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Spreading Disease through Social Groupings in Competition
J. theor. Biol. (2001) 212, 253}269 doi:10.1006/jtbi.2001.2368, available online at http://www.idealibrary.com on
Spreading Disease through Social Groupings in Competition IVANA BEARDMORE*
AND
K. A. JANE WHITE
Department of Mathematical Sciences, ;niversity of Bath, Bath BA2 7A>, ;.K. (Received on 31 March 2001, Accepted in revised form on 11 June 2001)
Many animal populations live in social groups which avoid contact with other conspeci"c groups for at least part of the year. This may give rise to competition between groups for items such as shelter, land and mates. We couple intra-speci"c group competition with disease dynamics to investigate how infectious diseases may spread through population subgroups, particularly with reference to the contact rates between groups. Our model uses a nonlinear systems of ODEs for which steady-state analysis is carried out in the simplest two-group system. This indicates that coexistence of social groups is possible with the disease or that competitive exclusion occurs with one group dying out whilst the other retains disease. Moreover, we show that in certain circumstances the model can exhibit multistability and we discuss the ecological implications of this result in relation to contact between social groups. 2001 Academic Press
1. Introduction Competition acts to reduce the growth rate of a population. This is achieved either by depleting a limited and essential resource (e.g. food, shelter, mates, etc.) or by interference where an increase in one population group reduces availability of the resource (for discussion, see Sutherland, 1996). Competition manifests itself both at the within species, intra-speci"c level and at the between species, inter-speci"c, level. In the current study, we are interested in the intra-speci"c situation for the special case where species live in social groupings and competition arises between these groups. Many, if not most, animal populations live in social groupings for part of the year (MacDonald, 1984). This imposes some form of heterogeneous community structure on populations and since many microparasitic diseases require contact (at some level) for transmission, it * Author to whom correspondence should be addressed. E-mail: [email protected] 0022}5193/01/180253#17 $35.00/0
appears that social groupings may play a crucial role in disease transmission across a population. Many questions are raised in this context*at a global level, these can be embedded in the most general questions, namely: 1. How do intra-speci"c between-group interactions a!ect disease transmission dynamics? 2. How does disease a!ect intra-speci"c interactions between social groupings? This paper presents a simple model to investigate disease transmission between two social groupings of the same species. In doing this, we attempt to broadly address these general questions concentrating mainly on the e!ects of contact between social groups and its e!ects on disease dynamics. The nature of social groupings and their interaction is of crucial importance in considering the spread of infectious diseases through a population. This idea was suggested in the early 1930s by Collias (1944), who suggested that the 2001 Academic Press
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aggression between groups of animals of the same species, by regulating population levels, helps cut down the incidence of disease. Recently, models have been developed to study the spread of an infectious disease through a nonhomogenous population, in which there are several segregated groups. In particular, theoretical studies of the spread of human diseases, particularly sexually transmitted or childhood diseases, have considered how the spread of disease is a!ected by the population structure on which it is superimposed (see Anderson & May, 1991; Mollison, 1995). Our approach is di!erent in that we are interested in the spread of infectious disease through a feral population which lives in social groupings. The importance of this work lies in the fact that many feral species are segregated at least for part of the year (MacDonald, 1983). Examples include foxes, badgers and wolves which live in social groupings throughout the year, and rabbits and deer which live in social groupings for part of the year. These species can act as a reservoir for disease which is then transmitted to domestic animals with implications for public health. Previously, work has been carried out to investigate how disease a!ects coexistence between species and ultimately the species richness of community. Considering the case of directly transmitted diseases, as opposed to free-living transmission, Holt & Pickering (1985), developed a simple formal theory for predicting species coexistence and exclusion when the only inter-speci"c interaction occurring is a shared infectious disease. Subsequently, Begon & Bowers (1994) extended the model to incorporate intra-speci"c competiton. Most recently, Bowers & Turner (1997), investigated a system of two hosts sharing a common disease with inter- as well as intraspeci"c host competition in order to "nd out how full competition and shared diseases combine in determining community structure. The work presented here complements that which came previously and focuses on how the contact between groups of the same species a!ects both population and disease dynamics. For simplicity, we consider the simplest model structure where there are two competing groups both susceptible to disease. We begin by describing the general model assumptions and structure. After brie#y
considering a single group model we proceed by looking at the separate cases where the only intra-speci"c between-group interaction is "rstly, disease transmission and secondly, competition. Finally, we consider the joint case with both disease transmission and competition and discuss the "ndings in an ecological context. Analysis of the above-mentioned models is presented in Appendices A}E. 2. The Model The simplest structure for a deterministic compartmental model of an infectious disease divides the population into two compartments, susceptible individuals and infected individuals. For simplicity, we employ this structure and consider two groups of conspeci"cs each with two compartments. Therefore, we consider a model with state variables S "the number of suscepG tible individuals in group i and I "the number G of infected individuals in group i, i"1, 2. The basic assumptions underlying the dynamics of the system are as follows: E
E
E
E
E
E
All individuals can give birth at a constant per capita birth rate a and all newborns are susceptible (i.e. there is no vertical transmission of disease or parasitic castration). The natural death rate for susceptibles is a constant per capita rate b. Infected individuals do not recover (certain death) and the per capita mortality rate for infecteds is c. Since this includes both naturaland disease-induced mortality, we have c'b. Disease is transmitted under the mass action assumption. That is, the per capita rate of infection for susceptibles is directly proportional to the number of infecteds and the transmission coe$cient within a group is the constant j. Contact between groups occurs at the constant per capita rate g. This contact can either result in the transmission of disease (with constant probability q), mortality (with constant probability p) or has no e!ect on the structure of the group (with constant probability 1!p!q). We assume that both groups are identical, i.e. demographic parameters are the same for each group.
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DISEASE AND SOCIAL GROUPS IN COMPETITION E
We disregard the within-group competition (Bowers & Turner, 1997), concentrating only on between-group competition e!ects.
These assumptions lead to the model form:
Here we concentrate on the spread of disease within one population group. Then eqn (1) is reduced to dS "a(S#I)!jSI!bS, dt
dS "a(S #I )!jS I !gqS I dt !np(S #I )S !bS ,
3.1. SINGLE GROUP DISEASE DYNAMICS
(1a)
dI "jS I #gqS I !npI (S #I )!cI , dt (1b)
dI "jSI!cI, dt
which admits two steady states (S*, I*). The trivial steady state exists for all parameter values and is stable provided a(b.
dS "a(S #I )!jS I !gqS I dt !np(S #I )S !bS ,
(2)
(1c)
dI "jS I #gqS I !npI (S #I )!cI . dt (1d) To completely specify the problem, we give initial conditions S (0)"S, I (0)"I, where G G G G i"1, 2. 3. Model Analysis In this section, we consider brie#y spread of disease within a single group and show that disease is capable of regulating individual population groups. We proceed with the two-group model distinguishing between the following cases. Firstly, we look at the model where there is no competition between the groups and the only intra-speci"c between-group interaction is the spread of disease. Secondly, in the absence of any disease, with competition being the only intraspeci"c between group interaction we show that coexistence between two groups is not possible. Finally, we consider the full model with both competition between the groups and disease transmission and investigate how competition in#uences disease dynamics. Details of how the steady states and criteria for their stability were determined are given in Appendices A}D.
(3)
In other words, if the death rate of susceptible individuals is greater than the birth rate of the whole population neither group is able to increase when both of them are rare. The nontrivial steady state
c c(a!b) , , j j(c!a)
is stable provided that eqn (3) does not hold and a(c. Hence, if the death rate of susceptible individuals is less than the birth rate of the whole population which, in turn, is less than the death rate of infected individuals the population will coexist with the disease. In the absence of disease, the population grows exponentially and we have shown that introduction of disease could regulate population growth within a group. Detailed calculations are presented in Appendix A. 3.2. TWO-GROUP MODEL: DISEASE TRANSMISSION ONLY
In the case where the only interaction between groups is through disease transmission model (1) reduces to dS "a(S #I )!jS I !gqS I !bS , dt dI "jS I #gqS I !cI , dt
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I. BEARDMORE AND K. A. J. WHITE
dS "a(S #I )!jS I !gqS I !bS , dt dI "jS I #gqS I !cI . dt
(4)
This formulation is equivalent to the one for symmetrical transmission in Holt & Pickering (1985) and here we derive the results for the special case where both groups belong to the same species and therefore assumed to have the same demographic parameters. We concentrate on the role of between-group contact parameter, g, rather than the e$ciency with which one group infects the other, nq/j which was used in Holt & Pickering (1985). We assume that the disease can regulate each group in the absence of the other. This condition is imposed to allow for groups to increase when both of them are rare. Stable coexistence between two groups with disease is possible provided j g( , q
(5)
and the population levels for both infected and susceptible individuals at the coexistence state are inversely proportional to the contact rate between the groups (see Appendix B.1). In other words, the stronger the contact between two groups the lower are the population levels at which the two groups coexist. In this case, even small contact between groups is less bene"cial to both groups than in the scenario (2), where there is no contact. From eqn (5) it follows that both groups coexist only for su$ciently small between-group contact rate. Increasing the contact parameter will result in one group excluding the other. Both exclusion of group 1 and exclusion of group 2 steady states are stable provided that the coexistence state is unstable. If eqn (5) does not hold, both groups can increase when rare and we have a priority e!ect in which either group can exclude the other, the initial condition determining the outcome. The assumption that both groups belong to the same species has enabled us to obtain an analytic expression for the coexistence steady state and its stability criterion (see Appendix B).
