct or indirect contacts with infected individuals. The vertical transmission mechanism plays a major role in the maintenance of these infections and its effects are analyzed in detail. This same mechanism can act as a means for controlling the size of the infected host population and an analysis of this effect is also provided. The sensitivity of the threshold parameters and the endemic prevalence rates of the disease to variations in the basic infection transmission components are investigated, The transmission components that are considered include the ability to transmit the pathogen vertically as well as horizontally, the size of the host population, and survival probabilities of the hosts. 0 1988 Academic press, hc. Rickettsia
1.
INTRODUCTION
Rickettsia rickettsii causes Rocky Mountain spotted fever and is transmitted to humans and other mammals by a vector population which acts as the host for this pathogen. This pathogen is vertically transmitted in the host population; that is, it is passed on from infective parents to their newborn offspring, and this mode of transmission plays a major role in the maintenance of the disease it causes.Keystone virus is an infection that is also vertically transmitted in its host. The mechanism of vertical trans* Partially supported by NSF Grant DMS 8507193. 7 Partially supported by NSF Grant DMS 8603450.
181 0040-5809/88$3.00 Copyright 0 1988 by Academic Press. Inc. All rights of reproduction III any form reserved.
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mission occurs commonly and is discussed in Fine (1975) where many examples of vertically transmitted diseasesare given. Mathematical models of such diseaseshave been formulated and analyzed by Fine and Le Due (1978) Busenberg and Cooke (1982), Cooke and Busenberg (1982), Busenberg et al. (1983), Regniere (1984), and El Doma (1987). Anderson and May (1981) give an extensive discussion of the role that vertical transmission can play in the transmission of infections within invertebrate populations. We present and analyze a model for the transmission of Keystone virus and one for the transmission of R. rickettsii. Our aim is to emphasize the role of vertical transmission in maintaining these infections and its effect on the size of the vector population. In the case of Keystone virus we show that the diseasedisappears in the absence of a high enough rate of vertical transmission. We also give a quantitative analysis of the effect of changes in the number of gonotrophic cycles of the mosquito hosts and in the horizontal transmission rate of the infection. In the case of R. rickettsii we also study the effect that vertical transmission, through several generations, can have on the population size of the ticks. This effect occurs becauseeach successivegeneration in a vertically infected lineage of ticks may start life with a larger load of pathogen than did the parent generation (Burgdorfer and Brinton, 1975). After a certain number of such successive vertically infected generations, the pathogen load is large enough to prevent successful oviposition and essentially causes infertility in such ticks. Finally, the age-structure of the host population plays a significant role in the population dynamics of vertically transmitted diseasesas has been already shown by us (Busenberg and Cooke, 1982; Busenberg et al., 1983; Cooke and Busenberg, 1982) and by El Doma (1987). This fact will again appear in both of the models that we discuss here even though we use only discrete dynamic equations in these models. 2. A MODEL FOR THE TRANSMISSION OF KEYSTONE VIRUS A model for the transmission of Keystone virus in the mosquito Aedes atlanticus was suggested by Fine and Le Due (1978) on the basis of field observations and was reformulated as a dynamic system and partially analyzed by Cooke and Busenberg (1982). Here we present a special case of this dynamic model which we have been able to analyze completely. Recall that mosquitoes need a blood meal from a vertebrate host prior to oviposition and that there are several such gonotrophic cycles (blood meals followed by oviposition) in a given season. The mosquitoes overwinter in the egg stage, so vertical transmission is a primary mechanism for the perpetuation of the infection. Since emergence and oviposition by the
VERTICALLY
TRANSMITTED
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mosquitoes are roughly synchronized, a discrete time formulation of the dynamics of this infection is possible. We use the following notation which is in agreement with our previous work (Cooke and Busenberg, 1982): B, = prevalence rate of the virus among female mosquitoes that oviposit in the kth year, 0 < Bk < 1. (Thus, Bk is the fraction
of eggs laid in the kth year that carries the virus.) d = maternal vertical transmission rate, 0 < d c 1.
