Dynamics of a opulation Model Controlling the Spread of lague in rairie Dogs Catalin Georgescu The University of South Dakota Department of Mathematical Sciences 414 East Clark Street, Vermillion, SD USA Catalin.Georgescu@usd.edu Dan Van eursem The University of South Dakota Department of Mathematical Sciences 414 East Clark Street, Vermillion, SD 57069 USA dpeursem@usd.edu Abstract: The infestation of prairie dog population with the deadly Yersinia estis has been recently the object of investigation of numerous biologists due to the potentially high risk of contamination of the human population. In the present paper we construct and analyze a model describing the interaction between the prairie dog population as the host population) and the flea population as the vector population) and derive the existence under certain conditions impose to the main parameter) of an equilibrium that plays the role of a global attractor. Key Words: Stability of Equilibria, Global attractor, Bifurcation. 1 Background. In recent years, prairie dog populations have been hit hard by the plague caused by Yersinia pestis. The literature suggests that total colonies can be wiped out upon infestation and the disease is rapidly spread from one colony to the next see [1],[2],[6]). There are reasons to expect that the spread of plague from face to face contact or through respiratory methods is not the predominate mode of transferring the disease as Hoogland et al. have reported see [5]) that upon killing fleas with the insecticide yraperm the spread of the disease was stopped for a time. The agreed upon mode of transmission of the vector is through infected fleas hosting on the prairie dogs or other mammals. Seery et al. and Hoogland et. al have shown that by using insecticides on an infected colony one can stop the spread of an epizootic and can be used to save the colony. We present a mathematical model that will allow one to show analytically what kind of control one needs on the fleas in order to prevent an epizootic once the plague has been manifested in a colony. The model shows that when one starts with healthy and infected prairie dogs and fleas, there are three equilibrium points that can develop. We will consider α, the ratio between the normal death rates of flea population and prairie dog population, as the working parameter of the problem. We will show that if this parameter stays smaller than a certain value α 0, the dynamics of the system is governed by three equilibria: there is the case where healthy and infected prairie dogs and fleas can coexist and it is an asymptotically stable equilibrium, there is the case where all infected fleas and infected prairie dogs are wiped out, and there is the case where only the healthy prairie dogs exist, these last two equilibria being unstable. If the parameter exceeds the bifurcation point α 0, one witness a collapse of equilibria to only healthy prairie dogs existing and the change in stability of the last two mentioned above equilibria. 2 The Origins of the Model. The model will consist of a typical host-vector model with the prairie dogs acting as the host population and the flea population being the vector population. Let be the total prairie dog population, S being the number of infected or sick prairie dogs, F be the total flea population, and D be the number of diseased or infected fleas. The governing equations are: ) d dt = r p 1 kp d r, 1) df dt = r f 1 F ) F d f F, 2) m ) ds S dt = t p D c 1 d r S, 3) dd dt = t S f F D) c 2d f D. 4) ISSN: 1790-5117 78
The overall populations of the prairie dogs and fleas are governed by a logistic growth model where r p and r f are the growth rates of the prairie dog and flea populations respectively, k p, is the carrying capacity of the prairie dog population, m is the maximum number of fleas that would be found on a prairie dog, thus giving m as the carrying capacity for the fleas, and d r and d f being the normal death rate of the overall prairie dog and flea populations respectively. The number of sick prairies dogs grows at a rate that is proportional to the product of the proportion of healthy prairie dogs and the number of diseased fleas with t p being the proportionality constant. This is similar to the interactions among predators and prey in the standard Lotka-Volterra equationssee [7]). We make a simplifying assumption here that with the infected prairie dogs, the deathrate c 1 d r is a constant multiple of the over all death rate. The infected flea population is very similar