DYNAMICS OF AN INFECTIOUS DISEASE INCLUDING ECTOPARASITES, RODENTS AND HUMANS

Transcription

1 DYNAMICS OF AN INFECTIOUS DISEASE INCLUDING ECTOPARASITES, RODENTS AND HUMANS A DÉNES AND G RÖST Bolyai Institute, University of Szege, Arai vértanúk tere 1, Szege H-6720, Hungary enesa@mathu-szegehu We present a mathematical moel escribing the sprea of an infectious isease sprea by ectoparasites which are harboure by roents eg plague transmitte by fleas sprea by rats We ientify three reprouction numbers for the roent subsystem an show that these three reprouction numbers completely characterize the global ynamics of this subsystem Depening on which of the four equilibria of the roent subsystem is globally attractive, we etermine the possible equilibria of the human subsystem an show that one of the equilibria of this subsystem is always globally attractive Hence we completely escribe the global ynamics of the full system 1 Introuction Ectoparasites are parasites that live on or in the skin but not within the boy These parasites, eg lice, fleas, mites have long been known as vectors of several infectious iseases incluing epiemic typhus an plague It is also commonly known that in several cases, ectoparasites are transmitte to humans from animals, most often by roents A well known example for this is plague, cause by the bacterium Yersinia pestis: the fleas transmitting this isease were transmitte to humans by rats [5] Other notable examples are Omsk haemorrhagic fever, cause by a Flavivirus transmitte by ticks on water voles an muskrats [4]; rickettsialpox, cause by the bacteria Rickettsia akari transmitte by mites on mice [6]; murine typhus, cause by the bacteria Rickettsia typhi, transmitte by fleas, usually on rats [9]; scrub typhus cause by the parasite Orientia tsutsugamushi, transmitte by trombiculi mites, carrie by mice The latter isease is estimate to cause more than a million cases annually in Asia with more than a billion people being at risk, which makes scrub typhus the most meically important rickettsial isease [10] 1

2 2 Figure 1 The pathogen can jump from roents to humans via ectoparasites Figure: courtesy of Júlia Röst In this work, we consier an infectious isease cause by a pathogen sprea by ectoparasites which are harboure by roents We assume that ectoparasites sprea by the roents might be infectious or non-infectious A given roent or human can be infeste only by one type either infectious or non-infectious of the ectoparasite A human can be infeste an hence possibly infecte through aequate contact with an infeste infecte roent or another human We assume that the ectoparasites are not transmitte back from humans to the roents Due to isinfestation an/or treatment, infeste an infecte humans may become susceptible again The structure of the paper is the following In Section 2, we establish a compartmental moel escribing the sprea of the infestation an the isease In Section 3, we stuy the subsystem forme by the equations for the roent compartments, while, using the results of Section 3, we stuy the human subsystem in Section 4 2 The moel We enote by Rt the compartment of susceptible roents, T t stans for the roents infeste by non-infectious parasites, while Qt enotes the number of roents infeste by infectious parasites Similarly, we have three compartments for the humans: St enotes susceptibles, It those infeste by non-infectious parasites, an Jt those infeste by infectious parasites A an stan for the birth, resp eath rates of roents The notation β 1 stans for the transmission rate between the compartments R an T, while β 2 is the transmission rate between R an Q, resp T an Q B an δ stan for natural birth an eath rates for humans, an ρ enotes iseaseinuce eath rate for the infecte human compartment J The parameter ν 1 enotes transmission rate between the compartments S an I, while ν 2 enotes transmission rate from J to S an J to I The parameter η 1