This was not possible for the case of two di!erent species sharing a disease (Holt & Pickering, 1985), due to the lack of symmetry of the system. Moreover, the analytic conditions for the stability of the coexistence state in our case agrees with the hypothesis posed in Holt & Pickering (1985): that the coexistence state is stable if no other steady state is. 3.3. TWO-GROUP MODEL: INTRA-SPECIFIC BETWEEN-GROUP COMPETITION ONLY
Here we consider the model where the only dynamics between groups is competition. Model (1) becomes dS "aS !gpS S !bS , dt
(6a)
dS "aS !gpS S !bS . dt
(6b)
The between-group coexistence occurs when the birth rate is greater than the death rate and both groups coexist at the same level. The coexistence steady state is a saddle point which means that the two groups can coexist only if initially they are both at the same level. The group which is initially smaller will become extinct while the initially larger group will experience unbounded exponential growth. The trivial state is stable when it is the only steady state available. Stability analysis is given in Appendix C. 3.4. TWO-GROUP MODEL: BETWEEN-GROUP COMPETITION AND INFECTIOUS INTERACTIONS
Finally, we consider the full model (1) with disease transmission and intra-speci"c betweengroup competition. We study the long-term behaviour of the model to determine conditions for disease persistence in either one or both of the groups. There are seven steady states, of which "ve have a distinct form. Details of the steady states and their stability are given in Appendix D; here we discuss them in more general terms. 3.4.1. ¹rivial Steady State This steady state exists for all parameter values. The groups will not simultaneously
DISEASE AND SOCIAL GROUPS IN COMPETITION
257
become extinct when rare provided that per capita growth rate of the population is greater than per capita susceptible death rate. This assumption is vital since assuming the converse will force both groups to die out even without the disease. That case is of little interest since we are interested in the control of populations through disease. During the course of this section we make the assumption that b(a(c. In other words, the disease can regulate each group in the absence of the other. 3.4.2. Group Exclusion This is the case where group 1 can exclude group 2, expression (D.3) and vice versa expression (D.4). If the contact rate between groups is su$ciently large, both groups are able to increase when rare. This is similar to the Lotka}Volterra competition model where, if the inter-speci"c parameter is large enough group exclusion steady states are both stable. As in Section 3.2 we have a priority e!ect. Figure 1(a) illustrates how initial condition determines the outcome. 3.4.3. Disease Eradication In this case the steady state s given by expression D.5. The groups are not able to coexist without the disease unless initially there are no infected individuals in both groups. In that case groups coexist at the same level. 3.4.4. Pathogen-mediated Group Coexistence at the Same ¸evel Groups coexist with each other and the disease for all parameter values, but this state is stable only for su$ciently small between-group contact rate. When there is no intra-speci"c betweengroup competition, coexistence is stable if no other steady state is (see Section 3.2). Introduction of intra-speci"c between-group competition gives rise to tristability. In other words, for certain values of between-group contact parameter, stable coexistence is possible together with stable group exclusion steady states (D.4) and (D.3).
FIG. 1. Basins of attraction: illustrates conditions on initial population of group 1 for determining the model outcome for the following parameter values a"0.6, b"0.1, c"0.6, p"0.1, q"0.4 and j"0.5: In (a) g"0.2, S (0)"1, I (0)"6 (b) g"0.4, S (0)"8, I (0)"8.
Figure 1(b) illustrates how in this case the initial condition determines the outcome. 3.4.5. Pathogen-mediated Group Coexistence at Di+erent ¸evels This state occurs as long as tristability is present, but this type of coexistence is always unstable. If there is no contact between the groups, each group is able to coexist with the disease at level (A.2) (see Section 3.1). For small contact
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I. BEARDMORE AND K. A. J. WHITE
FIG. E1. Time-dependent numerical solution of eqn (1). Full line denotes S , }. denotes I , 2 denotes S and } denotes I (8, 10, 4, 9) as an initial condition, solution. In (a) g"0.1, with (3, 5, 10, 12) as an initial condition, (b) g"0.2, with (c) g"0.2, with (8, 10, 2, 4) as an initial condition, (d) g"0.2, with (3, 5, 10, 12) as an initial condition, (e) g"0.4, with (8, 10, 4, 9) as an initial condition and (f) g"0.4, with (3, 5, 10, 12) as an initial condition.
parameters, the coexistence levels (D.6) are higher for susceptible but lower for infected individuals. Appendix D.2 shows that under certain conditions small contact between the groups is more bene"cial for susceptible and less bene"cial for infected individuals. In order to illustrate the dynamics of system (1), we plot the steady states of the susceptible [see Fig. 2(a)] and infected [see Fig. 2(b)] populations of group 1 while varying the contact rate
between groups (Doedel et al., 1997). The contact rate between groups g is chosen as a bifurcation parameter to re#ect similarities between this model and the simpler Lotka}Volterra competition model. In that as well as in our case, as the competition increases, the propensity for coexistence is reduced. Moreover, when g"0 our model system collapses to a pair of uncoupled ODE systems representing spread of disease within a group. Hence, using g as a bifurcation
DISEASE AND SOCIAL GROUPS IN COMPETITION
259
For large contact rates the model exhibits bistability with only two stable steady states (D.3) and (D.4), present. Depending on the initial population and the number of infected individuals only one group is able to coexist with the disease while the other one becomes extinct (for example, see Fig EI (e) & (f). Since the above results need to be true for the wide range of parameter values we carried out identical calculations "xing the contact rate parameter and varying in turn the probability of contact rate between groups ending in infection and then in death (p, q). The results obtained in these cases were of the same nature as those described here. 3.5. PERIODIC ORBITS
FIG. 2. Steady states (D.3), (D.4), (D.6), (D.11) and (D.12) shown in (a) for susceptible and in (b) for infected population in Group 1 are continued in parameter g which denotes the contact rate between groups. The rest of the parameters have the following values: a"0.6, b"0.1, c"0.6, p" 0.1, q"0.4 and j"0.5. Full line denotes where steady state is stable while dashed line denotes where steady state is unstable.
parameter not only allows us to compare our model solutions with the well-known competition models but also shows how combining groups both subject to disease when alone can a!ect disease transmission and population levels. Figure 2 shows that if the contact rate between groups is su$ciently small, system (1) has a stable steady state where there is group coexistence. In other words, for small contact rates between groups both populations will coexist with the disease (for example, see Fig EI (a)). For intermediate contact rates the model exhibits tristability where steady states (D.6), (D.3) and (D.4) are stable. Depending on the initial populations and distributions three outcomes are possible: group 1 becomes extinct while group 2 coexists with the disease; group 2 becomes extinct while group 1 coexists with the disease or both groups coexist with each other and the diseases (for example, see Fig EI (b), (c) & (d)).
A common phenomenon in disease dynamics is that of periodic epidemic outbreaks (Murray, 1993). However, in the case of model (1) we were unable to detect any biologically relevant periodic solutions. Similar behaviour was observed by Holt & Pickering (1985). When the contact rate between groups is zero we were able to prove analytically the non-existence of periodic solutions for all parameter values. Moreover in that case, regardless of the initial population and the number of infected individuals (I'0, i"1, 2) G both groups will coexist with each other and the disease. That is the coexistence steady state is globally attractive. The mathematical details are presented in Appendix E. Our working hypothesis is that when contact between groups is non-zero, the model system does not admit biologically relevant periodic solutions. Numerical simulations to date con"rm this hypothesis. 4. Discussion The social groupings of animals, formed for at least some part of the year, impose on populations a form of heterogeneous community structure. This can play a crucial role in disease transmission across a population since many microparasitic diseases require contact for transmission. The following questions can be raised at the global level. 1. How do social groups a!ect disease transmission dynamics? 2. How does disease a!ect interactions between social groupings?
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This paper presents a simple model to investigate disease transmission between two social groupings of the same species. In the absence of disease, where the only intra-speci"c interaction is between-group competition, the non-trivial steady state is unstable. Depending on initial conditions, one of the groups will survive to grow exponentially while the other dies out. Moreover, the initial advantage in population numbers determines the winner. In other words, the model always exhibits competitive exclusion in the absence of disease. To interpret this in ecological terms, we propose that the realized niches of each group are not su$ciently di!erentiated. In this paper, we have used identical model parameters to recognize the fact that each group is from the same species and that we have no dominance of groups. We chose to work with this form because it would allow us to explore at a base level how disease acts to regulate the between-group interaction. The tristability which we discussed in the results above arises from this underlying structure since it has not been previously noticed in the models of similar type (Bowers & Turner, 1997). Disease within a single group regulates the population dynamics and gives rise to a stable non-trivial steady state. When we include disease transmission between the two groups, the above-mentioned steady state of the type (S*, I*, 0, 0) becomes unstable when the contact between groups is su$ciently weak, g small. In its place, there is a stable non-trivial steady state in which both populations coexist with the disease. This coexistence is stable provided that group exclusion steady states are unstable. In other words, stable coexistence is possible only if both groups are able to increase when rare. (This is a special case of the model presented in Holt & Pickering, 1985.) Extending the model to include intra-speci"c between-group competition gives rise to similar behaviour for small contact rate. Again, if g is su$ciently small the only stable steady state is the coexistence of both groups with the disease. Essentially, the within-group disease transmission (analogous to intra-speci"c competition in the classic Lotka}Volterra competition model (see Murray, 1993 for details)) is larger than the between-group competitive regulation, thus preventing competitive exclusion. The realized
niches for each group are su$ciently di!erentiated. The di!erence from the case with no intra-speci"c between-group competition arises as g increases. As the interaction between groups increases, the realized niches are reduced and there are three possible stable steady states*coexistence between groups and between susceptible and infected individuals within the groups or competitive exclusion between the groups but persistence of disease within the surviving group (see Fig. 2). The outcome of the interaction depends on the initial conditions (see Fig. 1). Finally, when the interaction between groups becomes too large (g large), one group is driven to extinction, whereby the initial condition determines the outcome. These results are similar in nature to the classic results for the Lotka}Volterra competition model, see Murray (1993), except that this model can exhibit tristability. In the Lotka}Volterra model system, regulation of each species is through a density-dependent growth rate. Here, regulation of each group is also through a density-dependent process, namely disease. Disease divides the population into di!erent classes. In the case presented here, we consider the simplest scenario in which there are two such classes: susceptibles and infecteds. This process introduces an implicit time delay into the model as the regulation acts by infection reducing the susceptible population. We suggest that this time delay allows more #exibility in model outcome and hence, for intermediate levels of interaction, tristability as shown in Fig. 2. One particularly interesting feature of the model which we present is the following: if the disease transmission is the only intra-speci"c interaction between the groups the between}group contact rate g is always more detrimental to the stable coexistence steady-state levels for both susceptible and infected populations, than it would be if the groups existed on their own. However, if the intra-speci"c between-group competition is introduced when g is small, the steady-state population levels for the susceptible population when there is coexistence between social groups is higher, Fig. 2(a), than it would be if the groups existed on their own whilst corresponding levels for the infected population are lower, Fig. 2(b). Note that this is true for the particular case which
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DISEASE AND SOCIAL GROUPS IN COMPETITION
we present and more generally for other values of p and q. Recall that these represent the probabilities that between-group interaction result in mortality and disease, respectively. This suggests that between-group competition is one possible mechanism for a species to escape parasitism. However, even in this simpler case, these results are somewhat counter-intuitive. Between-group interaction has both the potential to directly reduce the susceptible population levels through inter-group competition mortality and to introduce more disease into the group through intergroup disease transmission. Both of these act to reduce population levels within the susceptible class. Neither of these processes dominate the interaction and it is the reduction in infected population [Fig. 2(b)] which in turn reduces mortality of the susceptible population due to disease. Note that the total population level for each group (S #I , i"1, 2) is reduced as would G G be expected of the inter-group competition. As we have demonstrated above, the current model structure does not allow for sustained population oscillations. This agrees with the proposition in Collias (1944) that aggression between species helps reduce the intensity of cyclic #uctuation of the population. Nevertheless, many diseases do exhibit cyclic behaviour (see, for example, Anderson & May, 1991) and it would certainly be interesting to see how such periodic behaviour could arise with multi-group interactions. The assumptions that both groups belong to the same species and therefore have identical model parameters, enabled us to conduct a detailed investigation of a four-dimensional ODE model which are often very di$cult to analyse. We were able to obtain a clear picture of the role of the between-group contact parameter in disease transmission. Using a relatively simple model we have shown that social groupings and intra-speci"c between-group competition are one possible mechanism of disease control. Since an extreme example of spatial segregation between groups of the same species is the phenomenon of territoriality, our model could be extended in a spatial context and used to investigate the interesting hypothesis that one purpose of territoriality may be to reduce the impact of infectious disease on a population.