i = proportion of vertically infected mosquitoes that are infectious to vertebrates, 0 < i < 1. n = average number of mosquitoes per vertebrate host at the time of the first blood meal, n > 1. p = proportion of female mosquitoes surviving through one gonotrophic cycle, 0 < p < 1. r = number of gonotrophic cycles in 1 year, r > 1. f= probability that a susceptible mosquito which feeds upon a viremic (infected) vertebrate will become infective, 0
(2.1)
where R =fs(p + .** +p’-I)/(1
+p+
... +pr-‘)=fps(l-p’-‘)/(1-p’),
(2.2)
with r > 1. R is the net horizontal transmission rate in the mosquito population mediated by the vertebrate hosts on which they feed. For this model we have the following conclusions based on Theorem 4.1 which is stated and proved in the Appendix. Define the threshold parameter T = d(inR + 1), with R defined in (2.2). Then, if T-C 1, as time increases, the prevalence rate B, tends to zero and the proportion of infective mosquitoes tends to zero. If T> 1, the prevalence rate
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B, tends to the endemic level Bz which can be computed as shown in Theorem 4.1.
First, note that in the absence of vertical transmission (d= 0), we always have T= 0, hence, the disease dies out in this case. Second, in the absence of horizontal transmission (f= 0) the rate R is equal to 0, and T= d< 1, and again the disease dies out. So, the maintenance of this infection requires a coupling of the horizontal and vertical transmission mechanisms. Finally, T can be easily shown to be an increasing function of r, the number of gonotrophic cycles, and varies from the value T=d(infip/(l+p)+ 1) when r=2 to T=d(infsp+ 1) as r+oo. Consequently, the likelihood of the disease reaching an endemic level increases with the number of gonotrophic cycles. This is illustrated in the following graph of T versus r and d where we have taken f = p = s = 0.5, i = 0.8, and n=lOoO. The above mathematical conclusions based on the dynamic model (2.1) can be heuristically explained as follows. Because of vertical transmission, the mosquito population starts each season with a positive proportion of infectives. Since only infective mosquitoes can have infective offspring, and since the probability d of vertical transmission is less than one, the disease would eventuallly die out if this were the sole mechanism for its transmission. However, because the population goes through several gonotrophic cycles (r 2 2), it initiates the mechanism of horizontal transmission by infecting vertebrate hosts who, in turn, become the source 1.6-
d = 0.004
0.
2
I
4 Number
of
gonotropbic
6 cyclea
FIG. 2.1. Variation of the threshold parameter T with the number of gonotrophic cycles r and the vertical transmission rate d.
VERTICALLY TRANSMITTED INFECTIONS
185
of new infections in the mosquito population. The process of horizontal transmission is enhanced by an increase in the number of gonotrophic cycles, hence, by any prolongation of the period of fertility of the mosquitoes. It is the close coupling of all these mechanisms that makes an endemic level of the disease possible. The threshold parameter T = d[ in&( ( 1 -p’ - ’ )/( 1 -p’) + 1] is easily seen to be an increasing function of each of the basic epidemiological parameters. Its variation with respect to all of these parameters except p (the survival probability) and r (the number of gonotrophic cycles) is linear and is largest with respect to d, the maternal vertical transmission rate. T varies in a sublinear way with respect to p from the value d at p = 0 to the value d[infi(r - 1)/r + 1] when p = 1. With respect to r, T again increases in a sublinear way with value d[infp/( 1 +p) + 1] at r = 2 and reaching the value d[infsp + l] as r -+ co. The sensitivity of the prevalence rate B*, when T> 1, to variations in the epidemiological parameters is more dihicult to see analytically since Eq. (4.1) canot be explicitly solved for B* in the most interesting cases
FIG. 2.2. (a) Variation of the prevalence rate with time when T-c 1. the prevalence rate with time when Ts 1.
