[EN] In this paper, we address our study to analyze diseases whose transmission
occurs through a contaminated environment. Moreover, the individuals
are organized in compartments by age range. This epidemiological process with
indirect transmission of the disease has a mathematical representation by means
of a nonlinear discrete-time invariant system. A strategy to act when the system
is unstable is proposed. As a consequence of it, the new system is N-periodic and
its stability is given by the monodromy matrix of the system. The above matrix
analysis is applied to establish some conditions over the number of steps that we
can be without action and being the epidemiological process asymptotically stable.The authors would like to thank the referees and the editor for their comments and useful suggestions for improvement of the
manuscript. This research was partially supported by Spanish Grant MTM2013-43678-P.Cantó Colomina, B.; Coll, C.; Pagán Moreno, MJ.; Poveda-Giner, JJ.; Sánchez, E. (2016). Spectral properties with application to epidemic models. International Journal of Complex Systems in Science. 6(1):1-7. http://hdl.handle.net/10251/81091S176

Cantó Colomina, Begoña

Coll, Carmen

Pagán Moreno, Mª Jesús

Poveda-Giner, Joan Josep

Sánchez, Elena

[EN] In this paper, we address our study to analyze diseases whose transmission
occurs through a contaminated environment. Moreover, the individuals
are organized in compartments by age range. This epidemiological process with
indirect transmission of the disease has a mathematical representation by means
of a nonlinear discrete-time invariant system. A strategy to act when the system
is unstable is proposed. As a consequence of it, the new system is N-periodic and
its stability is given by the monodromy matrix of the system. The above matrix
analysis is applied to establish some conditions over the number of steps that we
can be without action and being the epidemiological process asymptotically stable.The authors would like to thank the referees and the editor for their comments and useful suggestions for improvement of the
manuscript. This research was partially supported by Spanish Grant MTM2013-43678-P.Cantó Colomina, B.; Coll, C.; Pagán Moreno, MJ.; Poveda-Giner, JJ.; Sánchez, E. (2016). Spectral properties with application to epidemic models. International Journal of Complex Systems in Science. 6(1):1-7. http://hdl.handle.net/10251/81091176