Although this is essentially a theoretical study, we envisage its application to feral populations with wider implications, including issues in public health, since feral populations may be reservoirs for disease in domestic animals. REFERENCES ANDERSON, R. M. & MAY, R. M. (1991). Infectious Diseases of Humans: Dynamics and Control. New York: Oxford University Press. BEGON, M. & BOWERS, R. G. (1994). Host}host}pathogen models and microbial pest control: the e!ect of host self regulation. J. theor. Biol. 169, 275}287. BOWERS, R. G. & TURNER, J. (1997). Community structures and the interplay between interspeci"c infection and competition. J. theor. Biol. 187, 95}109. COLLIAS, N. E. (1944). Aggressive behaviour among vertebrate animals. Phys. Zool. 17, 83}123. DOEDEL, E. J., CHAMPNEYS, A. R., FAIRGRIEVE, T. F., KUZNETSOV, Y. A., SANDSTEDE, B. & WANG, X. (1997). A;¹O 97: Continuation and bifurcation Software for Ordinary Differential Equations (with HomCont), February. HOLT, R. D. & PICKERING, J. (1985). Infectious disease and species coexistence: a model of Lotka}Volterra form. Am. Nat. 126, 196}211. MACDONALD, D. (1984). ¹he Encyclopedia of Mammals. London: Allen and Unwin. MACDONALD, D. W. (1983). The ecology of carnivore social behaviour. Nature 301, 379}384. MOLLISON, D. (1995). Epidemic Models: their Structure and Relation to Data. Cambridge: Cambridge University Press. MURRAY, J. D. (1993). Mathematical Biology. 3rd Edn. Heidelberg: Springer-Verlag. SUTHERLAND, W. J. (1996). From Individual Behaviour to Population Ecology. Oxford: Oxford University Press.
APPENDIX A Single Group Dynamics Model Analysis For eqn (2) to be at steady state we must have 0"a(S #I )!jS I !bS , 0"jS I !cI .
(A.1)
Solving the above equation gives us the trivial steady state (0, 0) which exists for all parameter values and the non-trivial steady state
c c(a!b) , , j j(c!a)
(A.2)
which is biologically relevant provided b(a(c or c(a(b.
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I. BEARDMORE AND K. A. J. WHITE A.1. LINEAR STABILITY ANALYSIS OF THE
From eqn (B.5b) we get S "c/j. Substitu ting into eqn (B.5a) gives a non-trivial steady state
SINGLE GROUP SYSTEM
The Jacobian J of system (A.1) is given by
J"
a!b!jI a!jS jI jS !c
evaluated at the steady state (S*, I*). For the non-trivial steady state we have
c(a!b) a!b! c c(a!b) (c!a) J , " j j(c!a) c(a!b) (c!a)
a!c 0
and the trivial steady state is unstable provided a!b'0.
B.1. FINDING STEADY STATES
For system (4) to be at steady state we must have
(a!b) S "I , c!a
(B.8a)
(a!b) S "I . c!a
(B.8b)
Substituting eqn (B.8a) into eqn (B.2) and eqn (B.8b) into eqn (B.4) we get the coexistence steady state
c c(a!b) c c(a!b) , , , , j#gq (j#gq)(c!a) j#gq (j#gq)(c!a)
0"a(S #I )!jS I !gqS I !bS , (B.1) 0"jS I #gqS I !cI ,
(B.2)
0"a(S #I )!jS I !gqS I !bS , (B.3) 0"jS I #gqS I !cI .
(B.4)
(B.7)
The coexistence steady state is obtained by adding eqn (B.1) to eqn (B.2) and eqn (B.3) to eqn (B.4) to get
Two-Group Model: Disease Transmission Only
c c(a!b) 0, 0, , . j j(a!c)
a!b a 0 !c
APPENDIX B
(B.6)
The above steady state is biologically relevant provided b(a(c or c(a(b. Due to the model symmetry we also have a non-trivial steady state
.
Hence, non-trivial steady state is stable provided b(a(c. For the trivial steady state the Jacobian is J(0, 0)"
c c(a!b) , ,0, 0 . j j(a!c)
(B.9) which is biologically relevant provided b(a(c or c(a(b. At g"j/q, substituting eqn (B.8a) into eqn (B.2) and eqn (B.8b) into eqn (B.4) we arrive at
The trivial steady state (0, 0, 0, 0) exists for all parameter values. Setting S "0 and I "0, we get a reduced second order system with the steady-state equations
c(a!b) . I #I " j(c!a)
0"a(S #I )!jS I !bS ,
(B.5a)
0"jS I !cI .
(B.5b)
Combining eqn (B.10) together with eqns (B.8a) and (B.8b) gives a conituum of steady states forming a vertical bifurcation from eqn (B.9)
(B.10)
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DISEASE AND SOCIAL GROUPS IN COMPETITION
which holds provided that b(a(c and g' j/q. The equivalent conditions holds for stability of the steady state (0, 0, c/j, c(a!b)/j(c!a)). Finally, for the coexistence steady state we have
B.2. STABILITY
The Jacobian for system (4) is given by J"
a!jI !gqI !b a!jS 0 !gqS jI #gqI jS !c 0 gqS 0 !gqS a!jI !gqI !b a!jS 0 gqS jI #gqI jS !c
evaluated at steady state (S*, I*, S*, I*). For the trivial steady state we have the following Jacobian:
J
c c(a!b) c c(a!b) , , , j#gq (j#gq)(c!a) j#gq (j#gq)(c!a)
J J , J J
"
where
a(a!b) jc ! a! c!a j#gq J " , c(a!b) gqc ! c!a j#gq
a!b a 0 0 0 !c 0 0 J(0, 0, 0, 0)" . 0 0 a a!b 0 0 0 !c The eigenvalues of the above matrix are a " a!b and a "!c. Hence, the trivial steady state is stable when a!b(0. For group 2 exclusion steady state we have J
c c(a!b) J J , , , 0, 0 " j j(c!a) 0 ; J
where
J "
(a!b)!gq
J "
a!b a!c c!a , a!b 0 c c!a
a!b!c
c a!b a j c!a
c gq a!bc!a j
.
!c
Then
gq !1 '0, j
gqc(a!b) tr(J )"(a!b!c)! (0, j(c!a)
a(a!b) ! !a a!c c!a det(J!aI)"det c(a!b) !a c!a
a(a!b) jc gqc ! !a a! # c!a j#gq j#gq ;det . c(a!b) cgq !2 !a c!a j#gq
Therefore, the eigenvalues a , i3+1, 2, 3, 4, satisfy G a#a
det(J )"!c(b!a)'0,
det(J )"c(a!b)
gqc 0 ! j#gq . J " gqc 0 j#gq
The conditions for stability of this steady state are
a!b (0, tr(J )"a c!a
a#a
;
a(a!b) #c(a!b)"0, c!a
a(a!b) cgq c(a!b) #2 # c!a j#gq c!a
1 (2agq#jc!gqc)!a "0. j#gq
Hence, the coexistence steady state is stable when b(a(c and g(j/q.
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I. BEARDMORE AND K. A. J. WHITE
APPENDIX C
APPENDIX D
Two-Group Model : Intra-Speci5c Between-Group Competition Only
The Complete Model Analysis
C.1. FINDING STEADY STATES
For system (1) to be at steady state we must have
For system (6) to be at steady state we must have
0"a(S #I )!jS I !gqS I
0"(a!b)S !gpS S ,
!gp(S #I )S !bS
0"(a!b)S !gpS S .
" : F (S , I , S , I ),
The trivial steady state (0, 0) exists for all parameter values. The coexistence steady state of the form
D.1. FINDING STEADY STATES
a!b a!b , gp gp
(C.1)
0"jS I #gqS I !gpI (S #I )!cI " : F (S , I , S , I ), 0"a(S #I )!jS I !gqS I !gp(S #I )S !bS
exists and is biologically relevant whenever a!b'0.
" : F (S , I , S , I ), 0"jS I #gqS I !gpI (S #I )!cI
C.2. STABILITY
" : F (S , I , S , I ).
The Jacobian for system (6) is given by
J"
(a!b)!gpS !gpS , !gpS (a!b)!gpS
evaluated at steady state (S*, S*). For the trivial steady state we have the following Jacobian:
J(0, 0)"
a!b 0 . 0 a!b
The eigenvalues of the above matrix are a " a!b. Hence, the trivial steady state is stable when a!b(0. For the coexistence steady state the Jacobian is
a!b a!b 0 b!a , " . J gp b!a 0 gp The eigenvalues of the above matrix are a "!(a!b) and a "a!b and hence the coexistence steady state is a saddle point.
(D.1)
The trivial steady state (0, 0, 0, 0) exists for all parameter values. Setting S "0 and I "0, we get a reduced second-order system with the steady-state equation 0"a(S #I )!jS I !bS ,
(D.2a)
0"jS I !cI .
(D.2b)
From eqn (D.2b) we get S "c/j. Substituting into eqn (D.2a) gives a non-trivial steady state
c b!a c , , 0, 0 , j a!c j
(D.3)
provided aOc and b(a(c. In the modelling process, we have assumed that both groups have the same characteristics, hence, we have a symmetrical system (1). Thus, the presence of a particular steady state of some form (x, y, 0, 0) will automatically result in a symmetrical steady state of the form (0, 0, x, y).
265
DISEASE AND SOCIAL GROUPS IN COMPETITION
Hence, from the steady state (D.3) we can immediately obtain another non-trivial steady state
c b!a c 0, 0, , . j a!c j
(S**, I**, S**, I**) (D.4)
Another steady state can be obtained by setting I "0 and I "0 in eqn (D.1) in which case we get
a!b a!b , 0, ,0 gp gp
(D.5)
provided a'b. Continuing with the idea of symmetry, suppose now that we have another non-trivial steady state in the form (S*, I*, S*, I*).
a non-trivial steady state of the form
(D.6)
and immediately, using the assumption that both groups are identical we obtain a symmetrical steady state (S**, I**, S**, I**).