(b) Variation of
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when n is large. However, it is quite easy to determine numerically and in the previous two graphs we show the variation of the prevalence rate with time for two different values of r(r = 2 and Y= 10) in two cases,one with T< 1 (Bk -+ 0) and one with T> 1 (B, + B* > 0). 3. A MODEL FOR THE TRANSMISSION OF Rickettsia rickettsii Rickettsia rickettsii causes Rocky Mountain spotted fever (RMSF) in human and other vertebrates and ticks are the primary vectors that transmit this disease. There is a significant mortality in humans infected with RMSF and consequently there have been extensive field and laboratory studies (Burgdorfer and Brinton, 1975; Burgdorfer, 1975; Garvie et al., 1978) of the transmission of this disease in the tick population. Elsewhere (Busenberg and Cooke, 1982) we have modeled and analyzed the coupling between the vertical and horizontal modes of transmission of this disease. Here we present a model of one of the most striking characteristics of this type of infection, a characteristic shared by a number of other vertically transmitted diseases,namely, the cumulative effect of the pathogen as it is vertically passeddown through several generations. This effect occurs when the pathogen load that is vertically transmitted by an infected parent increases as the pathogen load of the parent increases.Infected offspring of a parent with a large pathogen load commence life with a relatively large load and may reach their reproductive age with a pathogen load that is larger than that of their infected parent. Consequently, the pathogen load may increase monotonically in each successive generation of an infective lineage. In the case of R. rickettsii in the tick Dermacentor andersoni, it has been found (Burgdorfer and Brinton, 1975) that, in certain circumstances, after several generations in an infective lineage, the pathogen load at birth may reach a high enough level to impede successfuloviposition. As a result, vertical transmission can act as a means for controlling population size. This is the aspect of the transmission of this disease that we model here. The following assumptions are based on field and laboratory data (Burgdorfer and Brinton, 1975): (i) The Rickettsia are vertically transmitted in the tick population with transmission rate of 10 to 100%. (ii) The level of the Rickettsia load increases with each successive generation of vertically acquired infectivity. (iii) Female ticks of the fifth generation in a lineage of vertically acquired infection carry a large enough load of Rickettsia to be unable to oviposit successfully and, hence, are sterile. We use two independent discrete variables to model this situation. The
VERTICALLY
TRANSMITTED
INFECTIONS
187
first is the cohort number starting at some arbitrary time origin, and the second is an internal variable indicating the generation number in an infected lineage. The second variable takes on the values (0, 1, .... S}, where 0 denotes ticks that acquired the disease via horizontal transmission, 1 denotes those that acquired the diseasevertically from a parent of infective lineage 0, and so on, with 5 denoting ticks that are vertically infected by a parent of lineage 4. The notation used in this model is n = cohort number, n > 0. o!= the generation number in an infective lineage of vertical transmission, 0 < a < 5. i,(a) = the number of infective ticks of the nth cohort that are of the a th infective lineage. S, = the number of susceptible ticks in the n th cohort. b = the birth rate of female ticks adjusted for the proportion of female to male ticks in the population. q = the probability of an infected female tick of lineage a = 0, 1, ...) 4, to vertically transmit the infection to its offspring, O
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and the mathematics. We have chosen to present this simpler case since it illustrates the same basic phenomena. In a discrete model, horizontal transmission which occurs over an extended time interval needs to be idealized as occurring at a synchronized specific instant of time. Here we take this instant to be the end of the active blood meal-seeking period of the season. Since we are following cohort groups which appear at the beginning of a particular season,we distinguish between those ticks that are born with the infection and those that acquire it via horizontal transmission. We shall assume that the horizontally infective ticks in the n th cohort come from susceptible ticks of the n th cohort that are infected via contact with infective ticks of the (n - 1)th cohort, and not from the new infectives of their own cohort. This assumption allows for the maturation period of new born infectives before they take their first blood meal, and also takes into account the time lapse that is necessary before the vertically acquired pathogen in newborn infectives is large enough to be effectively transmitted via horizontal contacts. The force of horizontal infection on the n th cohort is given by a funtion
f(s,J-,)=f(s.~ i L(a))
(3.1)
LX=0
which can take a variety of forms that are mathematically described by hypothesis (H) in the Appendix. Here we consider two forms off that we will discuss in some detail and which are representative of the general forms of the force of infection that are analyzed in the Appendix. These special forms are the “mass-action” form f(xvZn-,)=QJn-1
(3.2a)
and the “saturated mass-action” form
f(s”,r,~,)=kS,z”-,/(k’+z,-,).
