Ferreri, Luca

Giacobini, Mario Dante Lucio

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Institutional Research Information System University of Turin

17 October 2023AperTO - Archivio Istituzionale Open Access dell'Università di TorinoOriginal Citation:Intrinsic Properties of Epidemiological Dynamics on Bipartite NetworksTerms of use:Open Access(Article begins on next page)Anyone can freely access the full text of works made available as "Open Access". Works made availableunder a Creative Commons license can be used according to the terms and conditions of said license. Useof all other works requires consent of the right holder (author or publisher) if not exempted from copyrightprotection by the applicable law.Availability:This is the author's manuscriptThis version is available http://hdl.handle.net/2318/92988 sinceIntrinsic Properties of EpidemiologicalDynamics on Bipartite NetworksLuca Ferreri, and Mario Giacobini{luca.ferreri,mario.giacobini}@unito.itComplex Systems Unit, Molecular Biotechnology CenterComputational Epidemiology Group - Department of Animal Production, Epidemiology and Ecology, Faculty of Veterinary MedicineUniversity of Torino, ItalyIntrinsic Properties of EpidemiologicalDynamics on Bipartite NetworksLuca Ferreri, and Mario Giacobini{luca.ferreri,mario.giacobini}@unito.itComplex Systems Unit, Molecular Biotechnology CenterComputational Epidemiology Group - Department of Animal Production, Epidemiology and Ecology, Faculty of Veterinary MedicineUniversity of Torino, ItalyBipartite NetworksIn bipartite networks nodes are divided in two families and edges are allowed only betweenindividuals belonging to different families.A A A AB B B B BUntil nowadays bipartite graphs have been used to describe different structures, as:I co-actor movie networks, having actor and movie nodes;I academic collaboration networks, where researchers bind with their articles;I heterosexual contact networks♂ ♂ ♂ ♂♀ ♀ ♀ ♀ ♀I vector-host contact networksVector-Borne DiseasesA disease is called vector-borne when the pathogen is transmitted from an infected individual toanother individual by a blood-sucking arthropod, called vector.Many vector-borne diseases are a particular kind of zoonosis, infectious diseases that could betransmitted from animals to humans, being of big importance for public health reasons.Vector-borne diseases depend upon three different classes of actors:I the pathogens, either viruses, bacteria, protozoa, or helminths (worms);I the vectors, commonly arthropods such as ticks or mosquitoes;I the hosts, usually vertebrate, which could be infected by vectors, thus becoming source ofinfection for vectors successively feeding on them.An example: Lyme-BorreliosisLyme Borreliosis (LB) is the most common vector-borne disease of humans in temperateclimates. It can affect at different threat levels several tissues in the human body: skin, heart,nervous system, eyes, kidneys, and liver.LB’s pathologic agent are spirochetal bacteria belonging to the Borrelia genus. Vectors are hardticks of the genus Ixodes (in Europe I. ricinus), while hosts are small-medium vertebrate such asreptiles, birds and mammals depending on the host-seeking tick stage.Nymphs(Infected)Larvae(Uninfected)Eggs(Uninfected)Females(Infected / Uninfected)Adults(Infected)Hosts / Reservoir(Infected)Hosts / Reservoir(Infected / Uninfected)Hosts / Reservoir(Infected / Uninfected)Infections Spreading on Bipartite NetworksThe transmission of pathogens over a bipartite networks was firstly analyzed byGómez-Gardeñes and colleagues for heterosexually transmitted diseases.Those results were generalized by Bisanzio and coauthors and applied to vector-borne diseases.In that paper an important result was proven: if until then the aggregative behavior of vectors onhosts was usually described by negative-binomial distributions, they showed on different datasetsthat the best fitting was power law with exponents around 2.5.This implies an important consequence: the epidemic threshold vanishes in the limit of largenetwork sizes. In other terms, the condition for an endemic behavior, observed in severalcountries around the world, seems to be well supported by field data and by the network models.Moreover,I on dynamical processes occurring on bipartite networks we observe two exciting phenomenapathogens become endemic for lower probability transmission than those need to spread overunipartite networks structure sharing the same topological proprieties;I pathogens show an oscillating behavior which has an attractive biological interpretation.Evolutionary Advantage?We compare the epidemic threshold derived on bipartite networks with those derived on samebut unipartite networks. In fact we compare the asymptotic behavior for enlarging populationhaving degree distributed with two power-laws of exponents αA and αB. Analytically wedemonstrate that there exist couples of exponents (αA, αB) for which the epidemic thresholdtends to zero for bipartite but not for unipartite networks.