The analysis of the SIRS alcoholism models with relapse on weighted networks

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1 DOI /s RESEARCH Open Access The analysis of the SIRS alcoholism models with relapse on weighted networks Hai Feng Huo * and Ying Ping Liu *Correspondence: hfhuo@lut.cn Department of Applied Mathematics, Lanzhou University of Technology, Lanzhou 735, Gansu, People s Repulic of China Astract Two SIRS alcoholism models with relapse on networks with fixed and adaptive weight are introduced. The spread of alcoholism threshold R is calculated y the next generation matrix method. For the model with fixed weight, we prove that when R < 1, the alcohol free equilirium is gloally asymptotically stale, then the drinking crowd gradually disappear. When R > 1, the alcoholism equilirium is gloal attractivity, then the density of alcoholics will remain in a stale value. For the model with adaptive weight, we only make some numerical simulations. We also give two effective strategies. Our results show that the treatment of recuperator for stopping relapsing and preventing the susceptile people to drink are two effective measures to eliminate alcoholism prolem, and preventing the susceptile people to drink is more effective when the proportion of recuperator to accept treatment is equal to the proportion of susceptile people to refuse drinking alcohol. Keywords: Alcoholism, Threshold, Fixed weight, Adaptive weight, Optimal control, Stale Background Alcohol use and ause have een part of human society for centuries. The College Alcohol Study defines alcoholism as male students who had five or more and female students who had four or more drinks in a row at least once in a 2-week period Wechsler 2. Similar to other drug addictions, alcoholism can cause to a series of serious consequences. The World Health Organization estimates that aout 14 million people around the world under the influence of alcohol-related prolems, such as eing sick, losing a jo and so on Saunders et al What is more serious is that chronic alcohol consumption damages almost all parts of the ody and contriute to a numer of human diseases including ut not limited to liver cirrhosis, pancreatitis, heart disease, and sexual dysfunction and eventually e deadly Glavas and Weinerg 25. Early recognized since the 18s that alcoholism produced not only impairment of the senses ut also higher predisposition for tuerculosis. William Osler reported in 195 that patients who misused alcohol had higher predisposition to pneumonia Giraldina et al And studies over the last 3 years have also demonstrated that chronic alcohol consumption impairs the functions of oth T cells and B cells Sumana et al Between 26 and 21 in the United States, alcohol ause resulted in approximately 88, deaths, and the 216 The Authors. This article is distriuted under the terms of the Creative Commons Attriution 4. International License which permits unrestricted use, distriution, and reproduction in any medium, provided you give appropriate credit to the original authors and the source, provide a link to the Creative Commons license, and indicate if changes were made.

2 Page 2 of 19 average death rate associated with alcoholism was 28.5 per 1, population Gonzales et al Importantly, the majority of alcohol-related deaths were among adults aged 2 64 years old Giraldina et al In view of the aove situation, alcoholism has ecome a issue that need to e solved urgently. Since mathematical model can mimic the process of alcoholism and provide useful methods to control the spread of drinking ehavior. Several different mathematical models for alcoholism have een formulated and studied recently. Sanchez et al. 27 presented a simple model for alcohol treatment. Their model were ased on studying inge drinking in a college system and assumed the same leaving rate. Manthey et al. 28 uilt a model to capture the dynamics of campus drinking and to study the spread of drinking on campus. Benedict 27 proposed an SIR model and used standard contact rate etween susceptile and alcoholics. Furthermore they otained the alcoholism reproductive numer and discussed the existence and staility of all the equiliria. Huo and Song 212 introduced a two-stage model for inge drinking prolem, which the youths with alcohol prolems were divided into those who admit the prolem and those who do not admit it. Wang et al. 214 presented a deterministic SATQ-type mathematical model for the spread of alcoholism with two control strategies and analyzed some properties of the solutions including positivity, existence and staility. Huo and Wang 214 developed a nonlinear mathematical model with the effect of awareness programs on the inge drinking. Their results showed that awareness programs is an effective measure in reducing alcohol prolems. Quit drinking is usually temporary. Some drinking people may relapse since contacting with alcoholics or weak self-control aility. Sharma and Samanta 215 developed an alcohol ause model y introducing a treatment programm in the population and considered all possile relapses. They assumed that the drinkers in treatment most commonly relapse due to contact with heavy drinkers who are not in treatment. For the other mathematical models for alcoholism or smoking, please see Wechsler and Nelson 28, Room et al. 25, Mushayaasa and Bhunu 211, Huo and Zhu 213 and references cited therein. They commonly assume that communities are homogeneous, that is, communities are made up of individuals who mix uniformly and randomly with each other in the aove models. These assumptions make the analysis tractale ut not realistic Bansal et al. 27. In contrast to classical compartment models, a lot of studies on complex networks have een investigated during the past years. Liu et al. 213 presented an SIR model with individual s irth and death on scale-free networks and analyzed the staility of three equiliria. They also gave out two immunization schemes. But they didn t consider recuperator s temporarily immune, that is to say, recuperator is likely to ecome infected or susceptile ecause of the loss of immune. Zhu et al. 212a investigated a new epidemic SIS model with nonlinear infectivity on heterogeneous networks. The gloal ehavior of the model is studied. Wang et al. 212 proposed a modified SIS model with an infective vector on complex networks. They treated direct human contacts as a social network and assumed spatially homogeneous mixing etween vector and human populations. Huo and Liu 216 proposed an alcoholism model on complex heterogeneous networks and proved staility of all the equiliria. Nodes usually represent individuals and links represent potential contacts among those individuals on the complex network Zhu et al. 212a. The connectivity of a node

