Original article Biomath 1 (2012), 1209256, 1–5 B f Volume ░, Number ░, 20░░ BIOMATH ISSN 1314-684X Editor–in–Chief: Roumen Anguelov B f BIOMATH h t t p : / / w w w . b i o m a t h f o r u m . o r g / b i o m a t h / i n d e x . p h p / b i o m a t h / Biomath Forum Structure of the Global Attractors in a Model for Ectoparasite-Borne Diseases Attila Dénes∗ and Gergely Röst∗ ∗ Bolyai Institute, University of Szeged, Aradi vértanúk tere 1, Szeged, Hungary Emails: denesa@math.u-szeged.hu, rost@math.u-szeged.hu Received: 15 July 2012, accepted: 25 September 2012, published: 19 October 2012 Abstract—We delineate a mathematical model for the dynamics of the spread of ectoparasites and the diseases transmitted by them. We present how the dynamics of the system depends on the three reproduction numbers belonging to three of the four possible equilibria and give a complete characterization of the structure of the global attractor in each possible case depending on the reproduction numbers. Keywords-ectoparasites; global dynamics; global attrac- tors I. INTRODUCTION Ectoparasites (e.g. lice, fleas, mites) cause a serious problem in several parts in the world [2], [4]. Ectopar- asite infestations are often connected to the lack of hygiene and poor economical conditions, however, their presence is increasing in developed countries as well. The three louse species which transmit diseases are the head louse, the body louse and the pubic louse. These species are responsible for the spread of trench fever, epidemic typhus and relapsing fever. The flea species which most commonly affect humans are the cat, the rat and the human flea. Fleas transmit plague, murine typhus, fleaborne spotted rickettsiosis. The transmission of these diseases is different from that of other vector- borne diseases, as it is carried out through the human contact network, which means that the spread of the vectors themselves is similar to that of a disease. In this paper we delineate a model for the dynamics of ectoparasite-borne diseases and we describe the struc- ture of the global attractors in the different situations depending on the reproduction numbers. We assume the presence of one disease and one ectoparasite species which is a vector transmitting this particular disease. The human population is divided into three compart- ments: susceptibles (i.e. those who can be infested by both infectious and non-infectous vectors, denoted by S(t)), those who are infected by non-infectious para- sites (denoted by T (t)) and those who are infested by infectious vectors (denoted by Q(t)). We assume that someone infested by non-infectious vectors can transmit the parasites to susceptibles, while an individual infested by infectious vectors transmits both the parasites and the disease to susceptibles. An individual infested by infectious vectors transmits the infection to individuals infested by non-infectious vectors, i.e. a member of com- partment T can move to compartment Q upon adequate contact with someone from compartment Q. We assume that a person is infected by the disease if and only he is infested by infectious parasites. We suppose that individuals infested by infected parasites transmit the disease at the same rate to susceptibles and to those who are infested by non-infected parasites. We denote this transmission rate by βQ, and βT denotes the transmission rate for non-infectious vectors (to susceptibles). The rate of disinfestation is denoted by µ for the infected compartment and by θ for the non-infected compartment. We denote by b the natural birth and death rates, and we assume the disease is not