D.2. THE ROLE OF g
We now consider the coexistence steady state (S*, I*, S*, I*) as a function of between-group contact parameter g. In Appendix D.1, we have found that k#(k!4ac(j#gq) , S*" 2(j#gq)
0"a(S*#I*)!jS*I*!gqS*I*
(D.7b)
where k"c(j#gq#gp)#gp(a!b)#a(j #gq !gp). It is clear that S* (g)Pc/j and I* (g)P (c/j) (a!b)/(c!a) as gP0. We want to show that for small g
(D.8)
c S* (g)' , j
giving [from eqn (D.7b)] S* (j#gq!gp)!c I* " . gp
S* (j#gq!gp)!c , I* " gp
(D.7a)
0"jS*I*#gqS* I*!gpI* (S*#I*)!cI*
c (a!b) I* (g)( . j (c!a)
Substituting eqn (D.8) into eqn (D.7a) rearrange to get a quadratic equation for S* !(j#gq)S*#S* (c(j#gq#gp)#gp(a!b) #a(j#gq!gp))!ac"0,
(D.9)
which gives only one realistic solution when combined with (D.8), namely k#(k!4ac(j#gq) , S* " 2(j#gq)
(D.12)
However, it is not possible to obtain explicit forms for these steady-state values.
Substituting eqn (D.6) into eqn (D.1) we get
!gp(S*#I*)S*!bS*,
(D.11)
(D.10)
where k"c(j#gq#gp)#gp(a!b)#a(j#gq !gp). For certain parameter values there exists
We consider dS* dg
d k#(k!4ac(j#gq) " dg 2(j#gq) g"0
g"0
ja " (qc#pc#pb!qa!2pa)'0 (c!a) provided qc#pc#pb!qa!2pa'0.
(D.13)
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I. BEARDMORE AND K. A. J. WHITE
steady state is stable when a!b(0. For the steady state (D.3) we have the Jacobian
Also, dI* dg
d S* (j#gq)!S*gp!c " gp dg
g"0
d k#(k!4ac(j#gq)!2c(gp) " 2gp(j#gq) dg
dS* ! dg
dS* "!j(c!a)p! dg g"0
g"0
(0
g"0
c c !g (q#p) j j M " c b!a b!a g q!p !gpc j a!c a!c
The Jacobian J of system (D.1) at the steadystate level is (*F /*x ), where x "S , x "I , G G x "S , x "I . Hence,
where
(D.14)
J "
a!jI !gqI !gp(S #I )!b a!jS , jI #gqI jS !gp(S #I )!c
!gpS !gqS !gpS , J " !gpI gqS !gpI
M "
a!b!g
evaluated at steady state (S*, I*, S*, I*). For the trivial steady state we have the following Jacobian:
c b!a b!c q #p j a!c a!c c (b!a) gq j (a!c)
,
a !gp
c (b!c) !c j (a!c)
a!b tr(M )"a (0, a!c
a!jI !gqI !gp(S #I )!b a!jS , jI #gqI jS !gp(S #I )!c
det(M )"!c(b!a)'0,
J "
a!b a 0 0 0 !c 0 0 . J " 0 0 a!b a 0 0 0 !c
!gp
and 0 ; is just a 2;2 zero matrix. The eigen values of J are exactly the eigenvalues of M and M . We "rst consider M . In order for eigenvalues of M to be negative
!gpS !gqS !gpS , J " !gpI gqS !gpI
b!a a!c a!c , b!a 0 c a!c
a!b!c
M "
D.3. STABILITY
J J , J J
(D.16)
where
whenever (D.13) holds.
J"
M M , J " 0 ; M
g"0
(D.17a) (D.18b)
which requires b(a(c.
(D.19)
Next, we need to consider matrix M . M "
(D.15)
The eigenvalues of the above matrix are a "a!b and a "!c. Hence, the trivial
a a!b!gqI*!gp(S*#I*) gqI* !gp(S*#I*)!c,
recalling that S*"c/j and I*"c(b!a)/ j (a!c). As in the case of M in order for the
DISEASE AND SOCIAL GROUPS IN COMPETITION
eigenvalues of M to be negative we need the two conditions det(M ) :"(!gp(S*#I*)!c)(a!b!gqI* !gp(S*#I*))!agqI*'0, tr(M ) :"a!b!c!gqI*!2gp(S*#I*)(0. In order for the above to hold, recalling that S '0, I '0 and assumption (D.19) it is su$ cient that g'gJ ,
(D.20)
where gJ is a positive solution of a quadratic gJ p
c(c!b) (q(a!b)#p(c!b)) j(c!a)
#gJ
a!b!c c!b (q(a!b)!p j c!a
(D.21)
!c(a!b)"0.
Hence, steady state (D.3) will be stable if eqns (D.19) and (D.20) hold. For steady state (D.4), the Jacobian will be
M 0 ; , J " M M where the block entries of the matrix are the same as we have de"ned for J . In this case J is a lower triangular matrix. Equivalent to the case for J , the eigenvalues of J will be the eigen values of M and the eigenvalues of M . Hence, the condition for stability is the same. For steady state (D.5) we have
J "
a!j
0 0
j
gp
gp
!a#b!c a!b !a#b
gp a!b gq gp
a!b !a#b gp a!b gq gp a!b a!j gp a!b j !a#b!c gp
b!a !gq
a!b
b!a !gq 0
a!b
0 0 0
.
267
The eigenvalues of the above matrix are a " !(a!b), a "a!b, a "(j(a!b)/gp)!a# b!c#q(a!b)/p and a "j(a!b)/gp!a# b!c!q(a!b)/p. Hence, if the steady state is biologically relevant, i.e. a'b it is unstable. In principle, we could determine the local stability of the steady state (D.6) by applying, for instance, the Routh}Hurwitz criteria to the Jacobian matrix (D.14). In practice, for a "xed parameter values a, b, c, p and q with b(a(c we can show that there exists gL (a, b, c, p, q), gJ (gL (R, where gJ is a positive solution of quadratic (D.21), such that steady states (D.8) and (D.10) are stable if g(gL (see Doedel et al., 1997 and Fig. 2.) We can also show that the steady states (D.11) and (D.12) exist for gJ (g(gL , in other words when steady states (D.6), (D.4) and (D.3) are locally stable. Whenever they exist steady states (D.11) and (D.12) are always unstable. Using standard bifurcation theory we can demonstrate the existence of gL (a, b, c, p, q) s.t. gJ (gL (R, for nearby parameter values, upto genericity assumptions. Note that although pitchfork bifurcations are not preserved under generic perturbations, they are preserved under symmetric perturbations. Since our system posseses re#ective symmetry about the two groups, we except the pitchfork bifurcation to persist. APPENDIX E Non-existence of Periodic Orbits When g"0, system (1) decouples to a pair of SI systems dS G"a(S #I )!jS I !bS , G G G G G dt dI G"jS I !cI , G G G dt
(E.1)
where i"1, 2, with one locally stable steady state (c/j, ((a!b)/(c!a)) c/j, c/j, ((a!b)/(c!a))c/j). For ease of computation, we set u "I ! G G ((a!b)/(c!a) c/j, v "S !c/j, i"1, 2 so that G G
268
I. BEARDMORE AND K. A. J. WHITE
system (E.1) becomes
From eqn (E.4) it follows that ¸ is a decreasing function bounded below and hence has a limit as tPR
du G"bv #ju v , G G G dt
d¸ P0, tPR. dt From eqn (E.3) and for i"1, 2 N
dv G"!cu !dv !ju v , (E.2) G G G G dt where i"1, 2, b"c(a!b)/(c!a), c"c!a and d"a(a!b)/(c!a) and steady state u "v "u "v "0. We propose
ju jv ¸"j(u #v )!b ln 1# !c ln 1# b c
ju jv #j(u #v )!b ln 1# !c ln 1# b c
(E.3) as a Lyapunov function for system (E.2) which is true provided jdv jdv d¸ ! (0. "! c#jv c#jv dt
(E.4)
When c#jv '0 and c#jv '0, i.e. S 'a/j and S 'a/j, eqn (E.4) will hold and hence we can now divide the space into two regions (Fig. E2) as follows: Region I: I '0, I '0, S 'a/j and S 'a/j 8 b#ju , c#jv '0 for i"1, 2. G G Since at the region boundaries b u P! N ¸P!b ln0P#R, G j c v P! N ¸P!c ln0P#R, G j
ju u PRN¸"ju !b ln 1# G +ju P#R, G G G b jv v PRN¸"jv !c ln 1# G +jv P#R, G G G c where i"1, 2, we can conclude that (u (t), v (t), u (t), v (t)) cannot escape Region I. Also ¸(u , v , u , v ) has a minimum at ¸(0, 0, 0, 0)"0.
(E.5)
N v P0, tPR, G N
dv GP0, dt
tPR,
N !cu !bv #ju v P0, tPR, G G G G N !cu P0, G
tPR,
N u P0, tPR. G Hence (u , v , u , v ) P (0, 0, 0, 0) as tPR, in other words (0, 0, 0, 0) is a globally stable steady state in Region I. Region II: I '0, I '0, 0(S )a/j and 0(S )a/j 8 b#ju '0*c#jv and v * G G G !c/j for i"1, 2. From system (E.2) it follows that for i"1, 2 dv dc G"!(c#jv )u !bv *!bv * . G G G G dt j
(E.6)
Hence, dc c dc v (t)*v (0)# t*! # t. G G j j j
(E.7)
Using eqn (E.7) we can conclude that at some time t*, v (t*) will be outside Region II G (c!a) , 8 v (t*)'! G j (c!a) c dc , 8! # t*'! j j j a 8 t*' . dc From the above analysis we can conclude that starting in Region II, (u , v , u , v ) will leave it after a time t*)a/dc and enter Region I (Fig. E2). Since we have also proved that Region I
DISEASE AND SOCIAL GROUPS IN COMPETITION
269
FIG. E2. The S '0, I '0, S '0 and I '0 space is divided into two regions. Since g"0 four-dimensional system can be decoupled into two two-dimensional subsystems. In (a) S '0, I '0 while in (b) S '0 and I '0. The regions are: I '0, I '0, 0(S )a/j and 0(S )a/j. Region I: I '0, I '0, S 'a/j and S 'a/j and Region II:
is invariant and that (0, 0, 0, 0) is a globally stable steady state in that region it follows that (0, 0, 0, 0) is globally stable in the whole space
I , I , S , S '0. Therefore, system (E.2) does not have periodic solution for any parameter values.