3 Page 3 of 19 is defined as the numer of the links connected to the node, represent y k. The degree distriution of a network is defined as the proaility of a randomly chosen node to have a degree k, represent y Pk. Many networks Zhu et al. 212; Liu and Zhang 211; Zhang and Jin 211; Liu et al. 24 have een found to e scale-free networks, that is to say the degree distriution follows a power law distriution Pk = ck γ, 2 <γ 3, where c is any constant satisfy the equation n k=1 Pk = 1. Notice that in many real contact networks, there are groups of nodes with a high density of edges within them and a lower density of links etween groups. The differences etween links within a contact network can e descried y link weights, which can represent the amount of time two individuals interact or the intimacy etween individuals. The larger the weight is, the more the two nodes communicate, while, the more possile a susceptile individual will e infected through the edge Chu et al. 29. The usual assumption is that weights are constant and driven y the network connectivity, which is fixed as time goes on. For example, the weight etween two nodes with degrees i and j are represented y a function of their degrees Barrat et al. 24a,, c. However, as the disease ecomes severe, individuals tend to e more cautious in social contacts and make some reflection such as decreasing the out going visits, cutting down the meeting time and reducing the intimacy. Such ehaviors will change the strengths of nodes and the weights of links, which can e seen as an adaptive weight network. Further, it was found that the infectivity exponent has a stronger effect on the epidemic threshold and the epidemic prevalence than the weight exponent Chu et al For the other mathematical models on network with weights, please see Macdonald et al. 25, Zhu et al. 213 and references cited therein. Motivated y the Liu et al. 213, Zhu et al. 212, 213, we introduce the individuals irth and death in our model, and set up an SIRS alcoholism model in complex network. Furthermore, we study the impact of the fixed weight and adaptive weight on the spread of alcoholism. We not only introduce general forms of the weight function to account for different cases of transmission ut also to analyze the influence of weights on alcoholism spreading. In addition, we add the group of recuperator and study the relapse of the recuperator. We also give some control strategies against drinking, our results show that the treatment of recuperator for stopping relapsing and preventing the susceptile people to drink are two effective control strategy, and the latter has more effective than the former when the proportion of recuperator to accept treatment is equal to the proportion of susceptile people to refuse drinking alcohol. The paper is organized as follows: in Model formulation section, we set up the model via differential equations. Then we present a gloal analysis of the model in Gloal dynamics of the model section. In Control strategy section, we perform two control strategies. In Sensitivity analysis and numerical simulations section, we perform sensitivity analysis and numerical simulations. We finally conclude the paper and give some measures to control alcoholism in Conclusion and discussions section. Model formulation In epidemic model, the total population N generally is divided into susceptile, represented y S, infections, represented y I and recovery, represented y R. An SIRS model, susceptile people usually infected y infections and ecome infected individual, after

4 Page 4 of 19 infection acquired immunity, it will e recovery. It allows memers of the recovered class to e free of infection and rejoin the susceptile class. Zhu et al. 212 introduced an SEIRS epidemic model with the incuation period on complex network. Zhu et al. 213 proposed a modified epidemic SIS model on an adaptive and weighted contact network, they introduced the general forms of the weight function and presented a new weight called adaptive weight. Liu et al. 213 presented an SIR model with individual s irth and death on scale-free networks and analyzed the staility of three equiliria. Motivated y these work, we set up a new SIRS alcoholism model on complex network. First, we introduce the individual s irth and death in our model, Second, we take into account adaptive weight in our model. At last, we studied the effect of alcohol relapse on the spread of the alcoholism. In our model, we divide the whole population into three compartments: the susceptile St, denote the people who do not drink or drink limited; the prolem alcoholic It, denote the people who drink more than daily and weekly limit; the recuperator Rt, denote the people who recover from alcoholism after treatment. In order to reflect the heterogeneity of the contact network, it is necessary to consider the node with different degree. Let S k t, I k t and R k t denote the densities of susceptile individuals, prolem alcoholics and recuperator individuals with degree k at time t respectively. Then St = k PkS kt, It = k PkI kt and Rt = k PkR kt are the average densities of susceptile individuals, prolem alcoholics and recuperator individuals respectively, where Pk is the proaility that a randomly chosen node has degree k. On complex networks, as alcoholism spread in a crowd, every site of N is empty or occupied y only one individual. Just as Liu et al. 213, we give each site a numer:, 1, 2, 3. We interpret the four states as: state : vacant; state 1: a susceptile individual occupied; state 2: a prolem alcoholic occupied; state 3: a recuperator individual occupied. The states of the system at time t can e descried y a set of numers {, 1, 2, 3}. Each site can change its state at a certain rate. We assume that a irth event occurs at a vacant node at rate. A susceptile individual can e infected through contact with a prolem alcoholic. While a prolem alcoholic can e cured at rate α or lead to relapse at rate β through contact with prolem alcoholics or other reasons. All individuals death rate is µ and we assume that alcoholism is not fatal. If a person is dead, the corresponding side ecomes vacant. Therefore, the dynamics of S k t, I k t and R k t are descried y the following differential equations ds k t di k t dr k t = 1 S k t I k t R k t ks k t t + σ R k t µs k t, = ks k t t + βr k t αi k t µi k t, = αi k t βr k t σ R k t µr k t, 1 with initial conditions ={S k t, I k t, R k t R+ 3n S kt 1, I k t 1, R k t 1, k = 1, 2,..., n}, 2 where t = i ϕi λ ik Pi ki i t, i 3