fatal, thus the population size is constant. In the model equations we use mass action incidence. We have the following system of differential equations Citation: A. Dénes, G. Röst, Structure of the Global Attractors in a Model for Ectoparasite-Borne Diseases, Biomath 1 (2012), 1209256, http://dx.doi.org/10.11145/j.biomath.2012.09.256 Page 1 of 5 http://www.biomathforum.org/biomath/index.php/biomath http://dx.doi.org/10.11145/j.biomath.2012.09.256 A. Dénes et al., Structure of the Global Attractors in a Model for Ectoparasite-Borne Diseases (with all the parameters assumed to be positive): S′(t) = −βT S(t)T (t) − βQS(t)Q(t) + θT (t) + µQ(t) + b − bS(t), T ′(t) = βT S(t)T (t) − βQQ(t)T (t) − θT (t) − bT (t), Q′(t) = βQS(t)Q(t) + βQQ(t)T (t) − µQ(t) − bQ(t). (1) It can easily be seen that any solution with non- negative initial values remains non-negative for all for- ward time. We can suppose that N (t) = S(t) + T (t) + Q(t) = 1 holds for the total population. The phase space of our system is X := {(S, T, Q) ∈ R3+ : S + T + Q = 1}. II. EQUILIBRIA, REPRODUCTION NUMBERS By solving the algebraic equations 0 = −βT S∗T ∗ − βQS∗Q∗ + θT ∗ + µQ∗ + b − bS∗, 0 = βT S ∗T ∗ − βQQ∗T ∗ − θT ∗ − bT ∗, 0 = βQS ∗Q∗ + βQQ ∗T ∗ − µQ∗ − bQ∗, we can determine the four equilibria of system (1): ES = (1, 0, 0), ET = ( b + θ βT , 1 − b + θ βT , 0 ) , EQ = ( b + µ βQ , 0, 1 − b + µ βQ ) , EQT = ( θ − µ + βQ βT , b + µ βQ − θ − µ + βQ βT , 1 − b + µ βQ ) . Reproduction numbers have a clear biological interpreta- tion. We can obtain them by multiplying the number of new infections and the average length of the infectious period of an infectious agent newly introduced into a population currently being in one of the equilibria. By introducing an infested, non-infectious individual into a population in the equilibrium ES , we obtain the reproduction number R1 = βT b + θ , by introducing an infested and infectious individual into the same equilibrium we obtain the reproduction number R2 = βQ b + µ . Calculating the expected number of secondary infections caused by the introduction of an infectious infested individual into a population in the equilibrium ET gives the same reproduction number R2. If we introduce a non-infectious infested individual into a population in the equilibrium EQ, we obtain the reproduction number R3 = βT (b + µ) βQ(βQ − µ + θ) . The following lemma is taken from [3]. Lemma 2.1: The equilibrium ES always exists. The equi- librium ET exists if and only R1 > 1. The equilibrium EQ exists if and only if R2 > 1. The equilibrium EQT exists if and only if R2 > 1 and R3 > 1. III. STRUCTURE OF THE GLOBAL ATTRACTOR Here we recall the main result of [3]. Theorem 3.1: Let XQ := {(S, T, 0) ∈ R3+ : S + T = 1} and XT := {(S, 0, Q) ∈ R3+ : S + Q = 1} denote the extinction sets for Q and T , respectively. The four equilibria have the following global stability properties depending on the reproduction numbers: (i) Equilibrium ES is globally asymptotically stable if R1 ≤ 1 and R2 ≤ 1. (ii) Equilibrium ET is globally asymptotically stable on X \ XT if R1 > 1 and R2 ≤ 1. On XT , ES is globally asymptotically stable. (iii) If R2 > 1, R3 ≤ 1 and R1 ≤ 1, then EQ is globally asymptotically stable on X \ XQ and ES is globally asymptotically stable on XQ. (iv) If R2 > 1, R3 ≤ 1 and R1 > 1, then EQ is globally asymptotically stable on X \ XQ and ET is globally asymptotically stable on XQ. (v) If R2 > 1, R3 > 1, then EQT is globally asymptotically stable on X \(XQ ∪XT ), ET is globally asymptotically stable on XQ and EQ is globally asymptotically stable on XT . An equilibrium E is said to be globally asymptotically