(3.2b)
Note that the ratio f(x, y)/x has generic graphs in these two cases. Figure 3.1 is the graph of f(x, v)/x for f given by (3.2a) and (3.2b). We shall be able to derive results on general forms off which essentially behave like (3.2a) or (3.2b) and use the two definitions (H,) and (Hb), respectively, in the Appendix for these general forms. The mass-action form (3.2a) of the force of horizontal infection f is very familiar and is based on the hypothesis of uniform mixing, hence random contacts, between infectives and susceptibles. The saturation form (3.2b) has the property that f saturates at the value kS, as I,- i + co. This reflects the possibility of susceptible ticks engorging on a non-viremic vertebrate host even when the number of infective ticks is large. This is an important
VERTICALLY
TRANSMITTED
189
INFECTIONS
mass-action
. 0
FIG.
Y
3.1. Generic graphs of the mass-action and the saturated mass-action nonlinearities.
consideration because a substantial proportion of the offspring of infective ticks may be susceptible, and the intense viremia in the vertebrate hosts from which these susceptible ticks may be horizontally infected lasts only a few days. Hence there is a definite possibility of having susceptible ticks feed on non-viremic vertebrates and thus avoiding horizontal infection. Based on the above assumption, we have the following dynamic model. i,(O)=cf~,,Zn-1) i,,(a) = cbqi,- ,(a - l), S,=cb
p i
a = 1, 2, ...) 5
i,-,(a)+S,-,
(3.3)
- cfaz, zn- 1).
a=0
Even though we use exactly five vertically infected generations, all of the results that we shall prove and describe hold true when 5 is replaced by any natural number Na 2 in the above model equations. In particular, note that the expression for S, has no term of the form cbpi” _ 1(5) since we have assumed that the ticks of infective lineage 5 are not fertile. For this model we have two sets of conclusions, the first of which is based on Theorem 4.2 in the Appendix and gives explicit threshold conditions for the feasibility of an endemic level of the infection. Here we suppose that the force of infection f is of either of the two forms (3.2a) or (3.2b), or more generally satisfies hypothesis (H) in the appendix. Zf the net reproduction rate cb < 1 then the total population tends to zero with time and the zero population level is stable when cb < 1 and unstable when cb > 1. Moreover, a positive endemic equilibrium can exist only if 1~ cb < (1 - qbc)/p( 1 - (qbc)‘) = l/R,.
(3.4)
As we shall see below, the exact range of parameters where an endemic
190
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equilibrium is feasible and stable depends on the forms of the force of infection; however, in all casesthe restriction imposed by (3.4) must apply. Now note that in the absence of vertical transmission (q= 0,p = 1) condition (3.4) cannot hold if cb # 1 and a non-trivial feasible solution does not exist. In fact, in this case, i,(u) = 0 for n > 1, a = 1, 2, .... 5; and S,+I,=P,=cbP,_,,
hence, P, -+ 0 if cb < 1 and P, + co when cb > 1. So, the tick population either decays to zero or becomes unbounded. If cb = 1 and q = 0, then i(a) = 0 for cf= 1, .... 5 and i(O)= cf(S, I(O)). Also, R, = 1, and it is shown below that S=k,, t(0) = k, is a steady state solution for any constants k, and k2 for which k2 = cf(k,, k2). In the absence of horizontal transmission (f=O), i,(a)=0 for n> 1, a=O, l,..., 5, and P, = S, = cbS,- 1 for n > 2. Again we have P,+O if cb
Zf the force of infection f has the mass-action form (3.2a) (or more generally, obeys hypothesis (H,) in the Appendix) then an endemic equilibrium exists when and only when the threshold conditions in (3.4) hold. However, tf f has the saturated mass-action form (3.2b) then a unique endemic equilibrium exists when and only when the threshold conditions in both (3.4) and k>
(l-cb)
c( cbR, - 1) are satisfied. A similar condition applies for the more general forms of the force of infection which satisfy condition (Hb) in the Appendix.