5 Page 5 of 19 and the parameters are all positive constants. The model structure is shown in Fig. 1. The meanings of the parameters and variales in model 1 and 3 are as follows 1 S k t I k t R k t represents the new orn susceptile individuals per unit time, which is proportional to vacant nodes irth rate and the density of vacant nodes 1 S k t I k t R k t. ks k t t represents the new prolem alcoholic individuals per unit time, which is proportional to the degree k, the density of susceptile individuals S k t and the proaility that alcoholism transmits through a link t, while λ ik is the transmission rate from nodes with degree i to nodes with degree k, ϕi is the infectivity of the prolem alcoholic nodes with degree i. So ϕi i is the link s average infectivity of the prolem alcoholic nodes with degree i. P i k is the proaility that a node of degree k connected to a node of degree i. In this paper, we focuses on degree uncorrelated networks. Hence, Pi k = ipi/ k, where k = i ipi is the average degree of the network. µ is the natural death rate. Since the disease of alcoholism is assumed not fatal, so there is no disease related death. We assumed that if an individual dies, the corresponding side will ecome empty. α represents the recovery rate of the prolem alcoholics. Some recuperators are likely to recur drinking. The density of relapse alcoholics is βr k, where β means the recurrence rate. σ represents the transfer rate from recuperator to susceptile people. There is little literature aout the network model with links or nodes weights, ut the weighted patterns on complex networks have various formats. Weighted patterns are used to represent the different intensities of infection y contact. Usually, the weight etween two nodes with degree i and j are measured y a function of their degrees ωi, j = ω ij m Barrat et al. 24a,, c; Macdonald et al. 25, where ω and m depend on the specific network. In the Escherichia coli metaolic network m =.5; in the US airport network m =.8; in the scientist collaoration network m =. Here, we use a different expression for the weight function ωi, j = gigj Zhu et al. 213, where gk is an increasing function of k, ecause the nodes with more connections will e more influential and gain larger weights. Since ωi, j estimates the links weight, the weight of each node k can also e measured y summing up the weights of links connected to it. Thus, k = k i Pi kωi, k. On uncorrelated networks, k = kgk kgk / k. We assume that the node with degree i has a fixed transmission rate given y λi, and the transmission y the link from the i-degree node to a k-degree node is measured y the proportion of this link s weight accounting for the k-degree nodes weight Chu et al So, we have Fig. 1 Transfer diagram for alcoholism model

6 Page 6 of 19 ωi, k λ ik = λi = λgk k i kgk 4 In this paper, we also consider people s health-conscious ehavior, so the value of weight function will ecome less and less as the alcoholism progresses. In particular, if a person has more neighors, it will e more cautious, therefore the weight will decrease more oviously. Thus, the weight function can e expressed as g k, t = gk exp hkit, where hk is an increase function of k. The corresponding λ ik ecomes λ ik λ k gk exp hkit = kgk exp hkit 5 where kgk exp hkit = i igi exp hiitpi. Sustituting 4 and 5 into 3 respectively, we get two t respectively t = λgk λgk ϕipii i t, i 6 and t = λgk exp hkit λgk exp hkit i ϕipii i t 7 it is clear that when hk =, 7 reduces to 6. Then, sustituting 6 into 1, we otain the fixed weight system ds k t = 1 S k t I k t R k t + σ R k t µs k t kgk λgk ks ktθ, di k t = λgk dr k t kgk ks ktθ + βr k t αi k t µi k t, = αi k t βr k t σ R k t µr k t, 8 and sustituting 7 into 1, we otain the adaptive weight system ds k t di k t = dr k t = 1 S k t I k t R k t λgk exp hkit kgk exp hkit ks ktθ + σ R k t µs k t, λgk exp hkit kgk exp hkit ks ktθ + βr k t αi k t µi k t, = αi k t βr k t σ R k t µr k t, where θt = i ϕii itpi in 8 and 9. Let N k t = S k t + I k t + R k t e the density of the whole individuals with degree k, k = 1, 2,..., n. Then adding the three equations in 8 or 9 gives 9 dn k t = + µn k t. 1