3 3 enotes the transmission rate from roents infeste by non-infectious parasites to susceptible humans, while η 2 is the transmission rate from roents infeste by infectious parasites to humans We enote by θ 1, resp θ 2 the isinfestation, resp recovery rate from compartments I, resp J Figure 2 Transmission iagram representing transitions between the roent an the human compartments Using the above notations, our equations take the following form: R t = A β 1 RtT t β 2 RtQt Rt, T t = β 1 RtT t β 2 T tqt T t, Q t = β 2 RtQt + β 2 T tqt Qt, S t = B η 1 StT t η 2 StQt ν 1 StIt ν 2 StJt δst + θ 1 It + θ 2 Jt, 1 I t = η 1 StT t + ν 1 StIt η 2 ItQt ν 2 ItJt δit θ 1 It, J t = η 2 StQt + η 2 ItQt + ν 2 StJt + ν 2 ItJt δjt ρjt θ 2 Jt, with positive initial conitions R0, T 0, Q0, S0, I0, J0 0 The

4 4 phase space R 6 + = {R, T, Q, S, I, J R 6 : R, T, Q, S, I, J 0} is clearly invariant to system 1 3 The roent subsystem 31 Equilibria, local stability The first three equations of 1 can be ecouple from the remaining ones The subsystem for the sprea among roents, given by R t = A β 1 RtT t β 2 RtQt Rt, T t = β 1 RtT t β 2 T tqt T t, Q t = β 2 RtQt + β 2 T tqt Qt has a similar structure as the moel given by Dénes an Röst [1, 2], though, in the present case, birth an eath rates are not equal in contrary to the cite papers To calculate the equilibria of the full system, we start by calculating those of the roent subsystem 2, which are easily obtaine by solving the algebraic system of equations 0 = A β 1 RT β 2 RQ R, 0 = β 1 RT β 2 T Q T, 0 = β 2 RQ + β 2 T Q Q, resulting in the four possible equilibria E R = A, 0, 0, E T = β 1, A β 1, 0 E Q = β 2, 0, A β 2, E TQ = Aβ2 β 1, β 2 Aβ2 β 1, A β 2 By introucing a single infeste/infecte iniviual into one of the equilibria E R, E T an E Q, we obtain three ifferent reprouction numbers If we introuce a roent infeste by the non-infectious parasites into the isease- an infestation-free equilibrium, we obtain the reprouction number r 1 = Aβ1 2 Introucing a roent infeste by the infectious parasites into the equilibrium E R, we obtain the reprouction number r 2 = Aβ2 2 If we introuce a roent infeste by the infectious parasites into the equilibrium E T, we obtain again the same reprouction number r 2 Finally, let us introuce a roent infeste by the non-infectious parasites into the equilibrium E Q In this case, the expecte sojourn time of an iniviual 2 3

5 5 infecte with the first strain in the T -compartment is β 2 Q + 1, an the number of new infections generate by this iniviual is β 1 R, where R an Q stan for the first, resp thir coorinates of the equilibrium E Q This way we obtain the reprouction number r 3 = β12 β2 2A It is obvious that the equilibrium E R always exists, E T exists if an only if r 1 > 1, E Q exists if an only if r 2 > 1, while E T Q exists if an only if r 2 > 1 an r 3 > 1 The following proposition on the local stability of the four equilibria can easily be checke, see [3] Proposition 31 The isease-free equilibrium E R is locally asymptotically stable if r 1 < 1 an r 2 < 1 an unstable if r 1 > 1 or r 2 > 1 The equilibrium E T is locally asymptotically stable if r 1 > 1 an r 2 < 1 The equilibrium E Q is locally asymptotically stable if r 2 > 1 an r 3 < 1 The equilibrium E T Q is locally asymptotically stable if r 2 > 1 an r 3 > 1 32 Persistence Before we can state our results on the persistence of the three compartments, we will nee some notions an theorems from [8] Definition 31 Let X be a nonempty set an ρ: X R + A semiflow Φ: R + X X is calle uniformly weakly ρ-persistent, if there exists some ε > 0 such that lim sup t ρφt, x > ε for all x X, ρx > 0 Φ is calle uniformly strongly ρ-persistent if there exists some ε > 0 such that lim inf t ρφt, x > ε for all x X, ρx > 0 A set M X is calle weakly ρ-repelling if there is no x X such that ρx > 0 an Φt, x M as t System 2 generates a continuous flow on the phase space X := { R, T, Q R 3 +} Theorem 31 Rt is always uniformly persistent T t is uniformly persistent if r 1 > 1 an r 2 < 1 as well as if r 2 > 1 an r 3 > 1 Qt is uniformly persistent if r 2 > 1 Proof To show uniform persistence of the the susceptible compartment, we will use the metho of fluctuation see eg [7, Lemma A1] We enote by R the limit inferior of Rt, while T an Q enote the limit superior of T t, resp Qt as t Using the fluctuation lemma we know that