stable on a set Y if it is stable and for all y ∈ Y the solution starting from y converges to E as t → ∞. Following the notation of [1, 1.1.7], by M t we denote the set consisting of the states at time t of the solutions started from all of the points x ∈ M . Definition 3.2: Let A ∈ X be a compact invariant set. If A attracts each bounded subset of X, i.e. for any bounded subset M ⊂ X and any neighbourhood U of A there exists a T < ∞ such that M t ⊂ U for all t > T , then A is called the global attractor. Definition 3.3: The ω-limit set of a point x ∈ X, denoted by ω(x) consists of those elements y of X for which there exists a real sequence {tn} such that tn ↗ ∞ and xtn → y as n → ∞. The α-limit set is defined similarly with tn ↘ −∞. In the following theorem we describe the structure of the global attractor for system (1) in the five cases listed in Theorem 3.1. Theorem 3.4: The global attractor A for system (1) has the following structure: (i) A = {ES}. Biomath 1 (2012), 1209256, http://dx.doi.org/10.11145/j.biomath.2012.09.256 Page 2 of 5 http://dx.doi.org/10.11145/j.biomath.2012.09.256 A. Dénes et al., Structure of the Global Attractors in a Model for Ectoparasite-Borne Diseases Fig. 1. Representation of the flow and the attractors on the T Q-plane. (ii) A = {ES , ET }∪γ1, where γ1 is a connecting orbit from ES to ET , which is actually the segment between ES and ET in the extinction space XQ. (iii) A = {ES , EQ}∪γ2, where γ2 is a connecting orbit from ES to EQ, which is actually the segment between ES and EQ in the extinction space XT . (iv) A = {ES , ET , EQ} ∪ γ1 ∪ γ2 ∪ γ3 ∪ A1, where γ3 is a connecting orbit from ET to EQ, and A1 is the domain surrounded by ES , ET , EQ, γ1, γ2 and γ3 in the T Q- plane consisting of connecting orbits from ES to EQ. (v) A = {ES , ET , EQ, EQT } ∪ γ1 ∪ γ2 ∪ γ4 ∪ γ5 ∪ A2, where γ4 is a connecting orbit from ET to EQT , γ5 is a connecting orbit from EQ to EQT , and A2 is the domain surrounded by ES , ET , EQ, EQT , γ1, γ2, γ4 and γ5 in the T Q-plane consisting of connecting orbits from ES to EQT . Proof: (i) As proved in Theorem 3.1, ES is globally asymptotically stable in case (i), which means that the global attractor is the singleton ES in this case. For the proof of the remaining cases we reduce the sys- tem to two dimensions by substituting S with 1−T −Q. We get the system T ′(t) = βT (1 − T (t) − Q(t))T (t) − βQQ(t)T (t) − θT (t) − bT (t), Q′(t) = βQ(1 − T (t) − Q(t))Q(t) + βQQ(t)T (t) − µQ(t) − bQ(t) (2) and the four equilibria ES = (0, 0), ET = ( 1 − b + θ βT , 0 ) , EQ = ( 0, 1 − b + µ βQ ) , EQT = ( b + µ βQ − θ − µ + βQ βT , 1 − b + µ βQ ) . By standard linearization, we calculate the eigenvalues and eigenvectors of the Jacobian of the linearized system in the four equilibria. The details of the calculations are straightforward thus omitted, here we only discuss the results and implications. The eigenvalues of the Jacobian Biomath 1 (2012), 1209256, http://dx.doi.org/10.11145/j.biomath.2012.09.256 Page 3 of 5 http://dx.doi.org/10.11145/j.biomath.2012.09.256 A. Dénes et al., Structure of the Global Attractors in a Model for Ectoparasite-Borne Diseases of the linearized equation around the equilibrium ES are λ1 = −b−θ + βT = (b + θ)(R1 −1) with corresponding eigenvector (1, 0) and λ2 = −b−µ+βQ = (b+µ)(R2− 1) with corresponding eigenvector (0, 1). Linearizing at the equilibrium ET , one finds the eigenvalues λ1 = b + θ − βT = (b + θ)(1 − R1) with the eigenvector (1, 0) and λ2 = −b − µ + βQ = (b + µ)(R2 − 1) with the eigenvector( (βQ + βT )(b + θ − βT ) βT (βQ + βT − 2b − θ − µ) , 1 ) . Linearization around the steady