Spreading Disease through Social Groupings in Competition IVANA BEARDMORE*
AND
K. A. JANE WHITE
Department of Mathematical Sciences, ;niversity of Bath, Bath BA2 7A>, ;.K. (Received on 31 March 2001, Accepted in revised form on 11 June 2001)
Many animal populations live in social groups which avoid contact with other conspeci"c groups for at least part of the year. This may give rise to competition between groups for items such as shelter, land and mates. We couple intra-speci"c group competition with disease dynamics to investigate how infectious diseases may spread through population subgroups, particularly with reference to the contact rates between groups. Our model uses a nonlinear systems of ODEs for which steady-state analysis is carried out in the simplest two-group system. This indicates that coexistence of social groups is possible with the disease or that competitive exclusion occurs with one group dying out whilst the other retains disease. Moreover, we show that in certain circumstances the model can exhibit multistability and we discuss the ecological implications of this result in relation to contact between social groups. 2001 Academic Press
1. Introduction Competition acts to reduce the growth rate of a population. This is achieved either by depleting a limited and essential resource (e.g. food, shelter, mates, etc.) or by interference where an increase in one population group reduces availability of the resource (for discussion, see Sutherland, 1996). Competition manifests itself both at the within species, intra-speci"c level and at the between species, inter-speci"c, level. In the current study, we are interested in the intra-speci"c situation for the special case where species live in social groupings and competition arises between these groups. Many, if not most, animal populations live in social groupings for part of the year (MacDonald, 1984). This imposes some form of heterogeneous community structure on populations and since many microparasitic diseases require contact (at some level) for transmission, it * Author to whom correspondence should be addressed. E-mail: [email protected] 0022}5193/01/180253#17 $35.00/0
appears that social groupings may play a crucial role in disease transmission across a population. Many questions are raised in this context*at a global level, these can be embedded in the most general questions, namely: 1. How do intra-speci"c between-group interactions a!ect disease transmission dynamics? 2. How does disease a!ect intra-speci"c interactions between social groupings? This paper presents a simple model to investigate disease transmission between two social groupings of the same species. In doing this, we attempt to broadly address these general questions concentrating mainly on the e!ects of contact between social groups and its e!ects on disease dynamics. The nature of social groupings and their interaction is of crucial importance in considering the spread of infectious diseases through a population. This idea was suggested in the early 1930s by Collias (1944), who suggested that the 2001 Academic Press
254
I. BEARDMORE AND K. A. J. WHITE
aggression between groups of animals of the same species, by regulating population levels, helps cut down the incidence of disease. Recently, models have been developed to study the spread of an infectious disease through a nonhomogenous population, in which there are several segregated groups. In particular, theoretical studies of the spread of human diseases, particularly sexually transmitted or childhood diseases, have considered how the spread of disease is a!ected by the population structure on which it is superimposed (see Anderson & May, 1991; Mollison, 1995). Our approach is di!erent in that we are interested in the spread of infectious disease through a feral population which lives in social groupings. The importance of this work lies in the fact that many feral species are segregated at least for part of the year (MacDonald, 1983). Examples include foxes, badgers and wolves which live in social groupings throughout the year, and rabbits and deer which live in social groupings for part of the year. These species can act as a reservoir for disease which is then transmitted to domestic animals with implications for public health. Previously, work has been carried out to investigate how disease a!ects coexistence between species and ultimately the species richness of community. Considering the case of directly transmitted diseases, as opposed to free-living transmission, Holt & Pickering (1985), developed a simple formal theory for predicting species coexistence and exclusion when the only inter-speci"c interaction occurring is a shared infectious disease. Subsequently, Begon & Bowers (1994) extended the model to incorporate intra-speci"c competiton. Most recently, Bowers & Turner (1997), investigated a system of two hosts sharing a common disease with inter- as well as intraspeci"c host competition in order to "nd out how full competition and shared diseases combine in determining community structure. The work presented here complements that which came previously and focuses on how the contact between groups of the same species a!ects both population and disease dynamics. For simplicity, we consider the simplest model structure where there are two competing groups both susceptible to disease. We begin by describing the general model assumptions and structure. After brie#y
considering a single group model we proceed by looking at the separate cases where the only intra-speci"c between-group interaction is "rstly, disease transmission and secondly, competition. Finally, we consider the joint case with both disease transmission and competition and discuss the "ndings in an ecological context. Analysis of the above-mentioned models is presented in Appendices A}E. 2. The Model The simplest structure for a deterministic compartmental model of an infectious disease divides the population into two compartments, susceptible individuals and infected individuals. For simplicity, we employ this structure and consider two groups of conspeci"cs each with two compartments. Therefore, we consider a model with state variables S "the number of suscepG tible individuals in group i and I "the number G of infected individuals in group i, i"1, 2. The basic assumptions underlying the dynamics of the system are as follows: E
E
E
E
E
E
All individuals can give birth at a constant per capita birth rate a and all newborns are susceptible (i.e. there is no vertical transmission of disease or parasitic castration). The natural death rate for susceptibles is a constant per capita rate b. Infected individuals do not recover (certain death) and the per capita mortality rate for infecteds is c. Since this includes both naturaland disease-induced mortality, we have c'b. Disease is transmitted under the mass action assumption. That is, the per capita rate of infection for susceptibles is directly proportional to the number of infecteds and the transmission coe$cient within a group is the constant j. Contact between groups occurs at the constant per capita rate g. This contact can either result in the transmission of disease (with constant probability q), mortality (with constant probability p) or has no e!ect on the structure of the group (with constant probability 1!p!q). We assume that both groups are identical, i.e. demographic parameters are the same for each group.
255
DISEASE AND SOCIAL GROUPS IN COMPETITION E
We disregard the within-group competition (Bowers & Turner, 1997), concentrating only on between-group competition e!ects.
These assumptions lead to the model form:
Here we concentrate on the spread of disease within one population group. Then eqn (1) is reduced to dS "a(S#I)!jSI!bS, dt
dS "a(S #I )!jS I !gqS I dt !np(S #I )S !bS ,
3.1. SINGLE GROUP DISEASE DYNAMICS
(1a)
dI "jS I #gqS I !npI (S #I )!cI , dt (1b)
dI "jSI!cI, dt
which admits two steady states (S*, I*). The trivial steady state exists for all parameter values and is stable provided a(b.
dS "a(S #I )!jS I !gqS I dt !np(S #I )S !bS ,
(2)
(1c)
dI "jS I #gqS I !npI (S #I )!cI . dt (1d) To completely specify the problem, we give initial conditions S (0)"S, I (0)"I, where G G G G i"1, 2. 3. Model Analysis In this section, we consider brie#y spread of disease within a single group and show that disease is capable of regulating individual population groups. We proceed with the two-group model distinguishing between the following cases. Firstly, we look at the model where there is no competition between the groups and the only intra-speci"c between-group interaction is the spread of disease. Secondly, in the absence of any disease, with competition being the only intraspeci"c between group interaction we show that coexistence between two groups is not possible. Finally, we consider the full model with both competition between the groups and disease transmission and investigate how competition in#uences disease dynamics. Details of how the steady states and criteria for their stability were determined are given in Appendices A}D.
(3)
In other words, if the death rate of susceptible individuals is greater than the birth rate of the whole population neither group is able to increase when both of them are rare. The nontrivial steady state
c c(a!b) , , j j(c!a)
is stable provided that eqn (3) does not hold and a(c. Hence, if the death rate of susceptible individuals is less than the birth rate of the whole population which, in turn, is less than the death rate of infected individuals the population will coexist with the disease. In the absence of disease, the population grows exponentially and we have shown that introduction of disease could regulate population growth within a group. Detailed calculations are presented in Appendix A. 3.2. TWO-GROUP MODEL: DISEASE TRANSMISSION ONLY
In the case where the only interaction between groups is through disease transmission model (1) reduces to dS "a(S #I )!jS I !gqS I !bS , dt dI "jS I #gqS I !cI , dt
256
I. BEARDMORE AND K. A. J. WHITE
dS "a(S #I )!jS I !gqS I !bS , dt dI "jS I #gqS I !cI . dt
(4)
This formulation is equivalent to the one for symmetrical transmission in Holt & Pickering (1985) and here we derive the results for the special case where both groups belong to the same species and therefore assumed to have the same demographic parameters. We concentrate on the role of between-group contact parameter, g, rather than the e$ciency with which one group infects the other, nq/j which was used in Holt & Pickering (1985). We assume that the disease can regulate each group in the absence of the other. This condition is imposed to allow for groups to increase when both of them are rare. Stable coexistence between two groups with disease is possible provided j g( , q
(5)
and the population levels for both infected and susceptible individuals at the coexistence state are inversely proportional to the contact rate between the groups (see Appendix B.1). In other words, the stronger the contact between two groups the lower are the population levels at which the two groups coexist. In this case, even small contact between groups is less bene"cial to both groups than in the scenario (2), where there is no contact. From eqn (5) it follows that both groups coexist only for su$ciently small between-group contact rate. Increasing the contact parameter will result in one group excluding the other. Both exclusion of group 1 and exclusion of group 2 steady states are stable provided that the coexistence state is unstable. If eqn (5) does not hold, both groups can increase when rare and we have a priority e!ect in which either group can exclude the other, the initial condition determining the outcome. The assumption that both groups belong to the same species has enabled us to obtain an analytic expression for the coexistence steady state and its stability criterion (see Appendix B).