6 6 there exists a time sequence t k such that Rt k R an R t k 0 as k If we apply this to the equation for Rt, we obtain R t k + β 1 Rt k T t k + β 2 Rt k Qt k = A It is easy to see that for the total roent population we have Rt + T t + Qt A, thus, 0 T A an 0 Q A Using this an letting k we get R β 1+β 2 To show persistence of the infeste compartments, we nee some theory from [8] We use the notation x = R, T, Q X for the state of the system an the usual notation ωx for the ω-limit set of a point x efine as ωx := {y X : {t n } n 1 s t t n an Φt n, x y as n } We first show the persistence of T t Let ρx = T Let us consier the invariant extinction space of T, efine as X T := {x X : ρx = 0} We follow [8, Chapter 8] an examine the set Ω x XT := x XT ωx Applying the Benixson Dulac criterion with Dulac function 1/Q an the Poincaré Benixson theorem, we obtain that all solutions in the extinction space X T ten to an equilibrium Let us first consier the case r 1 > 1 an r 2 1 Clearly, in this case Ω = {E R } As a first step, we prove weak ρ-persistence In orer to apply [8, Theorem 817], we let M 1 = {E R } Then Ω is a subset of M 1, which is isolate, compact, invariant an acylic We have to show that M 1 is weakly ρ-repelling, from which we obtain persistence Let us suppose that this oes not hol, ie there exists a solution such that lim t Rt, T t, Qt = A, 0, 0 an T t > 0 Then for any ε > 0, for sufficiently large t, we have Rt > A ε an Qt < ε For such t, we can give the following estimation for T t: T t = T tβ 1 Rt β 2 Qt > T t β 1 A β 1ε β 2 ε, which is positive if ε is sufficiently small as Aβ1 > follows from r 1 > 1 This contraicts T t 0 In the secon case, when r 2 > 1 an r 3 > 1, also E Q exists, so we have Ω = {E R, E Q } Now we let M 1 = {E R } an M 2 = {E Q } Clearly, Ω M 1 M 2 an {M 1, M 2 } is acyclic an M 1 an M 2 are invariant, compact an isolate We have to show that M 1 an M 2 are weakly ρ- repelling Suppose first that M 1 is not weakly ρ-repelling Then there exists a solution such that lim t Rt, T t, Qt = A, 0, 0 an T t > 0 Again,