in that it grows at a rate proportional to the product of the proportion of diseased prairie dogs and the number of healthy fleas. The proportionality constant is t f and again we make the simplifying assumption that the death rate, c 2 d f, is a constant multiple of the overall death rate for the total flea population.this system is very similar to the governing equations of [3] where they were looking at ticks as being the vector population on deers. There are however some simplifying assumptions we made here that will allow for full analytical analysis of the system. One simplification involves the elimination of the interaction between the infected and healthy hosts due to the evidence that face to face and airborne transmissions seemed to be minor as mentioned earlier. We also eliminate the interaction between infected and healthy fleas since a lack of competitiveness has been noticed at this type of population. In scaling the equations we will let = /k p, t = t d r, F = F/mk p ), S = S/k p, and D = D/mk p ). These are the natural scalings where we scale the populations by their carrying capacity and time is scaled by the reciprocal of the prairie dog death rate. Introducing the positive dimensionless parameters of k 1 = r p /d r, k 2 = r f /d r, k 3 = mt p /d r, k 4 = t f /d r, and α = d f /d r, upon dropping the bars the scaled equations are as follows: d dt = k 11 ), 5) df dt = k 21 F )F αf, 6) ds dt = k 31 S )D c 1S, 7) dd dt = k S 4 F D) c 2αD. 8) We will use α as a control parameter for the system as it is essentially the death rate of the vector population. We will investigate controlling the population by varying this parameter. It is noted that the only physical solutions are for, F, S, and D to all be positive. The parameter α will play a particulary important role, since bifurcation occurs with respect to this parameter and it is essentially our control on how effective we are in killing the fleas. 3 Dynamical Analysis of the Model. Since the first two equations 5) and 6) of the original system are decoupled from the other two, it is natural to start with the analysis of the phase portrait in the first quadrant of the plane, F ). An immediate computation shows that in order to have equilibria in the first quadrant, the following two conditions have to be true: k 1 > 1 9) 0 < α < k 2, 10) and in this case the two nonzero equilibria will be, 0) and, F 0 ), where: ) = 1 1k1, F 0 = 1 α ). 11) k 2 Linearizing the system at an arbitrary point, F ) leads to the Jacobian: J, F ) = k 1 1) 2k 1 k 2 F 2 2 k 2 α) 2k 2F Under the assumptions 9) and 10) the equilibrium, F 0 ) will be a sink since the eigenvalues of the linearized system will be λ 1 = 1 k 1 < 0 and λ 2 = α k 2 < 0 and the equilibrium, 0) is a saddle with λ 1 = 1 k 1 < 0 and λ 2 = k 2 α > 0. Notice also that F = 0 and = are two invariant lines and on the line F = 0 we have d dt > 0 if < ISSN: 1790-5117 79
F 1.0 0.75 0.5 0.25 0.0 0.0 0.25 0.5 0.75 Figure 1: hase ortrait in F-plane,, F 0. The values k 1 = 2, k 2 = 1 and α = 0.5 were used. and d dt < 0 if >, so the phase portrait will look as in Figure 1. Remark 1. When α = k 2, the two equilibria will coincide with, 0) which becomes a sink in fact an attractor for all orbits in the first quadrant since df < 0 and the invariant line F = 0 stops the orbits dt from exiting the first quadrant). For α > k 2 the equilibrium, F 0 ) moves out of the first quadrant with, 0) continuing to be a sink. In order to analyze the general case, we will show that if one restricts themselves to the plane =, F = F 0 in the space of coordinates, F, S, D), all orbits in the first quadrant) starting in this plane will converge to an equilibrium S 0, D 0 ) in this plane. Since projections on the, F ) plane of all orbits except those starting with F = 0) converge to, F 0 ) as we have seen from above, we will have that almost all orbits will converge to a global attractor due to the following result: Lemma 2. Consider the following system of differential equations: dx dt = fx) dy = gx, y), dt where x R n and y R m, and the functions f and g are considered of class C 1. Assume that there is 1.0 an x 0 such that fx 0 ) = 0 and any solution xt) of the system dx/dt = fx) has the property that lim t xt) = x 0. Assume also that if x 0, yt)) is a solution of the above system, then lim t yt) = y 0, for some y 0 R m. Then for any solution xt), yt)) of the system we have that lim t xt), yt)) = x 0, y 0 ). We will apply this lemma for x =, F ) and y = S, D). We now proceed to