7 Page 7 of 19 By Eq. 1, we get that N k t = +µ + N ke +µt, where N k represents the initial density of whole population with degree k. Hence, lim t sup N k t = +µ, then N k t = S k t + I k t + R k t +µ for all t. Due to the limit system and original system have the same dynamic ehaviors for a long time. And S k t = +µ A kt I k t R k t at steady-state, it is sufficient to study the limiting systems di k t dr k t = kgk λkgk +µ I kt R k t θt + βr k t αi k t µi k t, = αi k t βr k t σ R k t µr k t, 11 and di k t = dr k t λgk exp hkit kgk exp hkit +µ I kt R k t θt + βr k t αi k t µi k t, = αi k t βr k t σ R k t µr k t, 12 It is easy to otain that I k t +µ and R kt +µ for t. So the region ={I k, R k I k t +µ, R kt +µ, k = 1, 2,..., n} is the positive invariant for oth 11 and 12. Gloal dynamics of the model The asic reproduction numer R Here, we first calculate the fix weight model s asic reproduction numer. Using the next generation method in Driessche and Watmough 22, it is clear that model 11 has an alcohol free equilirium E =,,..., 2k. System 11 can e written as and dx = Fx Vx, x = I k, R k T, where the rate of appearance of new infections is Fx = λkgkθ kgk +µ I k R k, and the rate of transfer of individuals out of compartments is βrk + α + µi Vx = k. αi k + β + σ + µr k The Jacoian matrices of Fx and Vx at the alcohol free equilirium E are F11 F = DFE = 13

8 Page 8 of 19 α + µe βe V = DV E = αe β + σ + µe, 14 where g1ϕ1p1 g1ϕ2p2 g1ϕnpn λ F 11 = + µ kgk 2g2ϕ1P1 2g2ϕ2P2 2g2ϕnPn , ngnϕ1p1 ngnϕ2p2 ngnϕnpn and E is identity matrix, is zero matrix. It is clear that V is a nonsingular M-matrix and F is a nonnegative matrix. According to the concept of next generation matrix and reproduction numer given in Driessche and Watmough 22, the reproduction numer of 11 equals to R = ρ FV 1 λβ + σ + µ kgkϕk = + µασ + µ + µβ + σ + µ kgk, 15 where kgkϕk = k i=1 iϕigipi. Coincidentally, we get that matrices F and V in model 12 are the same as that in model 11. Therefore, the reproduction numer R of model 12 is also given y 15, which implies that the adaptive weights cannot change the propagation threshold. According to the aove process and Theorem 2 in Driessche and Watmough 22, we otain the following results. Theorem 1 For the two alcoholism models 11 and 12, we have 1. Both of their asic reproductive numer are equal to R in If R < 1, the alcohol free equilirium E of 11 and 12 is locally asymptotically stale, ut unstale if R > 1, where R is defined y 15. Next, we will investigate the gloal staility of the alcohol free equilirium and the gloally attractive of the alcoholism equilirium of model 11. Since the staility of the equiliria in model 12 are difficult to demonstrate, so we only give some numerical simulations to discuss it at Sensitivity analysis and numerical simulations section. Uniqueness of the alcoholism equilirium We first give the following Lemma which guarantee that the density of population in each compartment cannot ecome negative or greater than +µ. Let I 1 = y 1, I 2 = y 2,..., I n = y n, R 1 = y n+1, R 2 = y n+2,..., R n = y 2n, we study the system for y 1,..., y n, y n+1,..., y 2n = ] 2n i=1 [,. +µ Lemma 1 The set is the positively invariant for system 11.

9 Page 9 of 19 Proof We will show that if y, then yt for all t >. Denote 1 = { y y i = for some i }, { 2 = y y i = where i = 1, 2,...,2n. Let the outer normals e denoted y ξi 1 =,..., 1,..., }{{} 2n ith and ξi 2 =,..., +1,..., }{{} 2n. We use the Nagumo s result in Yorke Since is a ith + µ for some i 2n-dimensional rectangle. From 11, it is easy to otain that for i = 1, 2,..., n. dy dy dy dy yi = yn+i = ξ 1 i yi = +µ yn+i = +µ ξ 1 n+i ξ 2 i = βy n+i, = αy i, }, = α + µ + µ, ξn+i 2 = β + σ + µ + µ. So, any solution that starts in y 1 2 stays inside. Furthermore, we will ascertain the uniqueness of the alcoholism equilirium. We give the following theorem. Theorem 2 There exists a unique alcoholism equilirium y = y 1,..., y n, y n+1,..., y 2n in model 11 when R > 1. Proof The alcoholism equilirium y = y 1,..., y n, y n+1,..., y 2n of system 11 is determined y equations { λigiθ kgk +µ y i y n+i + βy n+i α + µy i =, αy i βy n+i σ y n+i µy n+i =, a direct calculation yields y i = y n+i = λigiβ + σ + µθ + µ[α + µ + iθβ + σ + µ + αiθ αβ] kgk αλigiθ + µ[α + µ + iθβ + σ + µ + αiθ αβ] kgk Then we get a self-consistency equation of θ as follows θ = i = i ϕipiy i λigiβ + σ + µθ ϕipi + µ[α + µ + iθβ + σ + µ + αiθ αβ] kgk

