Evolution of increased susceptibility to infectious diseases in age structured populations

J. theor. Biol. (1990) 147, 133-142

Evolution of Increased Susceptibility to Infectious Diseases in Age Structured Populations SIGFRID LUNDBERG

Department of Ecology, Theoretical Ecology, Ecology Building, Lund University, S-223 62 Lund, Sweden (Received on 6 November 1989, Accepted in revised form on 14 May 1990) Current theory on the epidemiology of infectious diseases in genetically heterogeneous host populations ignores age structure. Thus, the possibility that microparasites might have negative effects on fitness in different ways during different phases of the lives of their hosts is not accounted for. For example, infections causing mortality and morbidity among juveniles might also have an impairing effect on reproduction among adults, as is the case in many so called childhood diseases. I demonstrate that when this is the case there may be a selective pressure for the host to evolve a higher level of susceptibility to an infection, provided it has the following properties: it should (i) provide life-long immunity, (ii) have a negative effect on reproduction and (iii) not cause too many mortalities. Introduction

An infectious disease is characterized by, a m o n g other things, its age-specific prevalence curve, i.e. a plot o f the fraction currently infected within each cohort against age. It is fruitful to regard this curve as a life history trait, which is evolutionarily adjusted in order to minimize the total detrimental effects o f the disease at different ages. A d o p t i n g this view, we can apply M e d a w a r ' s (1952) and H a m i l t o n ' s (1966) propositions, that if there is genetic variation for t h e age-specific expression o f a gene, through, e.g. other genes modifying its age o f onset, natural selection s h o u l d p r o m o t e an early expression if the gene is beneficial and, conversely, a late expression if it is not. By the same token, a trait which is favourable early in life but disfavourable later is likely to be selected for, since genetic variation for traits expressed late in life tends to be neutral. A gene decreasing the overall susceptibility to an infectious disease is in a sense equivalent with one for a late onset, because it shifts the m o d e o f the prevalence curve towards older ages. Current theory on the e p i d e m i o l o g y o f infectious diseases in genetically heterogeneous host p o p u l a t i o n s predicts that such genes will increase when rare (Holt & Pickering, 1985; A n d e r s o n & May, 1986). R o u g h g a r d e n (1979) a m o n g others has discussed genes improving r e p r o d u c t i o n at one age at the cost o f decreased survival at another. In life history theory it is generally taken for granted that an increased reproductive effort at one age leads to decreased survival at some older age. However, in the context o f infectious diseases it can be the other way around. C o n s i d e r the case o f an infection providing lifelong i m m u n i t y and having an impairing effect on r e p r o d u c t i o n when contracted by adults. Then the host might be selected for increased susceptibility to the infection, 133 0022-5193/90/021133+ 10 $03.00/0

O 1990 Academic Press Limited

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because the increased mortality among the juveniles caused by a shift of the prevalence curve towards the younger age classes can be compensated for by the lessened detrimental effect on reproduction. In the following this will be demonstrated using an age structured epidemiological model for a population in exponential growth. The Model

Consider a host population consisting of, in total, N(t) individuals at time t, of which Y(t) are infected. Further, let x(t, a), y(t, a) and z(t, a) be the number of susceptible, infected and immune individuals, respectively, of age a at time r After sexual maturation, which occurs at age a,, the per capita reproduction is b for those not infected, whereas it is reduced to a fraction p of this value for those in the infected class. Sexual maturation is the only age specificity which is included in this model. The dynamics of the population is described by the following set of first order partial differential equations (PDE) (McKendrick, 1926; Anderson & May, 1985):

Ox(t,a)

, Y(t) , fix(t,a)-~-(~-ixx(t,a),

Oa

Oy(t, a) kOy(t, a) Ot Oa

fix(t, a) Y(t) -~-(tx+a+y)y(t,a)

Oz(t, a) ~_Oz(t, am)_ yy(t, a)-i~z(t, a) Ot Oa

(la) (lb) (lc)

where ~ is the instantaneous death rate, due to other factors than the disease, ce is the incremental death rate for those infected and y the rate of recovery. /3 is the transmission parameter. The transmission is thus age independent and since fi Y ( t ) / N ( t ) is the per capita transmission rate it does not depend upon population size. This form enables the susceptible class to reach a stable age distribution when the population is in exponential growth. Stable age distributions for all of the three classes are required for obtaining the kind of trade-off between reproduction and survival aimed at. Anderson & May (1979) showed that if the transmission is modelled by a cross-product between the n u m b e r of susceptible and infected individuals, he former approaches a constant value when the rest of the poplation is in exponential growth. Their model had no age-structure, but the same results holds for models with the simple kind of age specific reproduction used here if the transmission rate is dependent upon population size. A boundary condition to the PDE system including the assumptions made above on the reproductive patterns is given by:

x(t,O)=b

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(2)

a.~

When simulating this model one finds that it always seems to approach a stable age distribution an exponential growth [Fig. 1 (a)-(c)], that is, the density functions with respect to age x, y and z are all growing exponentially with the same rate. So far, I have not found any exception to this rule, but I have not been able to prove