analyze the general system. First we determine the equilibria. Consider = in ds/dt = dd/dt = 0 equations and eliminate the term S D from the last two equations with =. This leads to: D = βs, where β = c 1 k 4 + k 3 k 4 F k 3 k 4 + k 3 c 2 α 12) Notice that 12) implies that S = 0 is equivalent to D = 0 due to positivity of coefficients. The implication D = 0 S = 0 is particulary important from observations in the field since this shows no transfer of disease from prairie dog to prairie dog, as prior research suggests. If S 0, then we obtain the equilibrium: S 0 = 1 c ) 1, D 0 = β 1 c ) 1 βk 3 βk 3 13) Hence we distinguish two cases: Case 1. If F 0 = 0, then 12) shows that 1 c 1 = βk 3 αc 2 /k 4 and hence the equilibrium S 0, D 0 ) has both components negative so it is not of interest for our model. Thus in this case in the hyperplane =, F = 0 only the equilibrium S = 0, D = 0) is of interest. The linearization at this point in the above hyperplane gives the Jacobian: J0, 0) = c1 k 3 0 c 2 α and hence the two eigenvalues are negative which shows this is a sink. Since the hyperplane F = 0 is invariant to the flow, in the 4 dimensional picture this equilibrium will have only one repelling direction, the line connecting, 0) with, F 0 ) in the S = 0, D = 0 plane. This reflects the fact that if there are no fleas, the prairie dog population tends to a constant level. Case 2. t) =, F t) = F 0. This is the interesting ) ISSN: 1790-5117 80
case since the plane t) =, F t) = F 0 is invariant and an attractor. We will denote by f 3 and f 4 the third and fourth components of the vector field in this plane, hence: f 3 S, D) = k 3 D k 3 SD c 1 S f 4 S, D) = k 4F 0 S k 4 SD c 2 αd Note that since the divergence f 3 S + f 4 is negative D in the first quadrant, the system will have no periodic orbits in the plane t) =, F t) = F 0. Before we analyze the nature of equilibria an important observation is necessary. Notice that from 13) we must have 1 c 1 /βk 3 > 0. Using 12) we obtain: 1 c 1 = 1 c 1 k 3 k 4 + c 2 α) βk 3 k 3 k 4 c 1 + k 3 1 α/k 2 )) = k 3k 4 αc 1 c 2 + k 3 k 4 /k 2 ) k 4 c 1 + k 3 1 α/k 2 )) so the parameter α should additionally satisfy: k 3 k 4 0 < α < α 0 =: = 14) k 3 k 4 /k 2 + c 1 c 2 = k 2 1 + c 1c 2 k 2 k 3 k 4 < k 2. We can see that if α satisfies 14) it will automatically satisfy 10). Unless otherwise stated, we will assume from now on that α < α 0. There are two equilibria in this case: 0, 0) and S 0, D 0 ). For the first one the Jacobian is: J0, 0) = c 1 k 3 k 4 F 0 15) c 2 α The trace of this matrix is obviously negative and the determinant, using 11), is D = αc 1 c 2 + k 3 k 4 /k 2 ) k 3 k 4 and hence from 14) we have D < 0 so this equilibrium is a saddle for the linear case and from Hartman-Grobman theorem it is a saddle for the original nonlinear) system. At the other equilibrium S 0, D 0 ), the Jacobian will be after some rearrangements): JS 0, D 0 ) = ) k3 D 0 +c 1 )/ k 3 S 0 )/ k 4 F 0 D 0 )/ k 4 S 0 +c 2 α )/ Again the trace of this matrix is obviously negative, but the sign of the determinant is less obvious. We clearly have: = 1 0 2 {k 3 D 0 + c 1 )k 4 S 0 + c 2 α ) k 3 k 4 S 0 )F 0 D 0 )} 16) Using 11) and 13)we will express in 16) D 0, S 0 and in terms of. An immediate computation leads to: = βk 3 k 4 +c 2 k 3 α) 1 c 1 k 4 c 1k 4 α + c2 1 k ) 4 β k 2 k 3 This implies that: > 0 β 2 > c 1 + c 1 α 1k 4 k 3 k 2 = k 3 k 4 + αc 2 = c 1k 4 k 2 k3 2 k2k 3 αk 3 + c 1 k 2 17) k 4 + αc 2 The expression of β when F = F 0 is derived from 12): β = k 4 k 2 c 1 + k 2 k 3 αk 3 k 2 k 3 k 4 + c 2 α After some rearrangements, the inequality in 17) becomes: k 3 k 4 α < = α 0. k 3 k 4 /k 2 + c 1 c 2 This shows that in the plane = and F = F 0 the equilibrium S = S 0 and D = D 0 is a sink if α < α 0 which is also the condition for which this nontrivial) equilibrium exists. We will now show that in the plane =, F = F 0 there is a connecting orbit between the two equilibria 0, 0) and S 0, D 0 ). Consider λ the positive eigenvalue of the linearized system at the equilibrium 0, 0). If 1, y) gives the direction of the unstable space, we will have that J0, 0) 1, y) t = λ1, y) t. The first equation of this system yields: c 1 + k 3 y = λ which implies y > 0. Hence the unstable manifold which will be tangent at the origin to this direction will enter in the first quadrant. The flow enters into a box situated in the first quadrant having two sides along the coordinate axes as it can also be seen from figure 2), so the ω-limit set of this unstable trajectories is inside this box. From the oincaré-bendixon criterion, since the system has no periodic orbits as we proved above, this orbit will have the equilibrium S 0, D 0 ) as a limit point. The arguments above can be summarized by the following: ISSN: 1790-5117 81