10 Page 1 of 19 Oviously, θ = satisfies aove equation, then y i = y n+i =, which is the alcohol free equilirium of 11. We transform the self-consistency equation form as θf θ =, where f θ = 1 i λigiβ + σ + µ ϕipi + µ[α + µ + iθβ + σ + µ + αiθ αβ] kgk since f θ = i λi 2 giβ + σ + µβ + σ + α + µ ϕipi + µ kgk [α + µ + iθβ + σ + µ + αiθ αβ] 2 > and lim f θ = 1 θ the equation f θ = has a unique non-trivial solution θ if and only if f <. f = 1 = 1 = 1 i ϕipiigiλβ + σ + µ + µ[α + µβ + σ + µ αβ] kgk λβ + σ + µ kgkϕk + µ[α + µβ + σ + µ αβ] kgk λβ + σ + µ kgkϕk + µ[ασ + µ + µβ + σ + µ] kgk < λβ+σ +µ kgkϕk then R = +µ[ασ +µ+µβ+σ +µ] kgk > 1, through which the alcoholism equilirium is admitted. The proof is completed. Gloal staility of the alcohol free equilirium Here, we use the method in Lajmanovich and Yorke 1976, d Onofrio 28 to demonstrate the gloal ehavior of the system 11. By letting y = y 1,..., y n, y n+1,..., y 2n T. Then, equations in 11 can e rewritten as a form dy = Ay + Hy, 16 where Ay is the linear part, H y is the nonlinear part and A11 βe A =, αe β + σ + µe λg1y1 +y Ny = n+1θ kgk... nλgny n +y 2nθ kgk... T, where E is unit matrix and

11 Page 11 of 19 A 11 = λg1ϕ1p1 +µ kgk 2λg2ϕ1P1 +µ kgk. nλgnϕ1p1 +µ kgk α + µ λg1ϕ2p2 +µ kgk 2λg2ϕ2P2 +µ kgk. nλgnϕ2p2 +µ kgk... α + µ... λg1ϕnpn +µ kgk 2λg2ϕnPn +µ kgk nλgnϕnpn +µ kgk α + µ Lemma 2 Let A = a ij n n e an n n matrix, and assume a ij whenever i = j. Then there exists an eigenvector ω of A such that ω, and the corresponding eigenvalue is SA. The staility modulus SA is defined y SA = max Reλ i, i = 1,..., n, where λ i are the eigenvalues of A. Proof Choose c R, such that c + a ii, for i = 1, 2,..., n. Then A + ce is an n n nonnegative matrix. Therefore, y Theorem 2.2 from Varga 2, there exists a nonnegative eigenvector ω with nonnegative real eigenvalue equal to its spectral radius ρa + ce. So we have A + ceω = ρa + ceω, where E is the unit matrix. Then, Aω = ρa + ce cω, so ω is also an eigenvector of A, and the corresponding eigenvalue is ρa + ce c. If λ is any eigenvalue of A, then λ + c is an eigenvalue of A + ce, so λ + c ρa + ce, then we have λ + c ρa + ce, and λ ρa + ce c, therefore Reλ ρa + ce c, that is to say ρa + ce c is the maximum real part of all eigenvalues of A, so SA = ρa + ce c. The proof is completed. Lemma 3 Lajmanovich and Yorke 1976 Consider the system dy = Ay + Ny, 17 where A is an n n matrix and Ny is continuously differentiale in a region D R n. Assume i the compact convex set C D is positively invariant with respect to the system 17, and C; ii lim y Ny / y =; iii there exist r > and a real eigenvector ω of A T such that ω y r y for all y C; iv ω Ny for all y C; v y = is the largest positively invariant set contained in H ={y C ω Ny = }. Then either y = is gloally asymptotically stale in C, or for any y C {} the solution φt, y of 17 satisfies lim t inf φt, y m, independent of y. Moreover, there exists a constant solution of 17, y = k, k C {}. Theorem 3 For system 11. When R < 1, there exists an alcohol free equilirium y = is gloally asymptotically stale in. When R > 1, there exists an alcoholism equilirium y is permanent in {}, that is to say, there exists an m satisfies lim t inf y m.