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FIG. 1. The change over time of the age distributions of (a) the susceptible [x(t, a)/N(t)], (b) the infected [y(t, a)/N(t)] and (c) the immune class [z(t, a)/N(t)]. The simulation was initiated with 90 susceptible and ten infected individuals spread evenly over the ages (0-3.75 time units of age). The system exhibits typical damped oscillations over the 12-5 time units the simulation lasted and the three distributions are at the end almost stable. The partial derivatives are approximate with difference ratios and the same step length for time and age is used, 0-0125 time units. The parameters are/7 = 6, p = 0.9, a = 0.05, a~ = 2, T = 1.25, b = 4 and #x = 0.1. (d) The convergence of d In N(t )/dt (solid curve) towards r (dashed and labelled r in the figure) and of the ratio Y(t)/N(t) (solid curve) towards 0 (dashed and labelled t7 in the figure). The method used is the same as for (a)-(c) but the step-length is 0.01 and fl = 4. analytically that this always holds. When approaching stable age distribution and exponential growth the ratio Y(t)/N(t)~ O, w h i c h is c o n s t a n t . I n t h e a s y m p t o t i c s t a t e t h e s y s t e m is l i n e a r a n d it is p o s s i b l e t o s e p a r a t e t h e v a r i a b l e s in e q n ( l a - c ) b y a s s u m i n g x( t , a ) = x ( a) e x p (rt), y( t, a ) = ~ , ( a ) e x p (rt) a n d z( t, a ) = to(a) e x p (rt), w h e r e r is t h e i n s t a n t a n e o u s r a t e o f p o p u l a t i o n i n c r e a s e , a n d X, tp a n d to a r e n o t normalized density functions describing the stable age distributions for the three classes. The PDE system then reduces to a set of linear ordinary differential equations (ODE): dx(a)

da dqJ(a)

da

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of this system into eqn (2) one obtains an E u l e r - L o t k a equation

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The direction of the selection pressure exerted by the pathogen upon the susceptibility trait in the host was tested using a procedure inspired by the concept of evolutionarily stable strategies (ESS) (Maynard Smith & Price, 1973), in the sense that a rare mutant host type is introduced, whose dynamics was supposed to be governed by a system of PDEs equivalent to eqn ( l a - c ) . The per capita rate of spread of the disease in the mutant population is

/3,,wYw(t)+/3,.,.Y"(t) N~(t)+ Nm(t) '

(3)

and in the wildtype population

/3wwY~( t) + /3~"Y,.( t) Sw(t)+Nm(t) '

(4)

where the subscripts m and w stands for mutant and wildtype, respectively. When the mutant is rare and both types are in exponential growth, eqn (3) reduces to ~3,.wOw and eqn (4) fl~.wOw where Ow= Yw(t)/Nw(t) is the stable fraction of infected individuals in the wildtype, rw, the rate of increase of the wildtype is calculated as above, and the results of these calculations are then used for the calculation of the rate of increase of the mutant, r~. Results and Discussion

The rate of exponential population growth (r), is a generally accepted fitness measure (Hamilton, 1966; Charlesworth, 1980). r decreases with the transmission p a r a m e t e r below a certain critical threshold, /3cri,, whereas above this threshold it increases, given that p < 1 and a is not too high [Fig. 2(a)]. By introducing a rare mutant host differing from the wildtype only as regards its susceptibility (i.e. assuming that /3ww=/3w,, and /3row=/3,.,.) and then calculating its initial rate of increase or decrease one can establish the direction of the selection pressure on both sides of the fitness trough. Below the threshold, the host population will evolve towards increased resistance, since a rare less susceptible host type (/3cr~t>/3ww> /3,.w) will increase in frequency and a more susceptible one will decrease. By the same token, there is a consistent, albeit relatively weak, selection pressure for increased susceptibility (the mutant increases when rare if/3or,
138