state EQ gives the fol- lowing eigenvalues of the Jacobian: λ1 = b + µ − βQ = (b + µ)(1 − R2) with the eigenvector (0, 1) and λ2 = −θ+µ−βQ +(b+µ)βT /βQ = (R3−1)βT /(R2R3) with the eigenvector (1, 0). Finally, if we linearize the system around the equilibrium EQT , we obtain the eigenvalues λ1 = b + µ − βQ = (b + µ)(1 − R2) with corresponding eigenvector( (βQ + βT )(βQ(βQ + θ) − bβT − µ(βQ + βT )) βT (b(βQ + βT ) − βQ(2βQ + θ) + (2βQ + βT )µ) , 1 ) and λ2 = θ − µ + βQ − (b + µ)βT /βQ = (1 − R3)βT /(R2R3) with corresponding eigenvector (1, 0). (ii) If R1 > 1 and R2 < 1 then ES has the stable eigenvector (0, 1) and the unstable eigenvector (1, 0). This means that ES has a one-dimensional stable manifold which coincides with the invariant extinction space XT and a one-dimensional unstable manifold which coincides with the segment (ES , ET ) of the extinction space XQ, while both of the eigenvectors at ET are stable. γ1 is the connecting orbit from ES to ET lying in XQ. If R2 = 1 then the second eigenvalue at ES is equal to zero. In this case, the equation for Q′(t) takes the form Q′(t) = −βQQ2(t) < 0 on XT , which means that all solutions started from XT tend to ES . Thus ES has the same one-dimensional stable and unstable sets as in the case R2 6= 1. From Theorem 3.1 we know that all solutions started from X \ XT tend to ET , thus ET has a two-dimensional stable set. (iii) If R1 < 1, R2 > 1 and R3 < 1 then ES has the stable eigenvector (1, 0), and the unstable eigenvector (0, 1), while (0, 1) and (1, 0) are both stable eigenvectors for EQ. If R1 = 1 then the equation for T ′(t) takes the form T ′(t) = −βT T 2(t) < 0, on the invariant extinction space XQ. This means that all solutions on the center manifold belonging to the zero eigenvalue (which coincides with XQ) tend to ES . If R3 = 1 then the Jacobian of the linearized system at EQ has a zero eigenvalue with eigenvector (1, 0). The line Q = 1 − b + µ βQ is invariant: if we substitute 1 − (b + µ)/βQ into the equation for Q′(t) we get Q′(t) = 0. This means that for R3 = 1 the center manifold belonging to the zero eigenvalue coincides with this line. For R3 = 1, the equation for T ′(t) has the form T ′(t) = −βT T 2(t) < 0 on this line, which means that all solutions started from this line tend to the equilibrium EQ. γ2 is the connecting orbit from ES to EQ lying in XT . This shows the statement of (iii). (iv) In this case, the first eigenvector belonging to ES loses its stability, while the same vector becomes a stable eigenvector for ET . Thus, ES has two unstable eigen- vectors and ET has the stable eigenvector (1, 0) and an unstable eigenvector. From Theorem 3.1 we know that any solution started from the one-dimensional unstable manifold of ET tends to EQ, from which the existence of a heteroclinic orbit γ3 from ET to EQ follows. The situation for EQ is the same as in case (iii). We have to show that A1 consists of heteroclinic orbits from ES to EQ. Let us take an arbitrary point p ∈ A1. From Theorem 3.1 we know that ω(p) = {EQ}. The negative limit set α(p) exists and is non-empty as the backward orbit is bounded by γ1 ∪ γ2 ∪ γ3. From the Poincaré–Bendixson Theorem we know that α(p) can only be an equilibrium point (as there are no periodic orbits). We can exclude EQ as it has a two-dimensional stable manifold. The unstable manifold of ET coincides with XQ, which is invariant and A1 ∩ XQ = ∅, thus α(p) = {ES}. (v) In this case, again, ES has two unstable eigenvectors and thus a two-dimensional unstable manifold. Similarly to the previous case, ET has a stable and an unsta- ble eigenvector, thus having a one-dimensional