This was not possible for the case of two di!erent species sharing a disease (Holt & Pickering, 1985), due to the lack of symmetry of the system. Moreover, the analytic conditions for the stability of the coexistence state in our case agrees with the hypothesis posed in Holt & Pickering (1985): that the coexistence state is stable if no other steady state is. 3.3. TWO-GROUP MODEL: INTRA-SPECIFIC BETWEEN-GROUP COMPETITION ONLY
Here we consider the model where the only dynamics between groups is competition. Model (1) becomes dS "aS !gpS S !bS , dt
(6a)
dS "aS !gpS S !bS . dt
(6b)
The between-group coexistence occurs when the birth rate is greater than the death rate and both groups coexist at the same level. The coexistence steady state is a saddle point which means that the two groups can coexist only if initially they are both at the same level. The group which is initially smaller will become extinct while the initially larger group will experience unbounded exponential growth. The trivial state is stable when it is the only steady state available. Stability analysis is given in Appendix C. 3.4. TWO-GROUP MODEL: BETWEEN-GROUP COMPETITION AND INFECTIOUS INTERACTIONS
Finally, we consider the full model (1) with disease transmission and intra-speci"c betweengroup competition. We study the long-term behaviour of the model to determine conditions for disease persistence in either one or both of the groups. There are seven steady states, of which "ve have a distinct form. Details of the steady states and their stability are given in Appendix D; here we discuss them in more general terms. 3.4.1. ¹rivial Steady State This steady state exists for all parameter values. The groups will not simultaneously
DISEASE AND SOCIAL GROUPS IN COMPETITION
257
become extinct when rare provided that per capita growth rate of the population is greater than per capita susceptible death rate. This assumption is vital since assuming the converse will force both groups to die out even without the disease. That case is of little interest since we are interested in the control of populations through disease. During the course of this section we make the assumption that b(a(c. In other words, the disease can regulate each group in the absence of the other. 3.4.2. Group Exclusion This is the case where group 1 can exclude group 2, expression (D.3) and vice versa expression (D.4). If the contact rate between groups is su$ciently large, both groups are able to increase when rare. This is similar to the Lotka}Volterra competition model where, if the inter-speci"c parameter is large enough group exclusion steady states are both stable. As in Section 3.2 we have a priority e!ect. Figure 1(a) illustrates how initial condition determines the outcome. 3.4.3. Disease Eradication In this case the steady state s given by expression D.5. The groups are not able to coexist without the disease unless initially there are no infected individuals in both groups. In that case groups coexist at the same level. 3.4.4. Pathogen-mediated Group Coexistence at the Same ¸evel Groups coexist with each other and the disease for all parameter values, but this state is stable only for su$ciently small between-group contact rate. When there is no intra-speci"c betweengroup competition, coexistence is stable if no other steady state is (see Section 3.2). Introduction of intra-speci"c between-group competition gives rise to tristability. In other words, for certain values of between-group contact parameter, stable coexistence is possible together with stable group exclusion steady states (D.4) and (D.3).
FIG. 1. Basins of attraction: illustrates conditions on initial population of group 1 for determining the model outcome for the following parameter values a"0.6, b"0.1, c"0.6, p"0.1, q"0.4 and j"0.5: In (a) g"0.2, S (0)"1, I (0)"6 (b) g"0.4, S (0)"8, I (0)"8.
Figure 1(b) illustrates how in this case the initial condition determines the outcome. 3.4.5. Pathogen-mediated Group Coexistence at Di+erent ¸evels This state occurs as long as tristability is present, but this type of coexistence is always unstable. If there is no contact between the groups, each group is able to coexist with the disease at level (A.2) (see Section 3.1). For small contact
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I. BEARDMORE AND K. A. J. WHITE
FIG. E1. Time-dependent numerical solution of eqn (1). Full line denotes S , }. denotes I , 2 denotes S and } denotes I (8, 10, 4, 9) as an initial condition, solution. In (a) g"0.1, with (3, 5, 10, 12) as an initial condition, (b) g"0.2, with (c) g"0.2, with (8, 10, 2, 4) as an initial condition, (d) g"0.2, with (3, 5, 10, 12) as an initial condition, (e) g"0.4, with (8, 10, 4, 9) as an initial condition and (f) g"0.4, with (3, 5, 10, 12) as an initial condition.
parameters, the coexistence levels (D.6) are higher for susceptible but lower for infected individuals. Appendix D.2 shows that under certain conditions small contact between the groups is more bene"cial for susceptible and less bene"cial for infected individuals. In order to illustrate the dynamics of system (1), we plot the steady states of the susceptible [see Fig. 2(a)] and infected [see Fig. 2(b)] populations of group 1 while varying the contact rate
between groups (Doedel et al., 1997). The contact rate between groups g is chosen as a bifurcation parameter to re#ect similarities between this model and the simpler Lotka}Volterra competition model. In that as well as in our case, as the competition increases, the propensity for coexistence is reduced. Moreover, when g"0 our model system collapses to a pair of uncoupled ODE systems representing spread of disease within a group. Hence, using g as a bifurcation
DISEASE AND SOCIAL GROUPS IN COMPETITION
259
For large contact rates the model exhibits bistability with only two stable steady states (D.3) and (D.4), present. Depending on the initial population and the number of infected individuals only one group is able to coexist with the disease while the other one becomes extinct (for example, see Fig EI (e) & (f). Since the above results need to be true for the wide range of parameter values we carried out identical calculations "xing the contact rate parameter and varying in turn the probability of contact rate between groups ending in infection and then in death (p, q). The results obtained in these cases were of the same nature as those described here. 3.5. PERIODIC ORBITS
FIG. 2. Steady states (D.3), (D.4), (D.6), (D.11) and (D.12) shown in (a) for susceptible and in (b) for infected population in Group 1 are continued in parameter g which denotes the contact rate between groups. The rest of the parameters have the following values: a"0.6, b"0.1, c"0.6, p" 0.1, q"0.4 and j"0.5. Full line denotes where steady state is stable while dashed line denotes where steady state is unstable.
parameter not only allows us to compare our model solutions with the well-known competition models but also shows how combining groups both subject to disease when alone can a!ect disease transmission and population levels. Figure 2 shows that if the contact rate between groups is su$ciently small, system (1) has a stable steady state where there is group coexistence. In other words, for small contact rates between groups both populations will coexist with the disease (for example, see Fig EI (a)). For intermediate contact rates the model exhibits tristability where steady states (D.6), (D.3) and (D.4) are stable. Depending on the initial populations and distributions three outcomes are possible: group 1 becomes extinct while group 2 coexists with the disease; group 2 becomes extinct while group 1 coexists with the disease or both groups coexist with each other and the diseases (for example, see Fig EI (b), (c) & (d)).
A common phenomenon in disease dynamics is that of periodic epidemic outbreaks (Murray, 1993). However, in the case of model (1) we were unable to detect any biologically relevant periodic solutions. Similar behaviour was observed by Holt & Pickering (1985). When the contact rate between groups is zero we were able to prove analytically the non-existence of periodic solutions for all parameter values. Moreover in that case, regardless of the initial population and the number of infected individuals (I'0, i"1, 2) G both groups will coexist with each other and the disease. That is the coexistence steady state is globally attractive. The mathematical details are presented in Appendix E. Our working hypothesis is that when contact between groups is non-zero, the model system does not admit biologically relevant periodic solutions. Numerical simulations to date con"rm this hypothesis. 4. Discussion The social groupings of animals, formed for at least some part of the year, impose on populations a form of heterogeneous community structure. This can play a crucial role in disease transmission across a population since many microparasitic diseases require contact for transmission. The following questions can be raised at the global level. 1. How do social groups a!ect disease transmission dynamics? 2. How does disease a!ect interactions between social groupings?
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I. BEARDMORE AND K. A. J. WHITE
This paper presents a simple model to investigate disease transmission between two social groupings of the same species. In the absence of disease, where the only intra-speci"c interaction is between-group competition, the non-trivial steady state is unstable. Depending on initial conditions, one of the groups will survive to grow exponentially while the other dies out. Moreover, the initial advantage in population numbers determines the winner. In other words, the model always exhibits competitive exclusion in the absence of disease. To interpret this in ecological terms, we propose that the realized niches of each group are not su$ciently di!erentiated. In this paper, we have used identical model parameters to recognize the fact that each group is from the same species and that we have no dominance of groups. We chose to work with this form because it would allow us to explore at a base level how disease acts to regulate the between-group interaction. The tristability which we discussed in the results above arises from this underlying structure since it has not been previously noticed in the models of similar type (Bowers & Turner, 1997). Disease within a single group regulates the population dynamics and gives rise to a stable non-trivial steady state. When we include disease transmission between the two groups, the above-mentioned steady state of the type (S*, I*, 0, 0) becomes unstable when the contact between groups is su$ciently weak, g small. In its place, there is a stable non-trivial steady state in which both populations coexist with the disease. This coexistence is stable provided that group exclusion steady states are unstable. In other words, stable coexistence is possible only if both groups are able to increase when rare. (This is a special case of the model presented in Holt & Pickering, 1985.) Extending the model to include intra-speci"c between-group competition gives rise to similar behaviour for small contact rate. Again, if g is su$ciently small the only stable steady state is the coexistence of both groups with the disease. Essentially, the within-group disease transmission (analogous to intra-speci"c competition in the classic Lotka}Volterra competition model (see Murray, 1993 for details)) is larger than the between-group competitive regulation, thus preventing competitive exclusion. The realized
niches for each group are su$ciently di!erentiated. The di!erence from the case with no intra-speci"c between-group competition arises as g increases. As the interaction between groups increases, the realized niches are reduced and there are three possible stable steady states*coexistence between groups and between susceptible and infected individuals within the groups or competitive exclusion between the groups but persistence of disease within the surviving group (see Fig. 2). The outcome of the interaction depends on the initial conditions (see Fig. 1). Finally, when the interaction between groups becomes too large (g large), one group is driven to extinction, whereby the initial condition determines the outcome. These results are similar in nature to the classic results for the Lotka}Volterra competition model, see Murray (1993), except that this model can exhibit tristability. In the Lotka}Volterra model system, regulation of each species is through a density-dependent growth rate. Here, regulation of each group is also through a density-dependent process, namely disease. Disease divides the population into di!erent classes. In the case presented here, we consider the simplest scenario in which there are two such classes: susceptibles and infecteds. This process introduces an implicit time delay into the model as the regulation acts by infection reducing the susceptible population. We suggest that this time delay allows more #exibility in model outcome and hence, for intermediate levels of interaction, tristability as shown in Fig. 2. One particularly interesting feature of the model which we present is the following: if the disease transmission is the only intra-speci"c interaction between the groups the between}group contact rate g is always more detrimental to the stable coexistence steady-state levels for both susceptible and infected populations, than it would be if the groups existed on their own. However, if the intra-speci"c between-group competition is introduced when g is small, the steady-state population levels for the susceptible population when there is coexistence between social groups is higher, Fig. 2(a), than it would be if the groups existed on their own whilst corresponding levels for the infected population are lower, Fig. 2(b). Note that this is true for the particular case which
261
DISEASE AND SOCIAL GROUPS IN COMPETITION
we present and more generally for other values of p and q. Recall that these represent the probabilities that between-group interaction result in mortality and disease, respectively. This suggests that between-group competition is one possible mechanism for a species to escape parasitism. However, even in this simpler case, these results are somewhat counter-intuitive. Between-group interaction has both the potential to directly reduce the susceptible population levels through inter-group competition mortality and to introduce more disease into the group through intergroup disease transmission. Both of these act to reduce population levels within the susceptible class. Neither of these processes dominate the interaction and it is the reduction in infected population [Fig. 2(b)] which in turn reduces mortality of the susceptible population due to disease. Note that the total population level for each group (S #I , i"1, 2) is reduced as would G G be expected of the inter-group competition. As we have demonstrated above, the current model structure does not allow for sustained population oscillations. This agrees with the proposition in Collias (1944) that aggression between species helps reduce the intensity of cyclic #uctuation of the population. Nevertheless, many diseases do exhibit cyclic behaviour (see, for example, Anderson & May, 1991) and it would certainly be interesting to see how such periodic behaviour could arise with multi-group interactions. The assumptions that both groups belong to the same species and therefore have identical model parameters, enabled us to conduct a detailed investigation of a four-dimensional ODE model which are often very di$cult to analyse. We were able to obtain a clear picture of the role of the between-group contact parameter in disease transmission. Using a relatively simple model we have shown that social groupings and intra-speci"c between-group competition are one possible mechanism of disease control. Since an extreme example of spatial segregation between groups of the same species is the phenomenon of territoriality, our model could be extended in a spatial context and used to investigate the interesting hypothesis that one purpose of territoriality may be to reduce the impact of infectious disease on a population.