12 Page 12 of 19 Proof We will confirm that the system 11 satisfies all the hypotheses of Lemma 3. Condition i: Lemma 3 is satisfied if we suppose that C = R 2n. Condition ii: using the mean inequality and limit rule can validate the conclusion. Condition iii: notice that A T is an 2n 2n matrix with a ij whenever i = j, then from Lemma 2, there exists an eigenvector ω = ω 1, ω 2,..., ω 2n of A T and the associated eigenvalue is SA T. Let r = min 1 i 2n ω i >, for y, 2n ω y r i=1 y 2 i, therefore ω y r y for all y. Condition iv: we know ω > and Ny, so it is clearly satisfied. Condition v: let H ={y ω Ny = }. If y H, then iλω i giy i +y 2i θ i kgk = for i = 1, 2,...,2n. But since each term of the sum is nonnegative, then we get that iλω igiy i +y 2i j ϕjpjy j kgk =. If this equation is estalished, then y j = or y i = y n+i =. From system 11, if y j = then y n+j = for j = 1, 2,..., n. That is y 1,..., y n, y n+1,..., y 2n =. So, the only invariant set respect to 11 contained in H is y =, condition v is satisfied. This completes the proof. Gloally attractive of the alcoholism equilirium Theorem 4 When R > 1, the only alcoholism equilirium y = y in model 11 is gloally attractive in {}. Proof We define the following functions, M : R and m : R for y, where My = max i y i y i, my = min i y i y are continuous and the right-hand deriva- i tive exists along solutions of 11. Let y = yt e a solution of 11, we may assume that Myt = y i t y i t, i = 1, 2,...,2n and t [t, t + ε]. For a given t and for sufficiently small ε > M 2.11 yt = y i t y i, for t [t, t + ε]. from 11, if 1 i n, we have y y i t i y i t = λi gi θ y kgk + µ y i i t y n+i t y i t y i + βy n+i t y i t α + µy i, or for i = 1, 2,..., n and i = n + i, we have y n+i y y n+i t n+i y n+i t = αy it y n+i t βy n+i σ y n+i µy n+i. According to the definition of M y t, we know y i t y y it i y, i = 1, 2,...,2n. i Then, if Myt > 1, for 1 i n we have

13 Page 13 of 19 y y i t λi gi θ y i y i t < i kgk + µ y i t y n+i t + βy n+i t α + µy i =, or for i = 1, 2,..., n and i = n + i y n+i y y n+i t n+i y n+i t <αy it y n+i t βy n+i σ y n+i µy n+i =, and since y i > and y i t >, we conclude that y i t <. Therefore, if Myt > 1, then M 2.11 yt <. Similarly, we can testify that if Myt = 1, y i t. And if myt < 1, then m 2.11 yt >. If myt = 1, then m 2.11 yt. Denote Qy = max { My 1, }, qy = max { 1 my, }. Both Q y and q y are continuous and non-negative for y. Notice that Q 2.11 yt, q 2.11 yt. Let H Q ={y Q 2.11 yt = } and H q ={y q 2.11 yt = }, then H Q ={y y i y i } and H q ={y y i y i +µ } {}. According to the LaSalle invariant set principle, any solution of 11 starting in will approach H Q H q ={y } {}. But if yt =, y Theorem 3 we know that lim t inf yt A >. Then we conclude that any solution yt of 11, such that y {}, satisfies lim t yt = y, so y = y is gloally attractive in {}. Control strategy Timely stopping recuperator recurrence drinking alcohol and stopping the susceptile people to drink are two important and effective ways to prevent the spread of the alcoholism. Next, we give two kinds of control strategies for the treatment of recuperator for stopping relapsing and preventing the susceptile people to drink, respectively. Due to the fixed weight model and adaptive weight model have the same asic reproductive numer, so we only study the control strategies of the fixed weight model. Proportion treatment Let ε e the treatment proportion for recuperator, <ε<1, then system 11 ecomes di k t dr k t = kgk λkgk +µ I kt R k t θt α + µi k t + β1 εr k t, = αi k t β1 εr k t σ R k t µr k t, 18 By the next generation method, the asic reproductive numer of 18 is R = λβ1 ε + σ + µ kgkϕk + µ[ασ + µ + µβ1 ε + σ + µ] kgk

14 Page 14 of 19 we can find that with the increase of the treatment proportion, R is smaller. Change the form of R, we get that R = = = λβ1 ε + σ + µ kgkϕk + µ[ασ + µ + µβ1 ε + σ + µ] kgk λ kgkϕk [ + µ ] ασ +µ kgk β1 ε+σ +µ + µ λ kgkϕk [ ασ +µ < ] kgk + µ β+σ +µ + µ λβ + σ + µ kgkϕk + µ[ασ + µ + µβ + σ + µ] kgk = R R is the asic reproductive numer of 11. So, we can see that proportion treatment to recuperator is a very effective control strategy, and the igger the proportion of recuperator to accept treatment, the alcoholism is more difficult to outreak. Proportion prevention Let ψ e the proportion of susceptile people who understand the harm of alcoholism and refuse to drink, <ψ<1, then system 11 ecomes di k t dr k t = kgk λkgk +µ I kt R k t 1 ψθt α + µi k t + βr k t, = αi k t βr k t σ R k t µr k t, 19 By the next generation method, we get the asic reproductive numer of 19 is ˆR = λβ + σ + µ1 ψ kgkϕk + µ[ασ + µ + µβ + σ + µ] kgk It is easy to know that ˆR < R, and the greater the proportion of susceptile people refuse to drink, the smaller the numer of alcoholics. Next, we are going to compare which strategy is more useful. Transform the form of R and ˆR, we have R = λ kgkϕk [ ] ασ +µ kgk + µ β1 ε+σ +µ + µ and ˆR = [ + µ λ kgkϕk ασ +µ β+σ +µ1 ψ + µ 1 ψ ] kgk When ψ = ε, we know that ˆR < R. That is to say, when the proportion of recuperator to accept treatment is equal to the proportion of susceptile people to refuse drinking