stable manifold and a one-dimensional unstable manifold. The eigenvector (1, 0) for EQ is unstable, which means that EQ has a one-dimensional stable manifold and a one-dimensional unstable manifold. EQT has two stable eigenvectors and thus a two-dimensional stable manifold. From Theorem 3.1 we know that all solutions started from X \ (XT ∪ XQ) tend to EQT , thus there exists a connecting orbit γ4 from ET to EQT and a connecting orbit γ5 from EQ to EQT . Similarly to case (iv) we can show that the domain A2 consists of connecting orbits from ES to EQT . If neither is available on your system, please use the font closest in appearance to Times. Avoid using bit-mapped fonts if possible. True-Type 1 or Open Type fonts are preferred. Please embed symbol fonts, as well, for math, etc. IV. CONCLUSION We described the global attractor in all possible cases. Depending on the three reproduction numbers, the global attractor might have the following structure: Biomath 1 (2012), 1209256, http://dx.doi.org/10.11145/j.biomath.2012.09.256 Page 4 of 5 http://dx.doi.org/10.11145/j.biomath.2012.09.256 A. Dénes et al., Structure of the Global Attractors in a Model for Ectoparasite-Borne Diseases • a singleton • a one-dimensional set consisting of two equilibria and a connecting orbit • a two-dimensional set consisting of three or four equlib- ria and connecting orbits between them. The biological interpretation of our results is the following. The reproduction numbers Ri (i = 1, 2, 3) completely deter- mine whether the infectious or the non-infectious parasites can invade a human population. This is mathematically expressed in the structure of the global attractors that we described. If R1 ≤ 1 and R2 ≤ 1, then the population is safe from any parasites. The implication for the control of the infection and infestation is that to eradicate the disease only, we have to decrease R2 to be less than 1, which is possible by reducing βQ or increasing µ. To eliminate all the parasites, besides decreasing R2 we also have to decrease R1 (possible by reducing βT or increasing θ). Decreasing only R1 is not enough for the elimination of the parasites. The reproduction number R3 is a threshold parameter which shows whether all the parasites become infectious or both infectious and non- infectious parasites can be present in the population. The transmission rates βQ and βT can be effectively reduced by vigorous monitoring and isolation of infested individuals, while µ and θ can be increased by disinfestation treatment of individuals. ACKNOWLEDGMENT Research supported by European Research Council Starting Grant Nr. 259559, OTKA K75517 and Bolyai Scholarship of Hungarian Academy of Sciences. REFERENCES [1] N. P. Bhatia, G. P. Szegö, Dynamical Systems: Stability Theory and Applications, Springer-Verlag, 1967. [2] P. Brouqui, D. Raoult, “Arthropod-Borne Diseases in Homeless", Ann. N.Y. Acad. Sci., vol. 1078, pp. 223–235, 2006. http://dx.doi.org/10.1196/annals.1374.041 [3] A. Dénes, G. Röst, “Global dynamics for the spread of ectoparasite-borne diseases", submitted. [4] L. Houhamdi, P. Parola, D. Raoult, “Les poux et les maladies transmises à l’homme", Med. Trop., vol. 65, pp. 13–23, 2005. [5] H. L. Smith, H. R. Thieme, Dynamical Systems and Population Persistence, Graduate Studies in Mathematics, vol. 118, AMS, Providence, 2011. Biomath 1 (2012), 1209256, http://dx.doi.org/10.11145/j.biomath.2012.09.256 Page 5 of 5 http://dx.doi.org/10.1196/annals.1374.041 http://dx.doi.org/10.11145/j.biomath.2012.09.256 Introduction Equilibria, Reproduction Numbers Structure of the Global Attractor Conclusion References