Although this is essentially a theoretical study, we envisage its application to feral populations with wider implications, including issues in public health, since feral populations may be reservoirs for disease in domestic animals. REFERENCES ANDERSON, R. M. & MAY, R. M. (1991). Infectious Diseases of Humans: Dynamics and Control. New York: Oxford University Press. BEGON, M. & BOWERS, R. G. (1994). Host}host}pathogen models and microbial pest control: the e!ect of host self regulation. J. theor. Biol. 169, 275}287. BOWERS, R. G. & TURNER, J. (1997). Community structures and the interplay between interspeci"c infection and competition. J. theor. Biol. 187, 95}109. COLLIAS, N. E. (1944). Aggressive behaviour among vertebrate animals. Phys. Zool. 17, 83}123. DOEDEL, E. J., CHAMPNEYS, A. R., FAIRGRIEVE, T. F., KUZNETSOV, Y. A., SANDSTEDE, B. & WANG, X. (1997). A;¹O 97: Continuation and bifurcation Software for Ordinary Differential Equations (with HomCont), February. HOLT, R. D. & PICKERING, J. (1985). Infectious disease and species coexistence: a model of Lotka}Volterra form. Am. Nat. 126, 196}211. MACDONALD, D. (1984). ¹he Encyclopedia of Mammals. London: Allen and Unwin. MACDONALD, D. W. (1983). The ecology of carnivore social behaviour. Nature 301, 379}384. MOLLISON, D. (1995). Epidemic Models: their Structure and Relation to Data. Cambridge: Cambridge University Press. MURRAY, J. D. (1993). Mathematical Biology. 3rd Edn. Heidelberg: Springer-Verlag. SUTHERLAND, W. J. (1996). From Individual Behaviour to Population Ecology. Oxford: Oxford University Press.
APPENDIX A Single Group Dynamics Model Analysis For eqn (2) to be at steady state we must have 0"a(S #I )!jS I !bS , 0"jS I !cI .
(A.1)
Solving the above equation gives us the trivial steady state (0, 0) which exists for all parameter values and the non-trivial steady state
c c(a!b) , , j j(c!a)
(A.2)
which is biologically relevant provided b(a(c or c(a(b.
262
I. BEARDMORE AND K. A. J. WHITE A.1. LINEAR STABILITY ANALYSIS OF THE
From eqn (B.5b) we get S "c/j. Substitu ting into eqn (B.5a) gives a non-trivial steady state
SINGLE GROUP SYSTEM
The Jacobian J of system (A.1) is given by
J"
a!b!jI a!jS jI jS !c
evaluated at the steady state (S*, I*). For the non-trivial steady state we have
c(a!b) a!b! c c(a!b) (c!a) J , " j j(c!a) c(a!b) (c!a)
a!c 0
and the trivial steady state is unstable provided a!b'0.
B.1. FINDING STEADY STATES
For system (4) to be at steady state we must have
(a!b) S "I , c!a
(B.8a)
(a!b) S "I . c!a
(B.8b)
Substituting eqn (B.8a) into eqn (B.2) and eqn (B.8b) into eqn (B.4) we get the coexistence steady state
c c(a!b) c c(a!b) , , , , j#gq (j#gq)(c!a) j#gq (j#gq)(c!a)
0"a(S #I )!jS I !gqS I !bS , (B.1) 0"jS I #gqS I !cI ,
(B.2)
0"a(S #I )!jS I !gqS I !bS , (B.3) 0"jS I #gqS I !cI .
(B.4)
(B.7)
The coexistence steady state is obtained by adding eqn (B.1) to eqn (B.2) and eqn (B.3) to eqn (B.4) to get
Two-Group Model: Disease Transmission Only
c c(a!b) 0, 0, , . j j(a!c)
a!b a 0 !c
APPENDIX B
(B.6)
The above steady state is biologically relevant provided b(a(c or c(a(b. Due to the model symmetry we also have a non-trivial steady state
.
Hence, non-trivial steady state is stable provided b(a(c. For the trivial steady state the Jacobian is J(0, 0)"
c c(a!b) , ,0, 0 . j j(a!c)
(B.9) which is biologically relevant provided b(a(c or c(a(b. At g"j/q, substituting eqn (B.8a) into eqn (B.2) and eqn (B.8b) into eqn (B.4) we arrive at
The trivial steady state (0, 0, 0, 0) exists for all parameter values. Setting S "0 and I "0, we get a reduced second order system with the steady-state equations
c(a!b) . I #I " j(c!a)
0"a(S #I )!jS I !bS ,
(B.5a)
0"jS I !cI .
(B.5b)
Combining eqn (B.10) together with eqns (B.8a) and (B.8b) gives a conituum of steady states forming a vertical bifurcation from eqn (B.9)
(B.10)
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DISEASE AND SOCIAL GROUPS IN COMPETITION
which holds provided that b(a(c and g' j/q. The equivalent conditions holds for stability of the steady state (0, 0, c/j, c(a!b)/j(c!a)). Finally, for the coexistence steady state we have
B.2. STABILITY
The Jacobian for system (4) is given by J"
a!jI !gqI !b a!jS 0 !gqS jI #gqI jS !c 0 gqS 0 !gqS a!jI !gqI !b a!jS 0 gqS jI #gqI jS !c
evaluated at steady state (S*, I*, S*, I*). For the trivial steady state we have the following Jacobian:
J
c c(a!b) c c(a!b) , , , j#gq (j#gq)(c!a) j#gq (j#gq)(c!a)
J J , J J
"
where
a(a!b) jc ! a! c!a j#gq J " , c(a!b) gqc ! c!a j#gq
a!b a 0 0 0 !c 0 0 J(0, 0, 0, 0)" . 0 0 a a!b 0 0 0 !c The eigenvalues of the above matrix are a " a!b and a "!c. Hence, the trivial steady state is stable when a!b(0. For group 2 exclusion steady state we have J
c c(a!b) J J , , , 0, 0 " j j(c!a) 0 ; J
where
J "
(a!b)!gq
J "
a!b a!c c!a , a!b 0 c c!a
a!b!c
c a!b a j c!a
c gq a!bc!a j
.
!c
Then
gq !1 '0, j
gqc(a!b) tr(J )"(a!b!c)! (0, j(c!a)
a(a!b) ! !a a!c c!a det(J!aI)"det c(a!b) !a c!a
a(a!b) jc gqc ! !a a! # c!a j#gq j#gq ;det . c(a!b) cgq !2 !a c!a j#gq
Therefore, the eigenvalues a , i3+1, 2, 3, 4, satisfy G a#a
det(J )"!c(b!a)'0,
det(J )"c(a!b)
gqc 0 ! j#gq . J " gqc 0 j#gq
The conditions for stability of this steady state are
a!b (0, tr(J )"a c!a
a#a
;
a(a!b) #c(a!b)"0, c!a
a(a!b) cgq c(a!b) #2 # c!a j#gq c!a
1 (2agq#jc!gqc)!a "0. j#gq
Hence, the coexistence steady state is stable when b(a(c and g(j/q.
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I. BEARDMORE AND K. A. J. WHITE
APPENDIX C
APPENDIX D
Two-Group Model : Intra-Speci5c Between-Group Competition Only
The Complete Model Analysis
C.1. FINDING STEADY STATES
For system (1) to be at steady state we must have
For system (6) to be at steady state we must have
0"a(S #I )!jS I !gqS I
0"(a!b)S !gpS S ,
!gp(S #I )S !bS
0"(a!b)S !gpS S .
" : F (S , I , S , I ),
The trivial steady state (0, 0) exists for all parameter values. The coexistence steady state of the form
D.1. FINDING STEADY STATES
a!b a!b , gp gp
(C.1)
0"jS I #gqS I !gpI (S #I )!cI " : F (S , I , S , I ), 0"a(S #I )!jS I !gqS I !gp(S #I )S !bS
exists and is biologically relevant whenever a!b'0.
" : F (S , I , S , I ), 0"jS I #gqS I !gpI (S #I )!cI
C.2. STABILITY
" : F (S , I , S , I ).
The Jacobian for system (6) is given by
J"
(a!b)!gpS !gpS , !gpS (a!b)!gpS
evaluated at steady state (S*, S*). For the trivial steady state we have the following Jacobian:
J(0, 0)"
a!b 0 . 0 a!b
The eigenvalues of the above matrix are a " a!b. Hence, the trivial steady state is stable when a!b(0. For the coexistence steady state the Jacobian is
a!b a!b 0 b!a , " . J gp b!a 0 gp The eigenvalues of the above matrix are a "!(a!b) and a "a!b and hence the coexistence steady state is a saddle point.
(D.1)
The trivial steady state (0, 0, 0, 0) exists for all parameter values. Setting S "0 and I "0, we get a reduced second-order system with the steady-state equation 0"a(S #I )!jS I !bS ,
(D.2a)
0"jS I !cI .
(D.2b)
From eqn (D.2b) we get S "c/j. Substituting into eqn (D.2a) gives a non-trivial steady state
c b!a c , , 0, 0 , j a!c j
(D.3)
provided aOc and b(a(c. In the modelling process, we have assumed that both groups have the same characteristics, hence, we have a symmetrical system (1). Thus, the presence of a particular steady state of some form (x, y, 0, 0) will automatically result in a symmetrical steady state of the form (0, 0, x, y).
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DISEASE AND SOCIAL GROUPS IN COMPETITION
Hence, from the steady state (D.3) we can immediately obtain another non-trivial steady state
c b!a c 0, 0, , . j a!c j
(S**, I**, S**, I**) (D.4)
Another steady state can be obtained by setting I "0 and I "0 in eqn (D.1) in which case we get
a!b a!b , 0, ,0 gp gp
(D.5)
provided a'b. Continuing with the idea of symmetry, suppose now that we have another non-trivial steady state in the form (S*, I*, S*, I*).
a non-trivial steady state of the form
(D.6)
and immediately, using the assumption that both groups are identical we obtain a symmetrical steady state (S**, I**, S**, I**).