15 Page 15 of 19 alcohol, the strategy of in proportion to prevent susceptile people drinking alcohol will e more effective. Sensitivity analysis and numerical simulations In this section, we perform some sensitivity analysis on the asic reproduction numer R in terms of the parameters. Our simulations take the scale-free networks with degree distriution is Pk = 18k 3 2 <γ 3. Let n = 4, gk = k r 1, ϕk = k r 2 and hk = k r 3, k = 1, 2,..., 4, where r 1, r 2 and r 3 are positive constants. Considering the influence of heavy alcoholics relapse and the weight etween individuals. We focus on simulate the relapse parameter β, the weight parameter r 1 and the nodes infectivity parameter r 2. From Fig. 2, it is clear that R presents growth trend with the increase of the relapse parameter β. It means that igger alcoholism recurrence rate are easy to cause outreaks of alcoholism. Figure 2a shows that the greater the weight parameter r 1 lead to greater R. It means that the greater the link s weight etween two nodes, the easier the alcoholism roke out. Figure 2 shows that R increases as the nodes infectivity parameter r 2 increases. That is to say, if a prolem alcoholic has ig infectivity, the alcoholism is more easy to roke out. Next, we perform some numerical simulations to illustrate our theoretical results of the models 11 and 12, so as to find etter control strategies. The parameters are used as =.2, µ =.4, σ =.6, α =.6, β =.2, r 1 = 1.1, r 2 = 1, r 3 = 1.2. It is clear that r 2 = means no weight model, r 2 =, r 3 = mean fixed weight model and r 2 =, r 3 = mean adaptive weight model. Figure 3a, descrie nodes without weight with λ =.5 and.9 respectively, Fig. 3a shows that when R =.8981 < 1, the alcoholism dies out quickly. Figure 3 shows that R = > 1, the prolem alcoholics population will maintain at a positive stationary level, which implies that the alcoholism will ecome endemic. Figure 4a, descrie nodes with fixed weight with λ =.2 and.5 respectively, Fig. 4a shows that when R =.6513 < 1, the alcoholism dies out quickly, which implies that the alcohol free equilirium of 11 is stale. Figure 4 shows that when R = > 1 the prolem alcoholics population will maintain at a positive stationary level of 11, which implies that the alcoholism will ecome endemic. Compared with R R β a.5 r Fig. 2 The relationship etween the asic reproduction numer R and the parameters on scale-free networks 1.5 β.2.4 r

16 Page 16 of 19 The relative density of I k k=1 k=15 k=2 k=25 k=3 k=35 k=4 The relative density of I k k=1 k=15 k=2 k=25 k=3 k=35 k= t t a Fig. 3 The densities of alcoholics with different degrees and without weight when R > 1 a and R < 1 Figs. 3a and 4, for same parameters, asic reproduction numer is.8981 and , respectively. So we know that R of model on networks with weights is larger than that on networks without weights. Figure 5a, descrie nodes with adaptive weight. We know that 12 and 11 have the same R. So we choose the same parameters as Fig. 4. From Fig. 5a,, we can see that since the adaptivity of weight, the alcoholics rapidly drop first and experience a valley then up to a small peak or dies out. This is caused y the ehavior of people s self-protection awareness. From this point, the adaptive model is more close to the actual situation. Figure 6a, shows the densities of prolem alcoholic individuals with different degrees and different the adaptive coefficient r 3. We know that the stronger the adaptive coefficient r 3, the greater the self-protection awareness of susceptile. It lead to alcoholics density maintain in a lower value. Conclusion and discussions In this paper, we proposed a modified SIRS alcoholism model with relapse on weighted networks to study the influences of individual s contact patterns on drinking dynamics. We construct two complex network models with fixed weight and adaptive weight, The relative density of I k k=1 k=15 k=2 k=25 k=3 k=35 k=4 The relative density of I k k=1 k=15 k=2 k=25 k=3 k=35 k= t t a Fig. 4 The densities of alcoholics with different degrees and fix weight when R > 1 a and R < 1