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# FIG. 2. (a) Host fitness measured as the rate (r) o f exponential growth'at stable age distribution vs. transmission parameter ft. The infected class' reproduction is reduced by 10% (p = 0.9), and the incremental death rate ( a ) ranges from 0 (top curve) to 0.05 (bottom curve) in steps of 0.01. The remaining parameters are: a s = 2 , y = 1.25, b = 4 a n d / ~ =0.1. (b) Host fitness measured as the rate of exponential increase for three wildtype populations (r~,; marked with dots and labelled 1, 2 and 3) and various

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the region of the genome coding for, among other things, antibodies and receptors on the surfaces of lymphocytes, the main actors in the immune response of all birds and mammals. Several mechanisms have been proposed for explaining the vast genetic variation in MHC. One is that parasites are adapted to the most common host genotypes, giving the rare ones the upper hand (Haldane, 1949). Another, more subtle one, is that host antigens often are included in coats of viruses when these are assembled inside host cells (Vogel & Motulsky, 1986) making them less conspicuous inside their present host, but more vulnerable when invading a new host individual which is of another genotype. As the previous host most likely was of a majority genotype, rarity again implies a lowered susceptibility. These and other related ideas have recently been the subject of much attention in evolutionary biology as it has been suggested that frequency dependent selection within the context of parasite-host systems has played a role in the origin of sexual reproduction (Hamilton, 1980) and that it is one mechanism for providing the non-additive genetic variation needed for the initial evolution of female choice (Hamilton & Zuk, 1982). Another framework for host diversity (be it genetic or on the ecological community level) is the concept of apparent competition between host species or strains sharing an infectious disease (Holt & Pickering, 1985). The main prediction from this theory is that different host types will coexist if the transmission rate is higher within than between the host types, which, in a metaphorical sense, are competing with each other for not getting the disease. The conclusion presupposes that the host population is in the state in which selection is favouring less susceptible genotypes. As rarity implies lower susceptibility coexistence between host strains or species are easily achieved [for example by setting/3cri,> flww =/3,.,. > fl,,w =/3w,. in Fig. 2(b)]. If the host population is in the state in which susceptibility is favoured, the reverse is true: there will be competition for getting the disease and coexistence is possible only if the transmission rate is higher between than within host types [let, e.g./3crit
mutants (r,,; dashed curves), which are supposed to be rare, interacting with these wildtypes vs. the transmission parameters/3ww and/3 ..... respectively. Since the transmission within the rare mutant faction can be ignored, the abscissa represents transmission rates of the disease f r o m the wildtype only, whereas for the c o m m o n wildtype only transmission within genotype is taken into account. The mutants are supposed to be biologically equivalent to the wildtypes except as regards the transmission parameter. The solid curve is a magnification o f the second curve in 2(a). At the point o f intersection between any dot and a dashed curve the rate of transmission from the wildtype to the mutant is the same as the rate of transmission within the wildtype, and so is fitness. The three wildtype populations are in three different states in which less susceptible mutants (population 1), more susceptible ones (population 3) and mutants (population 2) in either direction will propagate.

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affect r e p r o d u c t i o n in several ways. In the model I assume that infected individuals have their reproductive rate decreased to some fraction p. This assumption can be interpreted in two ways. The first possibility is that there is vertical transmission and that a p r o p o r t i o n 1 - p o f the foetuses are infected and that this leads to stillbirths. More detailed studies on how various m o d e s o f vertical transmission affects the evolution o f host susceptibility are badly needed. O f the classical c h i l d h o o d diseases only rubella and c h i c k e n p o x are regarded as health hazards during pregnancy. Congenital rubella causes a n u m b e r o f different disorders and can be serious if the m o t h e r contracted the disease during the first trimester o f the pregnancy (Kurent & Sever, 1977). A l t h o u g h there are reports o f various congenital malformations a m o n g children born to w o m e n w h o contracted chickenpox during early pregnancy, the most serious threat are the neonatal infections occurring if the m o t h e r is infected during the peripartum period (Stagno & Whitley, 1985). A s e c o n d possible interpretation o f the parameter p is that a certain fraction of the infected adults are temporarily reproductively disabled by the disease. This interpretation applies, for example, to m u m p s . A m o n g the complications reported for m u m p s , orchitis a m o n g adult males and oophoritis a m o n g females are interesting in this context. The latter condition is rare and presumably o f little importance, whereas the former can if bilateral lead to sterility (Feldman, 1976). Although, lasting sterility as a c o n s e q u e n c e o f orchitis is too rare to be considered as a serious public health problem, orchitis is painful e n o u g h to arrest any reproductive activities. The a s s u m p t i o n to make reproduction the only age-specific parameter is a gross over-simplification, since, e.g. the mortality and morbidity patterns o f most infectious diseases are highly age-specific, often, as in case o f measles (Black, 1976), with some m i n i m u m case mortality rate during late childhood. An obvious and very interesting extension o f the present model is thus to make the susceptibility some u n k n o w n age-specific function which might be f o u n d by applying optimal control theory. The fitness gained by the evolution o f increased susceptibility is relatively small in c o m p a r i s o n to what w o u l d be gained t h r o u g h the evolution o f a very high level o f resistance. Pathogens are a fact of life which their hosts have to cope with. A host to the right o f the fitness trough [Fig. 2 ( a ) - ( b ) ] is a sinecure for the virus. It m a y evolve towards an ever increasing level o f infectiousness and the best thing the host can do is to surrender; there is no benefit to be gained from a coevolutionary alms race.