αAαB1 2 3123Oscillation BehaviorWe generate a number of bipartite random networks with different sizes and connectionprobabilities p. Over these we let a disease spreads observing the following rules:I a sound individual at time t connected to at least one infected nodes will beinfectious at time t + 1I an infectious node at time t will be susceptible at time t + 1, no matter if it isconnect or not with infective individuals!FFWhat we observe is that after few iterations is that the system reaches an oscillating equilibrium.In fact we found a 2 periodic solution and even more amazingly we have a (anti-)synchronizationphenomena.Analytical resultsLet’s consider random bipartite networks, where A and B are the two families of nodes, and twonodes n ∈ A and m ∈ B are connected with probability p.For a node belonging to the A partition the degree distribution is nothing but the binomial:PA(k) =(NBk)pk (1− p)NB−k (1)and analogous for nodes belonging to B partition. Calling x(t) the prevalence of infected A-nodesat time t , and respectively y(t) the prevalence of infected B-nodes, and using eq. (1) we coulddetect the probability for a A-node to have k neighbors of which h infected is nothing but:(NBk)pk(1− p)NB−k(kh)y(t)h(1− y(t))k−hHowever, a susceptible node get the infection if at least one neighbor is infectious.Therefore, we sum previous for h = 1, . . . , k . Thus, at time t + 1 the prevalence in family A isnothing butx(t + 1) = (1− x(t))− (1− x(t))(1− py(t))NB (2)and similarly for the prevalence in family B.Now, in order to check the 2-periodic solutions, which mean oscillations of period 2, we imposethat concurrently xt+1 = xt−1 and yt+1 = yt−1. We numerically find the solutions using theNewton’s method. We observe they change as the value of p changes.pxyxyxyp = 10−4 p = 0.015 p = 0.04DiscussionsResults presented here are of some interests from a biological point of view.I The fact that there exist scale free random bipartite networks on which the transmissionprobability needed for a pathogen to became endemic tends to zero, in the limit of largepopulation size, while this does not happens on same but unipartite networks could mean thatpathogen could take advantage of the peculiar transmission route described by bipartite network.That could result in an evolutionary advantage for these pathogens as their transmissionprobabilities is less subject to selection pressure.I Moreover the oscillating behavior observed in infection spreading over synthetic bipartitenetworks is observed in real world: e.g. tick nymphs are active in Spring while larvae are activeduring the Summer of the same year. Hence, in order for the pathogen to be passed to a nextgeneration of larvae, hosts are required to keep the pathogen from Spring to Summer, havingbeen first infected by nymphs and then transmitting to larvae.Essential bibliographyI Pastor-Satorras, R., Vespignani, A. - Epidemic Spreading in Scale-Free Networks - Phys. Rev. Let., 86,14, 3200–3203 (2001)I Gómez-Gardeñes, J., Latora, V., Moreno, Y., Profumo, E.: - Spreading of sexually transmitted disease in heterosexually populations - PNAS, 105, 5, 1399–1404 (2008)I Bisanzio, Bertolotti, Tommasone, Amore, Ragagli, Mannelli, Giacobini, Provero: - Modeling the Spread of Vector-Borne Diseases on Bipartite Networks - PLoS ONE, 5, 10, 2010I Ferreri L., Venturino E., Giacobini M. - Do Disease Spreading on Bipartite Networks Have Some Evolutionary Advantages? - LNCS 6623 - 2011I Gray J., Kahl O., Lane R.S., and G. Stanek - Lyme Borreliosis. Biology, Epidemiology and Control - CABI 2002Acknowledgement: this project has been supported by the Lagrange Project on Complex Systems, CRT Foundation.

Intrinsic Properties of Epidemiological Dynamics on Bipartite Networks

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Intrinsic Properties of Epidemiological Dynamics on Bipartite Networks

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