7 7 for any ε > 0, for sufficiently large t, we have Rt > A an Qt < ε an for such t, we can give the following estimation for T t: T t = T tβ 1 Rt β 2 Qt > T t β 2 A β 1ε β 2 ε, where we use that β 1 > β 2, which follows from r 2 r 3 > 1 This expression is positive for ε small enough, which contraicts T t 0 Now let us suppose that M 2 is not weakly ρ-repelling Then there exists a solution such that lim t Rt, T t, Qt = β 2, 0, A β 2 Then, for any ε > 0, if t is large enough, then Rt > β 2 ε an Qt < A β 2 + ε an for such t we can give the following estimation for T t: T t = T tβ 1 Rt β 2 Qt [ > T t β1 A β 2 β 1 ε β 2 = T t β1 β 2 β 2 Aβ2 β 1 + β 2 ε ] + ε, which is positive for ε small enough as r 3 > 1 This contraicts T t 0 Let us now turn to the persistence of Qt in the case r 2 > 1 We set ρx = Q We have the equilibrium E R if r 1 1 an the two equilibria E R an E T if r 1 > 1 Similarly to the case of T t, we efine the extinction space of Q as X Q := {x X : ρx = 0} = {R, T, 0 R 3 +} In this case we have Ω = x XQ ωx = {E R } if r 1 1 an Ω = x XQ ωx = {E R, E T } if r 1 > 1 We efine M 1 = {E R } an M 2 = {E T } Just like in the proof of the persistence of T t, Ω is invariant, an M 1 an M 2 are isolate an acyclic To show that M 1 is weakly ρ-repelling, we can procee in an analogous way as in the case of T t In the case r 1 > 1, we have to show that M 2 is weakly ρ-repelling Suppose this oes not hol Then there exists a solution such that lim t Rt, T t, Qt = β 1, A β 1, 0 an Qt > 0 Then, for any ε > 0, if t is sufficiently large, then Rt > β 1 ε an T t > A β 1 ε an for such t we can give the following estimation for Qt: Q t = Qtβ 2 Rt + β 2 T t > Qt β 2 β 1 ε + β 2 = Qt Aβ2 2β 2 ε, A β 1 ε which is positive if ε is small enough, as r 2 > 1, which contraicts Qt 0 We have shown uniform weak persistence in all cases; to show uniform strong persistence, we apply Theorem 45 from [8] Our flow is clearly

8 8 continuous, the subspaces X T, X Q, X \ X T an X \ X Q are invariant The existence of a compact attractor is also clear, as all solutions enter a compact region after some time This means that all conitions of [8, Theorem 45] hol an thus we obtain uniform strong persistence 33 Global stability Theorem 32 1 Equilibrium E R is globally asymptotically stable if r 1 < 1 an r 2 < 1 2 Equilibrium E T is globally asymptotically stable on X \ X T if r 1 > 1 an r 2 < 1 E R is globally asymptotically stable on X T 3 Equilibrium E Q is globally asymptotically stable on X \X Q if r 2 > 1 an r 3 < 1 E R is globally asymptotically stable on X Q if r 1 < 1 an E T is globally asymptotically stable on X Q if r 1 > 1 4 Equilibrium E TQ is globally asymptotically stable on X \ X T X Q if r 2 > 1 an r 3 > 1 E T is globally asymptotically stable on X Q an E Q is globally asymptotically stable on X T Proof First we note that the roent subsystem 2 can be reuce to two imensions by introucing the notation F t := Rt + T t We obtain the system F t = A β 2 F tqt F t, Q t = β 2 F tqt Qt This system has two equilibria, A, 0 an β 2, A β 2, with the latter one only existing if r 2 > 1 We use the Dulac function 1/Q to show that there is no perioic solution of 4: A β 2 F Q F F Q + β 2 F Q Q = β 2 Q Q Q < 0 Thus, applying the Benixson Dulac criterion, we obtain that there is no perioic solution of 4, an by the Poincaré Benixson theorem we get that all solutions ten to an equilibrium In the first two cases, when r 2 < 1, only the first equilibrium exists Thus, in this case Qt 0 an F t A as t, an therefore, the secon equation of 2 takes the following form on the limit set: with γ = Aβ 1 4 T t = β 1 A T t T t T t = γt t β 1 T 2 t