The determination of the stability of the endemic state when it is feasible is difficult and we have treated this aspect of the problem only numerically. To this end we have used two independent methods for numerically studying the stability of the endemic state on a computer for several parameter ranges. These results have shown that the endemic level is stable whenever it is feasible when the force of infection takes the form (3.2b); while for the form (3.2a) there is a parameter region where this equilibrium is feasible and unstable. In this case, we have performed extensive numerical
VERTICALLY
TRANSMITTED
191
INFECTIONS
experiments which all show that a stable oscillatory solution exists and the subpopulations undergo bounded oscillatory changes which are quite complicated even though they are periodic. Finally, when cb exceedsthe upper threshold bound in (3.4) the total population grows without bound as time increases. This is the parameter region where the net reproduction rate is large enough to dominate the population-controlling effect due to the reduced fertility caused by the vertical transmission of the disease. Figures 3.2a and b show the dependence of the stability of the zero and the non-trivial steady states on the parameters b, c, and p obtained from the theorem and by numerically computing the eigenvalues of the linearized system about the non-trivial equilibrium forfgiven by (3.2a) and (3.2b). In the region where the zero equilibrium is unstable and the non-trivial
.075 t iI
nontrivial
,025
not
equilibrium feasible
not
equilibrium feasible
..
bc
nontrivial
.75
t a
zero equilibrium stable
.5 I
f .25t
0.
.S
1.
1.1
1.2
bc
FIG. 3.2. (a) Bifurcation boundaries and stability regions for the mass-action nonlinearity (plot independent of c). (b) Bifurcation boundaries and stability regions for the saturated mass-action nonlinearity f= ksi/(k’ + i).
f=ksi
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BUSENBERG AND COOKE
equilibrium is not feasible the population becomes unbounded. From this numerical computation it is clear that the region in the p vs bc plane where the disease controls the size of the tick population becomes larger as c increases.The boundary of this region, for fixed c, is monotone decreasing as a function of b and tends to infinity when bc tends to one. 4. APPENDIX: MATHEMATICAL ANALYSIS In this section we give the statements and proofs of the theorems which justify the conclusions presented in the previous sections. THEOREM 4.1. Let T = d(inR + 1) where R is given by (2.2). Then $ T < 1, B, + 0 as k -+ co and the proportion of infective mosquitoes tends to zero. If T > 1, Bk + B* > 0 as k -+ 00, where B* is the unique positive B satisfying
(l-d)B=R(l-dB)[l-(l-diB)“].
(4.1)
Hence, if T> 1, the prevalence rate tends to a constant endemic level as k increases. Proof of Theorem 4.1. We first consider the steady state solutions of (2.1) which must satisfy Eq. (4.1). Clearly, B = 0 is one such steady state, while if B # 0, the left hand side of (4.1), when viewed as a function of B, is a straight line with slope 1 - d. The right hand side of (4.1) has first derivative equal to dinR at B=O, its second derivative is negative when 0 < B c 1, and its value at B = 1 is less than 1 - d. So, there is a unique solution to (4.1): B= B* E [0, l] if and only if dinR> 1 -d. That is, the non-trivial feasible equilibrium exists if and only if T> 1. Next, writing (2.1) in the form B, = F(B,- i), we first note that, if 0 < B < 1, then 0 < F(B) < 1, so the dynamic system (2.1) maps the interval [0, l] onto itself and is well-posed. Computing the derivative of F we get F’(B)=d(l-R)+Rd(l-diB)“-‘[ni(l-dB)+(l-diB)”],
(4.2)
hence, F’(B) > 0 and F”(B) < 0 for BE [0, 11. Now, F(0) = T and we have the two situations depicted in Fig. 4.1. Because of the convexity of F and the fact that it is monotone increasing on [0, 11, it can be easily seen that, if T
O
Since F’(x) < 0, we get 0 < Bk < TB,- , < B,- , , so by iterating this
VERTICALLY TRANSMITTED INFECTIONS
Bk-l
193
1 B
Y
0 B
FIG. 4.1. (a) Plot of F vs B and the iteration scheme for T< 1. (b) Plot of F vs B and the iteration scheme for T> 1.
process: 0 < B, < TkBo+ 0 as k + co, and the proof is complete in this case. When T> 1, we consider the two situations where 0
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Theorems 4.2 and 4.3 require certain hypotheses on the form of the force of infection f which we collect below. (H). f is a monotone increasing function of S, and Z,,~ 1, with f(x, y ) = 0 if either x=0 or y=O and f(x,y)>O if x>O and y>O. f is also continuously differentiable and
Two particular forms of f that we shall consider in Theorem 4.3 generalize the mass-action and the saturated mass-action forms and are respectively defined via the following hypotheses: (H,). f satisfies hypothesis (H) and f (x, ax)/x is a convex function of x for every constant a > 0, and (H,,). f satisfies hypothesis (H) and f(x, ax)/x function of x for every constant a > 0.
is a concaue monotone
Note that, iff (x, ax)/x = g,(x) has a second derivative g:(x) with respect to x, then condition (H,) requires g:(x) 20 for all a >O and (Hb) requires g:(x) < 0 for all a 2 0.