17 Page 17 of 19 The relative density of I k k=1 k=15 k=2 k=25 k=3 k=35 k=4 The relative density of I k k=1 k=15 k=2 k=25 k=3 k=35 k= t t a Fig. 5 The densities of alcoholics with different degrees and adaptive weight when R > 1 a and R < 1 The relative density of I k k=1 k=15 k=2 k=25 k=3 k=35 k= t a Fig. 6 The densities of alcoholics with different adaptive coefficient r 3 = 1.4 in a and r 3 = 1.5 in The relative density of I k t k=1 k=15 k=2 k=25 k=3 k=35 k=4 respectively. We get the alcoholic propagation threshold R which determines the propagation dynamics. We also otain the existence of equiliria of 11. For the model with fixed weight, we prove that when R < 1, the model s alcohol free equilirium is gloal stale and alcoholism will disappear, otherwise, if R > 1, the alcoholism equilirium is gloal attractivity and alcoholism will persistence. For the model with adaptive weight, we only make some numerical simulations. By comparing the alcoholism models with no weight, fixed weight and adaptive weight, we have: 1. Fixed weight model have larger propagation threshold than no weight model; 2. The adaptive weight cannot change the propagation threshold, ut it can induce the alcoholism to decay quickly; 3. Strong adaptaility can inhiit the alcoholism population reach a high level; 4. In order to eliminate alcoholism prolem, first we should try to reduce the frequency of interaction etween susceptile and prolem alcoholic, that is to say decreasing the link s weight etween two nodes, this may e a effective measures. Second, through education or media to spread the dangers of alcohol ause in order to decrease the relapse β, it is also a very effective measures.

18 Page 18 of 19 The delay differential equations usually exhiit much more complicated dynamics than ordinary differential equations ecause the time delay can lead to instaility, oscillation, or ifurcation phenomena Bianca et al. 215; Bianca and Guerrini 214. There is a time delay during a susceptile individual ecomes the prolem alcoholic, so it is more realistic to consider the time delay in the modelling alcoholism process. We can modify 1 to the following model with delay ds k t di k t dr k t = 1 S k t I k t R k t ks k t τ t + σ R k t µs k t, = ks k t τ t + βr k t αi k t µi k t, = αi k t βr k t σ R k t µr k t, We leave this work for the future. Authors contriutions HFH and YPL designed the study, carried out the analysis and contriuted to writing the paper. Both authors read and approved the final manuscript. Acknowlegements This work is supported y the NNSF of China , the NSF of Gansu Province 148RJZA24, and the Development Program for HongLiu Outstanding Young Teachers in Lanzhou University of Technology. Competing interests Both authors declare that they have no competing interests. Received: 26 January 216 Accepted: 9 May 216 References Bansal S, Grenfell B, Meyers LA 27 When individual ehaviour matters: homogeneous and network models in epidemiology. J R Soc Interface 4: Barrat A, Barthelemy M, Pastor-Satorras R, Vespignani A 24a The architecture of complex weighted networks. Proc Natl Acad Sci 11: Barrat A, Barthelemy M, Vespignani A 24 Modeling the evolution of weighted networks. Phys Rev E 7:66149 Barrat A, Barthelemy M, Vespignani A 24c Weighted evolving networks: coupling topology and weight dynamics. Phys Rev Lett 92:22871 Benedict B 27 Modeling alcoholism as a contagious disease: how infected drinking uddies spread prolem drinking. Soc Ind Appl Math News 4: Bianca C, Guerrini L 214 Existence of limit cycles in the Solow model with delayed-logistic population growth. Sci World J 214:2786 Bianca C, Guerrini L, Riposo J 215 A delayed mathematical model for the acute inflammatory response to infection. Appl Math Inf Sci 9: Chu XW, Guan JH, Zhang ZZ, Zhou SG 29 Epidemic spreading in weighted scale-free networks with community structure. J Stat Mech Theory Exp 7:P743 Chu XW, Zhang ZZ, Guan JH, Zhou SG 211 Epidemic spreading with nonlinear infectivity in weighted scale-free networks. Phys A 39: d Onofrio A 28 A note on the gloal ehaviour of the network-ased SIS epidemic model. Nonlinear Anal Real World Appl 9: Giraldina TN, Jay KK, Marjolein DW 215 Alcohol use as a risk factor in infections and healing. Alcohol Res Curr Rev 37: Glavas MM, Weinerg J 25 Stress, alcohol consumption and the hypothalamic-pituitary-adrenal axis. In: Yehuda S, Mostofsky DI eds Nutrients, stress and medical disorders. Humana Press, NY, pp Gonzales K, Roeer J, Kanny D et al 214 Alcohol-attriutale deaths and years of potential life lost 11 states, Mor Mortal Wkly Rep 63: Huo HF, Liu YP 216 Staility of an SAIRS alcoholism model on scale-free networks. J Biol Dyn in review Huo HF, Song NN 212 Gloal staility for a inge drinking model with two-stages. Discrete Dyn Nat Soc 212:1 15 Huo HF, Zhu CC 213 Influence of relapse in a giving up smoking model. Astr Appl Anal 213:1 12 Huo HF, Wang Q 214 Modelling the influence of awareness programs y media on the drinking dynamics. Astr Appl Anal 214:1 8 Lajmanovich A, Yorke JA 1976 A deterministic model for gonorrhea in a nonhomogenous population. Math Biosci 28:

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