9 9 The solution starte from T t = 0 is the constant solution T t 0, while the nontrivial solutions take the form Ce γt 1 + β1 γ Ceγt 5 for C R + Clearly, for r 1 < 1 which is equivalent to γ 0, the solutions ten to zero on the limit set, therefore, for all solutions, T t 0 as t If r 1 > 1 ie γ > 0, we have lim t T t = A β 1 on the limit set; using the persistence of T t we obtain that for all solutions, T t A β 1 as t In the case r 2 > 1, also the secon equilibrium exists However, we know from the previous subsection that for r 2 > 1, the compartment Qt is uniformly persistent, so no solution with positive initial value in Qt can ten to the first equilibrium Thus, the limit of all such solutions is the secon equilibrium an Qt A β 2 as t We can procee in a similar way as in the case r 2 < 1: on the limit set, we can transform the secon equation of 2 to with γ = β 1 β 2 T t = β 1 β 2 A T t T t β 2 β 2 T t T t = γt t β 1 T 2 t Aβ2 Similarly as above, we can see that he solution starte from T t = 0 is the constant solution T t 0, while the nontrivial solutions take the form 5 In the case r 3 < 1 which is equivalent to γ 0, the solutions ten to 0, while if r 3 > 1 which is equivalent to γ > 0, we have lim t T t = β 2 Aβ2 β 1, an this is what we wante to show Remark 31 We note that changing global asymptotic stability to attractivity, the results of Theorem 32 also hol when the given reprouction numbers are equal to 1, instea of being smaller than 1 4 The human subsystem Let us now turn to the human subsystem of 1 consisting of the last three equations In the sequel, we assume that the roent subsystem is in a steay state, an substitute any of the equilibria of the roent subsystem into these

10 10 equations to obtain the system S t = B η 1 T St η 2 Q St ν 1 StIt ν 2 StJt δst + θ 1 It + θ 2 Jt, I t = η 1 T St + ν 1 StIt η 2 Q It ν 2 JtIt δit θ 1 It, J t = η 2 Q St + η 2 Q It + ν 2 StJt + ν 2 ItJt δjt ρjt θ 2 Jt, where T an Q are the secon, resp thir coorinates in any of the four equilibria 3 To fin all possible equilibria of 6, first we introuce the notation Gt := St + It to obtain the system 6 G t = B η 2 Q Gt ν 2 GtJt δgt + θ 2 Jt, J t = η 2 Q Gt + ν 2 GtJt δjt ρjt θ 2 Jt 7 We will apply the Benixson Dulac criterion with Dulac function 1/J an the Poincaré Benixson theorem to obtain that in this case, all solutions of system 7 ten to one of the equilibria Inee, we have B η 2 Q G ν 2 GJ δg + θ 2 J G J + J = η 2Q J η 2 Q G + ν 2 GJ δj ρj θ 2 J J ν 2 δ J η 2G J 2 < 0, from which we obtain the assertion above This equation may have two equilibria: D+Bν 2 D Bν Bη 2Q ν 2δ+ρ 2δν 2, D+Bν2+ D Bν Bη 2Q ν 2δ+ρ 2δ+ρν 2 an D+Bν 2+ D Bν Bη 2Q ν 2δ+ρ 2δν 2, D+Bν2 D Bν Bη 2Q ν 2δ+ρ 2δ+ρν 2 enote by E 1 an E 2, respectively, with D = δ 2 +Q η 2 ρ+δq η 2 +θ 2 +ρ The first coorinate of E 1 is always positive, since this coorinate may be rewritten as D+Bν 2 D+Bν 2 2 4Bδν 2δ+θ 2+ρ 2δν 2