As they stand, Eqs. (3.3) are not an explicit dynamic system expressing the variables at time n in terms of their values at (n - 1) because of the presence of the term S, in the force of infection. However, from the last equation in (3.3) we get S,+cf(S,,I,-,)=cb
[
p i
1
i,-,(cr)+S,-,
a=0
(4.3)
which, because of hypothesis (H), has a unique non-negative solution whenever i,- ,(a) and S,_ 1 are non-negative: S,=G
(
Znel,SSn-,,
i
i,-,(a)
Cr=O
>
.
(4.4)
Using (3.3) and (4.4) we obtain the explicit dynamic system: i,(O)=cf
((
G InPl,Ll,
i
Ll(a)
Or=0
a = 1, 2, ...) 5
l),
i,(a)=cbqi,_,(cr-
L,,
>
i Cl=0
indl(a) . >
,I,-,
>
(4.5)
VERTICALLY TRANSMITTED INFECTIONS
195
The function G depends on the specific form of J however, it is always non-negative and G(0, 0,O) = 0. For the particular form off given by (3.2b) we get the explicit dynamic system (4.5) with G given by
G=Ch(k’+I,-,)Cp~~=,i”-,(cr)+S,-,l k’+(l
(4.6)
+ck)Z,-,
The following result holds for the system (4.5) under hypothesis (H): THEOREM 4.2. Suppose that f satisfies hypothesis (H). Then the system (4.5) preserves positivity in the sense that non-negative initial data &(a) > 0, So 2 0 yield non-negative solutions. Moreover, if the net reproduction rate cb < 1, zero is the only feasible (non-negative) steady state of (4.5) and it is stable and attracts all non-negative solutions tf and only tf cb < 1; and it is unstable ifcb > 1. A non-trivialfeasible equilibrium solution to (4.5) can exist only if
1 d cb d (1 - qbc)/p( 1 - (qbc)‘).
(4.7)
(This is only a necessary but not a sufficient condition.)
The proof of this result is given at the end of this section. The non-trivial steady state which may exist under condition (4.7) can be easily seen to satisfy t(o)J’-W~ cbR,-
1’
a = 1, 2, .... 5,
I(a) = (cbq)’ t(O),
where
s,
1 - (cbq)6 l-cbq
1 - cb S .cbR,- 1 >
with
R =P(’ - (qbC)5) P l-qbc ’
(4.8) Note that, by hypothesis (H,), the equation for 3 in (4.8) has a unique positive solution whenever (1 - cb)/(cbR, - 1) > 0. However, the other components of the steady state given by (4.8), as we shall show below, are non-negative if and only if (4.7) holds. THEOREM 4.3. Zf f satisfies hypothesis (H,) then the conclusions of Theorem 4.2 hold and a non-trivial feasible equilibrium exists if and only tf (4.7) holds. Iff satisfies (H,,) then the conclusions of Theorem 4.2 hold and
(i) rf lim,, mf (x, Ax)/x = 00, where A = (1 - (cbq)6)/( 1 - cbq) . ( 1 - cb)/(cbR, - 1), then a unique non-trivial equilibrium exists if and only if (4.7) holds with the strict inequalities.
196
BUSENBERG
AND
COOKE
(ii) rf lim,, o.f(x, 2x)/x= K< Co, then a unique non-triuial equilibrium exists if and only if (4.7) with strict inequalities and K>(l
1)
-cb)/c(cb&-
(4.9)
both hold.