RiuNet

Int. J. Complex Systems in Sciencevol.6(1) (2016), pp. 1–7Spectral properties with application to epidemic modelsBegon˜a Canto´1,†, Carmen Coll1, Mar´ıa Jesu´s Paga´n2, JoanPoveda2 and Elena Sa´nchez 11 Instituto de Matema´tica Multidisciplinar, Universitat Polite`cnica de Vale`ncia.Camino de Vera, 14. 46022 Valencia. Spain.2 Grupo de Investigacio´n e Innovacio´n Alimentaria, Universitat Polite`cnica deVale`ncia. Camino de Vera, 14. 46022 Valencia. Spain.Abstract. In this paper, we address our study to analyze diseases whose trans-mission occurs through a contaminated environment. Moreover, the individualsare organized in compartments by age range. This epidemiological process withindirect transmission of the disease has a mathematical representation by meansof a nonlinear discrete-time invariant system. A strategy to act when the systemis unstable is proposed. As a consequence of it, the new system is N -periodic andits stability is given by the monodromy matrix of the system. The above matrixanalysis is applied to establish some conditions over the number of steps that wecan be without action and being the epidemiological process asymptotically stable.Keywords: Structured matrix, positive system, periodic system, controlMSC 2000: 93A30,93C55† Corresponding author: bcanto@mat.upv.esReceived: July 27th, 2016Published: November 30th, 20161. Model description and resultsIn recent years, concern has grown about foodborne illness because food isthe vehicle for the transmission of numerous diseases, such as salmonella [3,4, 7, 8]. Therefore, the source of food contamination is becoming a growingpublic health problem in industrialized countries. This contamination canbe endogenous (animals are fed on such a food) or exogenous (which takesplace during processing, transport or storage), [9]. According to the annualreport of Epidemiological Surveillance of Communicable Diseases issued bythe National Epidemiological Surveillance Network; in 2011, 271 Salmonella2 Spectral properties with application to epidemic modelsoutbreaks were reported in Spain, with 2861 sick people and 7 deaths. Morethan half of them were caused by salmonella (85% of outbreaks) followed byS. typhimurium (10%), [6]. Therefore, one of the strategies that can be raisedis how to prevent the disease in the initial production stageIn [2, 4] mathematical models in order to analyze the salmonella diseasein a henhouse are showed and, in [1] the amount of bacteria in the henhouseis also considered.In this paper, we consider a discrete-time mathematical model where thevariables S(t) and I(t) represent time dependent population densities of sus-ceptible and infected individuals at time t, t ∈ Z, t ≥ 0, respectively. Thismodel estimates the prevalence of salmonella infection in the henhouse. Weanalyze the epidemic process taking into account the salmonella density in theenvironment, being this density denoted by C(t). Then, the behavior of thesalmonella will be analyzed through the evolution of the infected individualand contaminant populations.Then, the model is described by the following nonlinear systemx(t+ 1) = F (x(t))where x(t) = (S(t) I(t) C(t))T and the parameters involved are the usual inthis type of models. That is, the parameters 0 < p, q, s < 1 shall representdensity survival rate of S(t), I(t) and C(t), respectively, the parameter σdenotes the exposition rate, the death removal rate at time t is denoted by µ(t)and P is the population size. Moreover, µ(t)P and βI(t), 0 < β < 1, representthe replacement rate and the density of pathogen produced by individualsinfected, respectively.A linear model may arise from the linearization of above system arounddisease-free equilibrium point, and this approach can provide sufficient accu-racy describing the evolution of the system.Linearizing around of the disease-free equilibrium point x∗ = F (x∗) de-noted by Pf = (Sf , O,O) we obtain a linear discrete-time systemxˆ(t+ 1) = Axˆ(t) + b,being xˆ(t) = x(t)−x?, A =( ∗ ∗O E)and we rewrite E = T +F. To analyzethe behavior of the solution around the Pf , we only consider the subsystemcontaining the variables causing infection, i.e, variables that correspond to theinfected and the contaminant. It is clear that this subsystem is not asymptot-ically stable if ρ(E) > 1. In our model, considering a population P , it easy tocheck that if ρ(E) > 1 the eigenvalues areλ1,2 =q + s±√(q − s)2 + 4βσP2B. Canto´ et al 3with λ1 > 1 and the sign of the second eigenvalue, λ2, depends on the size ofthe population.So, if P satisfies P <qsβσ, then λ2 > 0, with 0 < λ2 < 1 < λ1. And, if Psatisfiesqsβσ< P , then λ2 < 0, with −1 < λ2 < 0 < 1 < λ1.On the other hand, one of the most important parameters of epidemicmodeling is the basic reproduction number R0 which is a measure or indicatorto know whether the disease will disappear at all or not. In [5] the expressionof R0 is given for an epidemiological model with indirect transmission. Thus,R20 = βσP (1− q)−1(1− s)−1.The disease will die out in the long run if R0 < 1 and the disease will beable to spread through a population if R0 > 1.We consider the case R0 > 1, then the population P satisfiesP > K =(1− q)(1− s)βσNote that, K < qsβσ if and only if q + s > 1, this occurs when the densitysurvival rate of infected individual plus the density survival rate of pathogenis greater than one.If we focus on the fact that the infection does not disappear in the hen-house, we want to find strategies for infection fade. Since the henhouse isusually not clean, one strategy may be to perform a periodic cleaning of thehenhouse to eliminate the contaminant but we need to delay as much as pos-sible this cleaning action. We want to find the minimum number of steps thatcan be uncleaned, keeping the asymptotically stable system. That is, if we letanother step without cleaning the environment, then the disease grows andthe system is unstable in a neighborhood of free-disease point.Therefore, we start with the initial condition (S(0), I(0), 0) and