11 11 It can easily be seen that the first coorinate of E 2 is always positive Let us first consier the case Q > 0, ie when the roent subsystem tens to the equilibrium E Q or E TQ, which is equivalent to r 2 > 1 In this case, the secon coorinate of E 1 is always positive, while secon coorinate of E 2 is negative if Q > 0 Hence, in the case Q > 0, there is only one equilibrium an using the Poincaré Benixson theorem, we obtain that all solutions ten to E 1 In the case r 2 1, ie when Q = 0, the system takes the simpler form G t = B ν 2 GtJt δgt + θ 2 Jt, J t = ν 2 GtJt δjt ρjt θ 2 Jt 8 This system has the two equilibria B e 1 := δ, 0 an e 2 := δ + θ2 + ρ, Bν 2 δδ + θ 2 + ρ ν 2 ν 2 δ + ρ Now, it is easy to see that the first of these equilibria always exists, while the secon one only exists if R J 0 := Bν 2 δδ + θ 2 + ρ > 1 Just as above, we obtain that all solutions of 8 ten to one of these equilibria In the case R J 0 1, this equilibrium is clearly e 1 Using similar methos as for the roent subsystem, we will show that Jt is always uniformly strongly persistent if R J 0 > 1 To show this, we choose ρx = J Consier the extinction space X J efine as X J := {x R 2 + : ρx = 0}; now Ω = M 1 := e 1, which is obviously invariant, isolate an acyclic Let us suppose that M 1 is not weakly ρ-repelling, ie there is a solution which tens to e 1 such that Jt > 0 Then, given any ε > 0, for t sufficiently large, we can give the following estimate for J t: J t = ν 2 GtJt δjt ρjt θ 2 Jt > Jt ν 2 B δ ν 2ε δ ρ θ 2, which is positive as R J 0 > 1 Hence, in the case R J 0 > 1, all solutions of 8 starte with positive initial value J0 ten to the equilibrium e 2 We have now finishe the analysis of 7 an showe that in each case, epening on the reprouction numbers r 2 an R J 0, all solutions of this equation ten to an equilibrium Let us enote by J the secon coorinate of this equilibrium an substitute this value into the first two equations of

12 12 6 to obtain S t = B η 1 T St η 2 Q St ν 1 StIt ν 2 J St δst + θ 1 It + θ 2 J, I t = η 1 T St + ν 1 StIt η 2 Q It ν 2 J It δit θ 1 It This equation has a similar structure as 7 The two possible equilibria of system 9 are ν 1B+θ 2J +P ν 1B+θ 2J P 2 +H 2ν 1K, ν1b+θ2j P + ν 1B+θ 2J P 2 +H 2ν 1K an ν 1B+θ 2J +P + ν 1B+θ 2J P 2 +H 2ν 1K, ν1b+θ2j P ν 1B+θ 2J P 2 +H 2ν 1K enote by E 1 an E 2, respectively, where the notations K, P an H are efine as K = δ + η 2 Q + ν 2 J, P = Kδ + θ 1 + η 1 T + η 2 Q + ν 2 J an H = 4η 1 ν 1 T B + θ 2 J K Again, we can apply the Benixson Dulac criterion, in this case with the Dulac function 1/I, to show that all solutions ten to an equilibrium: B η 1 T S η 2 Q S ν 1 SI ν 2 J S δs + θ 1 I + θ 2 J S I η 1 T S + ν 1 SI η 2 Q I ν 2 J I δi θ 1 I I + I = η 1T η 2Q ν 1 ν 2J δ I I I I η 1T S I 2, which is negative for all I, S > 0 Similarly as in the case of the equilibria of system 7, it is easy to see that the first coorinates of E 1 an E 2 are always positive, while the secon coorinate of E 2 is negative if T > 0 ie when r 1 > 1 an r 2 < 1, meaning that E T is globally asymptotically stable or r 2 > 1 an r 3 > 1 meaning that E TQ is globally asymptotically stable Hence, in this case there is only one equilibrium, an by the Poincaré Benixson theorem, all solutions ten to this equilibrium From the above, we obtain that if T > 0 an Q > 0, ie when r 2 > 1 an r 3 > 1, then all solutions ten to the equilibrium E1 1, E1 2, E1, 2 where upper inex i enotes the ith coorinate of a given equilibrium In the case T > 0 an Q = 0, ie when r 1 > 1 an r 2 1, E 1 is the only equilibrium of 9, an the reprouction number R J 0 etermines which 9