D.2. THE ROLE OF g
We now consider the coexistence steady state (S*, I*, S*, I*) as a function of between-group contact parameter g. In Appendix D.1, we have found that k#(k!4ac(j#gq) , S*" 2(j#gq)
0"a(S*#I*)!jS*I*!gqS*I*
(D.7b)
where k"c(j#gq#gp)#gp(a!b)#a(j #gq !gp). It is clear that S* (g)Pc/j and I* (g)P (c/j) (a!b)/(c!a) as gP0. We want to show that for small g
(D.8)
c S* (g)' , j
giving [from eqn (D.7b)] S* (j#gq!gp)!c I* " . gp
S* (j#gq!gp)!c , I* " gp
(D.7a)
0"jS*I*#gqS* I*!gpI* (S*#I*)!cI*
c (a!b) I* (g)( . j (c!a)
Substituting eqn (D.8) into eqn (D.7a) rearrange to get a quadratic equation for S* !(j#gq)S*#S* (c(j#gq#gp)#gp(a!b) #a(j#gq!gp))!ac"0,
(D.9)
which gives only one realistic solution when combined with (D.8), namely k#(k!4ac(j#gq) , S* " 2(j#gq)
(D.12)
However, it is not possible to obtain explicit forms for these steady-state values.
Substituting eqn (D.6) into eqn (D.1) we get
!gp(S*#I*)S*!bS*,
(D.11)
(D.10)
where k"c(j#gq#gp)#gp(a!b)#a(j#gq !gp). For certain parameter values there exists
We consider dS* dg
d k#(k!4ac(j#gq) " dg 2(j#gq) g"0
g"0
ja " (qc#pc#pb!qa!2pa)'0 (c!a) provided qc#pc#pb!qa!2pa'0.
(D.13)
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I. BEARDMORE AND K. A. J. WHITE
steady state is stable when a!b(0. For the steady state (D.3) we have the Jacobian
Also, dI* dg
d S* (j#gq)!S*gp!c " gp dg
g"0
d k#(k!4ac(j#gq)!2c(gp) " 2gp(j#gq) dg
dS* ! dg
dS* "!j(c!a)p! dg g"0
g"0
(0
g"0
c c !g (q#p) j j M " c b!a b!a g q!p !gpc j a!c a!c
The Jacobian J of system (D.1) at the steadystate level is (*F /*x ), where x "S , x "I , G G x "S , x "I . Hence,
where
(D.14)
J "
a!jI !gqI !gp(S #I )!b a!jS , jI #gqI jS !gp(S #I )!c
!gpS !gqS !gpS , J " !gpI gqS !gpI
M "
a!b!g
evaluated at steady state (S*, I*, S*, I*). For the trivial steady state we have the following Jacobian:
c b!a b!c q #p j a!c a!c c (b!a) gq j (a!c)
,
a !gp
c (b!c) !c j (a!c)
a!b tr(M )"a (0, a!c
a!jI !gqI !gp(S #I )!b a!jS , jI #gqI jS !gp(S #I )!c
det(M )"!c(b!a)'0,
J "
a!b a 0 0 0 !c 0 0 . J " 0 0 a!b a 0 0 0 !c
!gp
and 0 ; is just a 2;2 zero matrix. The eigen values of J are exactly the eigenvalues of M and M . We "rst consider M . In order for eigenvalues of M to be negative
!gpS !gqS !gpS , J " !gpI gqS !gpI
b!a a!c a!c , b!a 0 c a!c
a!b!c
M "
D.3. STABILITY
J J , J J
(D.16)
where
whenever (D.13) holds.
J"
M M , J " 0 ; M
g"0
(D.17a) (D.18b)
which requires b(a(c.
(D.19)
Next, we need to consider matrix M . M "
(D.15)
The eigenvalues of the above matrix are a "a!b and a "!c. Hence, the trivial
a a!b!gqI*!gp(S*#I*) gqI* !gp(S*#I*)!c,
recalling that S*"c/j and I*"c(b!a)/ j (a!c). As in the case of M in order for the
DISEASE AND SOCIAL GROUPS IN COMPETITION
eigenvalues of M to be negative we need the two conditions det(M ) :"(!gp(S*#I*)!c)(a!b!gqI* !gp(S*#I*))!agqI*'0, tr(M ) :"a!b!c!gqI*!2gp(S*#I*)(0. In order for the above to hold, recalling that S '0, I '0 and assumption (D.19) it is su$ cient that g'gJ ,
(D.20)
where gJ is a positive solution of a quadratic gJ p
c(c!b) (q(a!b)#p(c!b)) j(c!a)
#gJ
a!b!c c!b (q(a!b)!p j c!a
(D.21)
!c(a!b)"0.
Hence, steady state (D.3) will be stable if eqns (D.19) and (D.20) hold. For steady state (D.4), the Jacobian will be
M 0 ; , J " M M where the block entries of the matrix are the same as we have de"ned for J . In this case J is a lower triangular matrix. Equivalent to the case for J , the eigenvalues of J will be the eigen values of M and the eigenvalues of M . Hence, the condition for stability is the same. For steady state (D.5) we have
J "
a!j
0 0
j
gp
gp
!a#b!c a!b !a#b
gp a!b gq gp
a!b !a#b gp a!b gq gp a!b a!j gp a!b j !a#b!c gp
b!a !gq
a!b
b!a !gq 0
a!b
0 0 0
.
267
The eigenvalues of the above matrix are a " !(a!b), a "a!b, a "(j(a!b)/gp)!a# b!c#q(a!b)/p and a "j(a!b)/gp!a# b!c!q(a!b)/p. Hence, if the steady state is biologically relevant, i.e. a'b it is unstable. In principle, we could determine the local stability of the steady state (D.6) by applying, for instance, the Routh}Hurwitz criteria to the Jacobian matrix (D.14). In practice, for a "xed parameter values a, b, c, p and q with b(a(c we can show that there exists gL (a, b, c, p, q), gJ (gL (R, where gJ is a positive solution of quadratic (D.21), such that steady states (D.8) and (D.10) are stable if g(gL (see Doedel et al., 1997 and Fig. 2.) We can also show that the steady states (D.11) and (D.12) exist for gJ (g(gL , in other words when steady states (D.6), (D.4) and (D.3) are locally stable. Whenever they exist steady states (D.11) and (D.12) are always unstable. Using standard bifurcation theory we can demonstrate the existence of gL (a, b, c, p, q) s.t. gJ (gL (R, for nearby parameter values, upto genericity assumptions. Note that although pitchfork bifurcations are not preserved under generic perturbations, they are preserved under symmetric perturbations. Since our system posseses re#ective symmetry about the two groups, we except the pitchfork bifurcation to persist. APPENDIX E Non-existence of Periodic Orbits When g"0, system (1) decouples to a pair of SI systems dS G"a(S #I )!jS I !bS , G G G G G dt dI G"jS I !cI , G G G dt
(E.1)
where i"1, 2, with one locally stable steady state (c/j, ((a!b)/(c!a)) c/j, c/j, ((a!b)/(c!a))c/j). For ease of computation, we set u "I ! G G ((a!b)/(c!a) c/j, v "S !c/j, i"1, 2 so that G G
268
I. BEARDMORE AND K. A. J. WHITE
system (E.1) becomes
From eqn (E.4) it follows that ¸ is a decreasing function bounded below and hence has a limit as tPR
du G"bv #ju v , G G G dt
d¸ P0, tPR. dt From eqn (E.3) and for i"1, 2 N
dv G"!cu !dv !ju v , (E.2) G G G G dt where i"1, 2, b"c(a!b)/(c!a), c"c!a and d"a(a!b)/(c!a) and steady state u "v "u "v "0. We propose
ju jv ¸"j(u #v )!b ln 1# !c ln 1# b c
ju jv #j(u #v )!b ln 1# !c ln 1# b c
(E.3) as a Lyapunov function for system (E.2) which is true provided jdv jdv d¸ ! (0. "! c#jv c#jv dt
(E.4)
When c#jv '0 and c#jv '0, i.e. S 'a/j and S 'a/j, eqn (E.4) will hold and hence we can now divide the space into two regions (Fig. E2) as follows: Region I: I '0, I '0, S 'a/j and S 'a/j 8 b#ju , c#jv '0 for i"1, 2. G G Since at the region boundaries b u P! N ¸P!b ln0P#R, G j c v P! N ¸P!c ln0P#R, G j
ju u PRN¸"ju !b ln 1# G +ju P#R, G G G b jv v PRN¸"jv !c ln 1# G +jv P#R, G G G c where i"1, 2, we can conclude that (u (t), v (t), u (t), v (t)) cannot escape Region I. Also ¸(u , v , u , v ) has a minimum at ¸(0, 0, 0, 0)"0.
(E.5)
N v P0, tPR, G N
dv GP0, dt
tPR,
N !cu !bv #ju v P0, tPR, G G G G N !cu P0, G
tPR,
N u P0, tPR. G Hence (u , v , u , v ) P (0, 0, 0, 0) as tPR, in other words (0, 0, 0, 0) is a globally stable steady state in Region I. Region II: I '0, I '0, 0(S )a/j and 0(S )a/j 8 b#ju '0*c#jv and v * G G G !c/j for i"1, 2. From system (E.2) it follows that for i"1, 2 dv dc G"!(c#jv )u !bv *!bv * . G G G G dt j
(E.6)
Hence, dc c dc v (t)*v (0)# t*! # t. G G j j j
(E.7)
Using eqn (E.7) we can conclude that at some time t*, v (t*) will be outside Region II G (c!a) , 8 v (t*)'! G j (c!a) c dc , 8! # t*'! j j j a 8 t*' . dc From the above analysis we can conclude that starting in Region II, (u , v , u , v ) will leave it after a time t*)a/dc and enter Region I (Fig. E2). Since we have also proved that Region I
DISEASE AND SOCIAL GROUPS IN COMPETITION
269
FIG. E2. The S '0, I '0, S '0 and I '0 space is divided into two regions. Since g"0 four-dimensional system can be decoupled into two two-dimensional subsystems. In (a) S '0, I '0 while in (b) S '0 and I '0. The regions are: I '0, I '0, 0(S )a/j and 0(S )a/j. Region I: I '0, I '0, S 'a/j and S 'a/j and Region II:
is invariant and that (0, 0, 0, 0) is a globally stable steady state in that region it follows that (0, 0, 0, 0) is globally stable in the whole space
I , I , S , S '0. Therefore, system (E.2) does not have periodic solution for any parameter values.