performcleaning every N stages leaving the henhouse free of contaminant. Then, theprocess is given in the following systemI(t+ 1) = qI(t) + σ(t)B(t)S(t)B(N) = s(t)B(t) + βI(t)where σ(t) y s(t) are two N -periodic functions defined byσ(t) ={0 t = 0σ t = 1, , 2, . . . , N − 1 , s(t) ={0 t = 0s t = 1, , 2, . . . , N − 1 ,σ(t+N) = σ(t) y s(t+N) = s(t).Linearizing around of its disease-free equilibrium point, we have the fol-lowing periodic system4 Spectral properties with application to epidemic models(I(t+ 1)B(t+ 1))=(q f(t)β s(t))(I(t)B(t))= E(t)(I(t)B(t)), (1)wheref(t) = σ(t)Sf and E(t) = T (t) + F (t) given byT =(q 00 s(t)), F (t) =(0 f(t)β 0).We obtain a parameter that quantifies the transmission potential of the diseaseby using the rates of transmissibility through of the pathogen produced by theinfected individuals, taking into account the “monodromy matrix” associatedwith the periodic system. This matrix is defined asΦE(k, k0) =k−1∏i=k0E(k0 + k − 1− i), k > k0, E(k0, k0) = In.In this case, it is easy to see that we can limit the study of the behavior of thespectral radius of the matrix of monodromy in s = 0. So, we want to get thegreatest value of N so that ρ(EN,0) < 1 y ρ(EN+1,0) > 1.The parameters q and s are important for controlling salmonella, and thepopulation size in the henhouse plays an important role because it determinesthe prevalence of salmonella. Now, we study the case mentioned above.If P satisfies that K < P <qsβσ, then λ2 > 0. With, 0 < λ2 < 1 < λ1.We search for a period N such thatXN =λN1 (q − λ2) + λN2 (λ1 − q)λ1 − λ2 < 1andXN+1 =λN+11 (q − λ2) + λN+12 (λ1 − q)λ1 − λ2 > 1being Xn = ρ(EN,0) and we denoteh(N) =λ1 − λ2 − (q − λ2)λN1λN1 (λ1 − q).Then, it follows the following result.Lemma 1 Consider the spectrum of matrix E. If R0 > 1 with P < qsβσthen(i) λ2 < q < λ1B. Canto´ et al 5(ii) h(N) is a decreasing function and there exits N˜ such that 0 < h(N) < 1for all N < N˜ .Proof. If R0 > 1 with P satisfying P < qsβσ, then λ2 > 0, with 0 < λ2 < 1 <λ1.(i) The proof is based on the following observation. λ1 + λ2 = q + s andfrom the expression of the eigenvalues it is easy to see that λ1 − q > 0 andq − λ2 > 0.(ii) The basic idea of the proof is to takeh(N) = λ−N1λ1 − λ2λ1 − q −q − λ2λ1 − q .By (i) it is clear thatλ1 − λ2λ1 − q > 0 andq − λ2λ1 − q and λ−N1 is a decreasing function,then h(N) is also a decreasing function with h(0) = 1. According to the aboveexpression of h(N), we haveh(1) =λ1(1− q) + λ2(λ1 − 1)λ1(λ1 − q) > 0.Since h(N) is a decreasing function there exists N˜ such that h(N˜) > 0 andh(N˜ + 1) < 0. Moreover, 0 < h(N) < 1 for all N < N˜. The behavior of the solution of the system around the free equilibriumpoint is now discussed. The prevalence of infection by excreting salmonella canbe controlled using cleaning strategies and the desired value of N is obtainedin the following result.Proposition 1 Consider that the trajectory of the N -periodic system given by(1) is in an environment of the disease free point with R0 > 1 and P < qsβσ.Take N ∈ N satisfying N < N˜ andln(h(N))ln(λ2λ1 )< N < N + 1 <ln(h(N + 1))ln(λ2λ1 )(2)then N is the maximum of the period values such that the N -periodic systemis asymptotically stable.Proof. If N satisfies the condition given in the proposition and using theexpression of the spectral radius we have h(N) > (λ2λ1)N >λ1λ2h(N + 1) andλ1 − λ2 > λN1 ((λ2λ1 )N (λ1 − q) + (q − λ2)) then ρ(EN,0) < 1.On the other hand, (λ2λ1 )N+1 > h(N +1) and λN+11 ((λ2λ1)N+1(λ1− q)+(q−λ2)) > λ1 − λ2 then ρ(EN+1,0) > 1.6 Spectral properties with application to epidemic modelsThus, N satisfies ρ(EN,0) < 1 and ρ(EN+1,0) > 1. The obtained results suggest that to reduce the level of salmonella in thehenhouse is necessary to reduce the quantity of excretions in the henhousebecause this decreases the transmission of salmonella. The given result showsthe reduction in salmonella level by using an effective control strategy everyN steps.Acknowledgements: The authors would like to thank the referees andthe editor for their comments and useful suggestions for improvement of themanuscript. This research was partially supported by Spanish Grant MTM2013-43678-P.References[1] C. Beaumont, J. B. Burie, A. Ducrot, P. Zongo Propagation ofSalmonella within an industrial hen house. SIAM J. Appl. Math. 72 41113-1148 (2012).[2] A. D. C. Berriman, D. Clancy, H. E. Clough, R. M. Christley,Semi-stochastic models for Salmonella infection within finishing pig unitsin the UK. Math. Biosci. 245 148-156 (2013).[3] K. Bollaerts, W. Messens, M. Aerts, J. Dewulf, D. Maes, K.Grijspeerdt, Y. Stede, Evaluation of scenarios for reducing humansalmonellosis trough household consumption of fresh minced pork meatRisk Analysis.30 853-865 (2010).[4] M. H. Brown, K. W. Davies, C. M. P. Billon, C. Adair, P. J.McClure, Quantitative microbiological risk assessment: principles ap-plied to determining the comparative risk of salmonellosis from chickenproducts Journal of Food Protection. 61 1446-1453 (1998).[5] B. Canto´, C. Coll, E. Sa´nchez Epidemic dynamics of an infectionthrough the pathogen density in the environment, Comptes rendus del’Acadmie bulgare des Sciences 69(7) 835-844 (2016).[6] S. Finstad, C. A. O’Bryan, J. A. Marcy, P. G. Crandall, S. C.Ricke, Salmonella and broiler processing in the United States: relation-ship to foodborne salmonellosis. Food Research International 45 789-794(2012).[7] M. C. Malpass, A. P. Williams, D. L. Jones, H. M. Omed, Mi-crobiological quality of chicken wings damaged on the farm or in theprocessing plant. Food Microbiol 27 521-525 (2010).B. Canto´ et al 7[8] N. Olaimat, R. A. Holley Factors influencing the microbial safety offresh produce: A review. Food Microbiol. 32 1-19 (2012).[9] Piedrola Gil, Medicina Preventiva y Salud Pu´blica Elsevier Masson,(2016).

Spectral properties with application to epidemic models

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Spectral properties with application to epidemic models

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