13 13 equilibrium is the limit of the solutions of 8 Hence, if r 1 > 1, r 2 1 an R J 0 1 then all solutions of 6 ten to the equilibrium E 1 1, E 2 1, 0, while if r 1 > 1, r 2 1 an R J 0 > 1 then all solutions of 6 ten to the equilibrium E1 1, E1 2, Bν 2 δδ + θ 2 + ρ ν 2 δ + ρ In the case T = 0 ie when r 1 < 1 an r 2 < 1, meaning that E R is globally asymptotically stable or r 2 > 1 an r 3 < 1, meaning that E Q is globally asymptotically stable, system 9 reuces to S t = B η 2 Q St ν 1 StIt ν 2 StJ δst + θ 1 It + θ 2 J, I t = ν 1 StIt η 2 Q It ν 2 ItJ δit θ 1 It, which has the two equilibria E 1 = η2q +ν 2J +δ+θ 1 ν 1, ν1b+θ2j η 2Q +ν 2J +δη 2Q +ν 2J +δ+θ 1 ν 1η 2Q +ν 2J +δ, resp B + θ 2 J E 2 = η 2 Q + ν 2 J + δ, 0 10 One may easily observe that the secon equilibrium always exists, while the sign of the secon coorinate of the first equilibrium epens on the parameters an the limits Q an J : the first equilibrium exists if an only if R I 0 := ν 1 B + θ 2 J η 2 Q + ν 2 J + δη 2 Q + ν 2 J + δ + θ 1 > 1 In the case R I 0 1, there is only one equilibrium, E 2, so it is clear from the above that all solutions of 6 ten to the equilibrium B + θ 2 J η 2 Q + ν 2 J + δ, 0, J In the case R I 0 > 1, we will again use persistence theory to show that all solutions of 10 ten to the equilibrium E 1 We now choose ρx = I an consier the extinction space X I := {x R 2 + : ρx = 0} It is clear that now Ω = M 1 := {E 2 }, which is invariant, acylic an isolate Let us suppose that M 1 is not weakly ρ-repelling, ie there exists a solution which

14 14 tens to E 2 such that It > 0 Then, for any ε > 0, for large enough t, we can estimate I t as I t = Itν 1 St η 2 Q ν 2 J δ θ 1 B + θ 2 J > It ν 1 η 2 Q + ν 2 J + δ + ε η 2 Q ν 2 J δ θ 1, which is positive as R I 0 > 1 From this we obtain that in the case R I 0 > 1, all solutions of 10 starte with positive initial value I0 ten to E 1 On the ω-limit set of solutions of 6, equation 10 hols, which has at most two equilibria Hence, the global attractor of 10 consists either of a single equilibrium or two equilibria an connecting orbits between them When there is only one equilibrium, then the solutions of 6 ten to this equilibrium When two equilibria exist, then Jt is uniformly persistent, hence, the ω-limit set of positive solutions of 6 can only be the equilibrium with the positive J coorinate Now we go through all possibilities regaring the value of Q an J to give a precise characterization In the case Q > 0 ie r 2 > 1, there is only one equilibrium of 7, hence Jt tens to E1 2 This means that in the case r 2 > 1 an R I 0 1 all solutions of 6 ten to the equilibrium B+θ 2E 2 1, 0, A η 2 β +ν 2 2E1 2+δ E2 1, while in the case r 2 > 1 an R I 0 > 1, all solutions of 6 ten to the equilibrium η2q +ν 2E 2 1 +δ+θ1, ν1b+θ2e2 1 η2q +ν 2E 2 1 +δη2q +ν 2E 2 1 +δ+θ1 ν 1η 2Q +ν 2E1 2+δ, E1 2 ν 1 with Q = A β 2 In the case r 2 1 ie Q = 0, the reprouction number R J 0 etermines the limit of Jt In the case r 1 1, r 2 1, R J 0 1, R I 0 1, all solutions of 6 ten to the equilibrium B δ, 0, 0 In the case r 1 1, r 2 1, R J 0 1, R I 0 > 1, all solutions of 6 ten to the equilibrium δ + θ1, B ν 1 δ δ + θ 1, 0 ν 1