FIG. 3. Stable age distributions of the susceptible (solid curve), infected (dashed curve) and immune classes (dotted curve), for three levels of the transmission parameter (/3): 3.0, 3.8 and 5-0. The incremental death rate (c~) is 0.01. The three cases correspond to combinations of parameters marked with dots in Fig. 1. The vertical lines correspond to the age at sexual maturation. Note that the proportion of the reproductive individuals that are immune increases with/3, such that in (c) almost all mature individuals belong to this class.

142

s. L U N D B E R G

This work was s p o n s o r e d by the Swedish Natural Science Research Council. I am grateful for discussions, help and s u p p o r t from friends and colleagues at D e p a r t m e n t o f Pure and A p p l i e d Biology, Imperial College, L o n d o n , and at D e p a r t m e n t o f Ecology, Lund University.

REFERENCES

ANDERSON, R. M. & MAY, R. M. (1979). Population biology of infectious diseases: Part I. Nature, Lond. 280, 361-367. ANDERSON, R. M. & MAY, R. M. (1985). Vaccination and herd immunity to infectious diseases. Nature, Lond. 318, 323-329. ANDERSON, R. M. & MAY, R. M. (1986). The invasion and spread of infectious diseases within animal and plant communities. Phil. Trans. R. Soc. Lond. B. 314, 533-570. BLACK, F. L. (1976). Measles. In: Viral Infections of Humans (Evans, A. S., ed.) pp. 297-316. New York: Plenum. CHARLESWORTH, B. (1980). Evolution in Age Structured Populations. Cambridge: University Press. FELDMAN, H. A. (1976). Mumps. In: Viral Infections of Humans (Evans, A. S., ed.) pp. 317-336. New York: Plenum. HALDANE, J. B. S. (1949). Disease and evolution. La Ricerca Sci. 19, (Suppl.), 68-76. HAMILTON, W. D. (1966). The moulding of senescence by natural selection. J. theor. Biol. 12, 12-45. HAMILTON, W. D. & ZUK, M. (1982). Heritable true fitness and bright birds: a role for parasites? Science 218, 384-387. HAMILTON, W. D. (1980). Sex versus non-sex versus parasite. Oikos 35, 282-290. HOLT, R. D. & PtCKERING, R. D. (1985). Infectious disease and species coexistence: a model of Lotka-Volterra form. Am. Nat. 126, 196-211. KURENT, J. E., & SEVER, J. L. (1977). Infectious diseases. In: Handbook of Teratology (Wilson, J. G. & Fraser, E. C., eds) pp. 225-259. New York: Plenum Press. MAYNARD SMITH, J. & PRICE, G. R. (1973). The logic and animal conflict. Nature, Lond. 246, 15-18. MCKENDRICK, A. G. (1926). Applications of mathematics to medical problems. Proc. Edinburgh Math. Soc. 44, 98-130. MEDAWAR, P. (1952). An unsolved problem of biology. Reprinted In: The Uniqueness of the Individual. pp. 28-54. New York: Dover, 1981. ROUGHGARDEN, J. (1979). Theory of Population Genetics and Evolutionary Ecology: An Introduction. New York: MacMillan. STAGNO, S. & WHITLEY, R. J. (1985). Herpesvirus infections of pregnancy. Part II: Herpes Simplex virus and Varicella Zoster virus infections. N. Engl. J. Med. 313, 1327-1330. VOGEL, F. & MOTULSKY,A. G. (1986). Human Genetics. Berlin: Springer Verlag.