d 0.125 0.1 0.075 0.05 0.025 0.0 0.0 0.025 0.05 0.075 S 0.1 0.12 Figure 2: hase ortrait in SD-plane, S, D 0. The values k 1 = 2, k 2 = 1, α = 0.5 and k 3 = k 4 = c 1 = c 2 = 1 were used. roposition 3. Let us consider k 1 > 1. If 0 < α < α 0, then there are only three equilibria of interest: E 1 =, 0, 0, 0), E 2 =, F 0, 0, 0) and E 3 =, F 0, S 0, D 0 ). Any solution of interest having initial condition, F 0, S 0, D 0 ) with F 0 strictly positive either tends to the equilibrium E 2 if S 0 = D 0 = 0) and hence if initially there are no diseased fleas or prairie dogs, the two populations coexist or to the equilibrium E 3 if S 0 > 0 and D 0 > 0). Therefore, except on a set of measure zero, solutions of interest tend to E 3, the equilibrium which plays the role of a global attractor. If F 0 = 0 then all solutions will tend toward E 1. For α = α 0 the equilibrium E 3 coincides with E 2 and all solutions of interest with F 0 > 0 will tend to this equilibrium. For α 0 < α < k 2 the equilibrium E 3 will have the last two coordinates negative so it is no longer of interest. In fact, the components f 3 and f 4 of the vector field will become negative for S > 0 and D > 0 so solutions of interest with F 0 > 0 will continue to approach E 2. They can not go below S = 0 and D = 0 since they would intersect then the solutions in the, F ) plane contradicting the uniqueness theorem which obviously holds for this system. When α gets closer to k 2 the equilibrium E 2 gets closer to E 1. When α = k 2, as explained in Remark 1, there will be only one equilibrium with all solutions of interest going toward E 1 = E 2 = E 3 for F 0 > 0. Solutions in the, F ) plane also go toward this equilibrium. We conclude with the following proposition. roposition 4. Consider the system 1) with k 1 > 1. If α [α 0, k 1 ], all solutions with initial condition having F 0 > 0 tend toward E 2 and those with F 0 = 0 tend toward E 1. In other words, the populations St) and Dt) become eventually depleted. 4 Conclusion The model presented in this paper generates a four dimensional system of differential equations depending on several parameters. Rescaling and fixing some proportionality constants leads to a system with one parameter and the bifurcation problem is described. The idea behind managing such a system was to take advantage of the fact that the first two equations are decoupled. We proved the existence of an attractor for all solutions starting in the first quadrant for the first two equations and using this we then prove the existence of an equilibrium that is a global attractor for solutions having positive initial conditions. This shows that for k 1 > 1 and 0 < α < α 0, we will have coexistence with healthy prairie dogs and fleas if there are no initial diseased fleas or prairie dogs. With initial diseased prairie dogs and fleas both prairie dogs and fleas will coexist with disease present in both populations. For α 0 α < k 2 only healthy prairie dogs and fleas will exist. Finally for α k 2 only healthy prairie dogs will exist. Acknowledgements: The research was partially supported by a University of South Dakota Travel Grant. References [1] Barnes,A., A Review of plague and its relevance to prairie dog population and the black-footed ferret, Washington, DC: United States Fish and Wildlife Service, 1993. [2] Cully,F.,Williams,S., Interspecific comparisons of sylvatic plague in prairie dogs, Journal of Mammology, 82, 894-905, 2001. [3] Gaff,H.D., Gross,L.J., Modeling Tick-Borne Disease: A Metapopulation Model, Bulletin of Mathematical Biology, 69, 265-288, 2007. [4] Guckenheimer,J., Holmes,., Nonlinear Oscillations, Dynamical Systems, and Bifurcation of Vector Fields, Springer-Verlag, 1983. ISSN: 1790-5117 82
[5] Hoogland,J.,Davis,S.,Benson- Amram,S.,Labruna,D.,Goossens,B.,Hoogland,M., yraperm Kills Fleas and Halts lague Among Utah rairie Dogs, The Southwestern Naturalist, 493), 376-383; [6] Seery,D.,Biggins,D.,Montenieri,J., Enscore,R.,Tanda,D.,Gage,K., Treatment of Blacked-Tailed rairie Dog burrows with Deltamethrin to Control Fleas Insecta: Siphonaptera) and lague, Journal of Medical Entomology, Vol.40, no.5, 2003. [7] Takeuchi,Y.,Global Dynamical roperties of Lotka-Volterra Systems Singapore, River Edge NJ: World Scientific, 1996. ISSN: 1790-5117 83