15 15 In the case r 1 1, r 2 1, R J 0 > 1, R I 0 1, all solutions of 6 ten to the equilibrium δ + θ2 + ρ, 0, ν 2B δδ + θ 2 + ρ ν 2 ν 2 δ + ρ In the case r 1 1, r 2 1, R J 0 > 1, R I 0 > 1, all solutions of 6 ten to the equilibrium ν2b+θ 1ρ+δθ 1 θ 2 ν 1δ+ρ, δθ2 ν2b ν + δ+θ2+ρ 1δ+ρ ν 2 θ1 ν 1, ν2b δδ+θ2+ρ ν 2δ+ρ 5 Discussion We have establishe a six-compartment moel to escribe the sprea of an infectious isease sprea by ectoparasites which are transmitte to humans by roents We have ientifie three reprouction numbers for the roent subsystem These threshol numbers etermine which of the four possible equilibria of the roent subsystem is globally attractive Assuming that the roent subsystem is alreay in a steay state, we stuie the human subsystem an calculate the possible equilibria of this subsystem epening on which of the roent equilibria is globally attractive We also etermine which equilibrium of the human subsystem is globally attractive Our results show that in each case, epening on the ifferent reprouction numbers, one equilibrium is globally attractive Our results show that if one type of the parasite infectious or noninfectious is present in the roent population, then the same type will also be present in the human population Using our results, we may stuy the possibilities of eraicating the isease There are three main ways to control the isease: we may ecrease the transmission rates η 1,2 between humans an roents, increase the isinfestation rates θ 1,2 of humans to shorten the uration of infestation of humans an we may reuce which means culling of the roents Controlling only the human population increasing the isinfestation rates θ 1,2 only results in a mitigation not sufficient to eraicate the isease The same hols for ecreasing the transmission rates from roents to humans, except the extreme case of ecreasing the transmission rates η 1,2 to zero In this latter case, one may ecrease the human reprouction numbers to be less then 1 by increasing the isinfestation rates θ 1,2 an thus eliminate the infestation By controlling the roent population increasing the eath rate, one can reuce the reprouction numbers r 1 an r 2 to be both less than 1 an this way one may eliminate the infestation among the roents Also

16 16 in this case, infestation from roents to humans can be eliminate an this way the human reprouction numbers etermine which equilibrium of the human subsystem will be globally attractive Hence, also in such a case, by increasing the isinfestation rate among humans may result in the elimination of the parasites an of the isease Acknowlegements A Dénes was supporte by Hungarian Scientific Research Fun OTKA PD an OTKA K G Röst was supporte by the EU-fune Hungarian grant EFOP an by Hungarian Scientific Research Fun OTKA K References 1 A Dénes an G Röst, Biomath 1, , A Dénes an G Röst, Nonlinear Anal Real Worl Appl 18, A Dénes an G Röst, Impact of excess mortality on the ynamics of iseases sprea by ectoparasites, in: M Cojocaru, I S Kotsireas, R N Makarov, R Melnik, H Shoiev, Es, Interisciplinary Topics in Applie Mathematics, Moeling an Computational Science, Springer Proceeings in Mathematics & Statistics, , M R Holbrook, J F Aronson, G A Campbell, S Jones, H Felmann, A D T Barrett, J Infect Dis 191, M J Keeling an C A Gilligan, Proc R Soc B 267, Ch D Paock an M E Eremeeva, Rickettsialpox, in: D Raoult, Ph Parola, Rickettsial iseases, CRC Press, H L Smith, An introuction to elay ifferential equations with applications to the life sciences, Springer, New York, H L Smith an H R Thieme, Dynamical systems an population persistence, AMS, Provience, Y Tselentis an A Gikas, Murine typhus, in: D Raoult, Ph Parola, Rickettsial iseases, CRC Press, G Watt an P Kantipong, Orientia tsutsugamushi an scrub typhus, in: D Raoult, Ph Parola, Rickettsial iseases, CRC Press, 2007