Condition (4.9) can also be written in the form c’“:_“:; 1)-g (0,O) + c & (0,O) > 1. ” n 1
(4.10)
We now give the proofs of Theorems 4.2 and 4.3. Proof of Theorem 4.2. The fact that system (4.5) preserves positivity follows from the non-negativity of G in (4.4). Now, if cb < 1, upon adding the first and last equation in (3.3) we get i,(O)+S,=cb
(
p f
i,-,(a)+S,-,
)
LX=0
(4.11)
so (4.5) cannot have a steady state with either S > 0 or i(0) > 0. But, if i(0) = 0, the second equation in (4.5) implies i(u) = 0 for c1= 1,2, .... 5 and (4.5) has only the zero steady state when cb < 1. In order to prove the stability of the trivial steady-state, we note that (4.11) yields the relation i,(O) + S, < cb[i,-
l(O) + S,- ,] Q (cb)“[i,(O)
+ So],
hence, if cb < 1, i,(O) + 0 and S, --f 0 as n + co for any initial data io(0), So. Since i,(O) + 0, the second equation in (4.5) implies i,(a) + 0 as n + co for a = 1, 2, ...) 5 and the zero steady state attracts all non-negative solutions when cb< 1. Next, when cb > 1 we use hypothesis (H) to linearize (4.5) about the zero equilibrium and obtain the linear system
i,(O iA1 L(5 s,
0
0
.
.
-00
cbq 0
0 cbq
. .
. ’
. .
0
0
0
0
cbq
0
0
cbp
cbp
.
.
cbp
0
cb
0 .
0 0
VERTICALLY
TRANSMITTED
INFECTIONS
197
One of the eigenvalues of this lower triangular matrix is cb > 1, and the others are all zero, so the trivial equilibrium is unstable in this case. It remains to establish condition (4.7). From the expression for i(0) in (4.8) we easily see that a non-trivial equilibrium with all subpopulations positive can exist only if (1 - cb)/(cbR, - 1) > 0, hence, only if either (a) 1
t&J51 = bcp - bcp(qbc)5 < , bcp+ 1 -bc
1 -qbc
whenever bc< 1. So, the strict inequality in (4.7) must hold - -in this case. If cb = 1 and cbR, < 1, then i(a) = 0, a = 0, 1, .... 5, hence,f(s, I) =f(S, 0) = 0 and any constant can be an equilibrium value of s. If cb > 1 and cbR, = 1, then S must be zero, hence, i(0) = cf(0, Z) = 0, and there is no non-trivial steady state. Finally, if cb = 1 and cbR, = 1, then q = 0, p = 1, z(cx)= 0 for a = 1, 2, ...) 5, and (3.3) reduces to z(O)= cf(s, i(0)). Then S=ki, i(O)= k,, k,, k, any constants such that k, = cf(k,, k2), is a steady state solution. So, steady state solutions can exist only if (4.7) holds, and Theorem 4.2 is proved. Proof of Theorem 4.3. Since both (H,) and (H,,) include (H) we only need to prove the existence and uniqueness of the steady state. When either one of the inequalities in (4.7) is not strict, we have seen in the previous proof that the non-trivial solutions, when they exist, are not unique. Hence, we can limit our considerations to the case where the inequalities in (4.7) are strict. In this case, by (4.8) we need to consider the existence of a unique S > 0 satisfying
p/c=f(s,ns)/s,
(4.12)
where /I = (1 - cb)/(cbR, - 1) and I = (( 1 - (cbq)6)/( 1 - cbq))( 1 - cb)/ (cbR,- 1) ~0. From hypothesis (H) the right hand side of (4.12) tends to zero as s-0, and by (H,) as well as by (Hb)(i) it tends to co as S+ co, consequently, (4.12) has at least one positive solution for any A > 0. When (H,,)(ii) holds then it is clear that (4.12) can be satisfied by a finite S>O if and only if (4.9) holds. Uniqueness follows by the monotonicity of the right hand side of (4.12). Since by Theorem 4.2 we already know that (4.7) is necessaryfor a non-trivial solution to exist, it follows that the conditions in Theorem 4.3 are boh necessary and sufficient for a unique non-trivial solution to exist and the proof is finished.
653 37 2.6
198
BUSENBERG AND COOKE ACKNOWLEDGMENTS
We thank Keith Saints and Mark Overly for producing the graphs used in this paper. This paper was completed when the first named author (S.B.) was visiting the Centre for Mathematical Biology at the University of Oxford and we are grateful for the support of the Centre during the writing of this paper.
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