SIR stochastic model of adaptive network of virus propagation

SummarySocial attention has been increasingly paid to the spread of Ebola virus, be-cause of its death rate up to 90%. President Obama has described the outbreakof Ebola in West Africa as a threat to global security. Nevertheless, the worldmedical association has announced that their new medication could stop Ebolafrom spreading and cure patients whose disease are not advanced. This is agospel for infected patients.2uuhuihjhhhThis paper presents four models to eradicate Ebola virus. Focusing on SIRstochastic model of adaptive networks of virus propagation, we establish themodel to simulate the spread of Ebola in a crowd.4 Thus we can predict thetrend of Ebola virus properly. Then, by improving the Susceptible-Infective-Removal(SIR) model and the Barabasi-Albert(BA) scale-free network model,we distribute the limited vaccines to the infected patients effectively. We al-so put forward a strategy about how to make full use of the limited medication.The Cellular Automata model is created for the simulation of viruss spreadingin adaptive networks. At last but not least, by using the Prims Algorithm, wefind a route to transport vaccines as effectively as possible.The four models in this paper consider the spread of the disease range, therequired quantity of medication, the speed of vaccine production, and the de-livery speed of medication. Through these models, the spread of the Ebola viruswill be controlled reasonably and effectively in the end.Keywords: Ebola virus SIR model BA scale-free network Cellular Au-tomata model Prims AlgorithmSIR stochastic model of adaptive network ofvirus propagationMCM Problem A Team # 41545January 27, 2016Contents1 Introduction 21.1 Restatement of the problem . . . . . . . . . . . . . . . . . . . . . . 21.2 Assumptions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.3 Strength . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.4 Weakness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.5 Survey of Previous Research . . . . . . . . . . . . . . . . . . . . . . 42 Model 1: A mathematical model of SIR 52.1 Assumptions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52.2 A mathematical model of SIR without considering the latency . . 52.2.1 Model Structure . . . . . . . . . . . . . . . . . . . . . . . . . 52.3 A mathematical model of SIR considering the latency . . . . . . . 72.3.1 Model Structure . . . . . . . . . . . . . . . . . . . . . . . . . 72.3.2 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . 82.4 Values . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92.4.1 strength . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92.4.2 Weakness . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93 Model 2:BA scale-free network model 93.1 Assumptions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93.2 Model Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93.3 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103.4 Strength and Weakness . . . . . . . . . . . . . . . . . . . . . . . . . 103.4.1 Strength . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103.4.2 Weakness . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104 Model 3 :The SIR stochastic model of the adaptive network of virus 114.1 Model Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134.2 Values . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14Team # 41545 Page 2 of 205 Model 4:Distribution strategy and transportations 165.1 Distribution strategy . . . . . . . . . . . . . . . . . . . . . . . . . . 165.2 Prims algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165.2.1 Assumptions . . . . . . . . . . . . . . . . . . . . . . . . . . 165.2.2 Algorithm structure . . . . . . . . . . . . . . . . . . . . . . 165.2.3 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . 175.2.4 Values . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 186 Conclusion 187 Announcement 18Reference 191 IntroductionIn 1995, a movie called ”extreme panic” describes the horrible influence of Ebo-la. Ebola virus ( Ebola virus, EBOV) is one of the worlds most contagious andhighest mortality virus. The virus is a potent virus that can cause Ebola hemor-rhagic fever occur (EBHF) between crowd and animals. Since EBOV was firstdiscovered in 1976, EBHF epidemic prevalent in Africa several times. Becausethe Ebola broke out in a village which is near to the Ebola River in a village inthe northern West Africa Zaire, EBOV got its name. On November 9th, 2014, theglobal EBOV infections are up to 14098, and 5160 people died from the Ebola,including many medical staffs involved in the treatment. The World MedicalAssociation has announced that their new medication could stop Ebola fromspreading and cure patients whose disease is not advanced. This is a gospel forall infected patients.1.1 Restatement of the problemTo simulate the spread of Ebola in a crowd, we establish a SIR stochastic modelof adaptive networks of virus propagation to send medication effectively. Thefinal target is to slow down the speed of the spread of Ebola virus.In order to meet the requirements of the target, we should break down theproblems.1. The vaccines can only cure the patients whose disease is not advanced.2. The number of vaccines is limited.3. Not all patients can be cured successfully by receiving the vaccines. Theymay become infections again in a period of time.Team # 41545 Page 3 of 204. This virus is both lethal and virulent, which means we should take the limit-ed medication into account. Distribution strategy is also a necessary wayto reduce the spread of Ebola virus.The delivery system should be efficient, and we also need to control a furtherspread of the disease. The limited vaccines should be sent to destination in theshortest time by choosing a best route.In the process of vaccines production and delivery, there will be new infec-tions in this period of time. We must take these new patients into consideration,so as to determine the distribution of vaccines.1.2 AssumptionsThe locations of delivery: We only consider three countries named Guinea,Sierra Leone and Liberia. Because t the situation is serious in these areas.The total number of people in three countries are N million constantly,considering there is no migration and death caused by other reasons.This paper doesnt consider the types of Ebola virus. Even though thereare five types of Ebola virus.This paper doesnt consider the primate animals infected.1.3 StrengthOur models combine the SIR model and BA scale-free network model in-novatively, and improved the SIR model effectively. Based on Cellularautomata, we simulate the spread of virus accurately.The four models are universal.Our models not only simulate the spread of the disease but also considerthe drugs of delivery. These models are more practical and comprehen-sive than previous models.1.4 WeaknessDue to limited available data on the Internet, there is a lack of accuracy ofthe total number of the national populations prediction.Because the symptoms are not obvious in the latent patients. When webuild the SIR model, its unable to distinguish the susceptible patientsand the latent patients.Team # 41545 Page 4 of 201.5 Survey of Previous ResearchAs for the study on mathematical model for Ebola virus, as early as 1996, theliterature (2) uses models SIR and SEIR, simulating two periods of Ebola out-break: The outbreak of Zaire in 1976, and the outbreak of Yambuku in 1995.They draw a conclusion: when the basic reproduction rate R0 satisfies 1.72 =R0 = 8.60, the infectiousness of Ebola virus is less powerful than before. Inrecent years, there are also some documents (see 1,3)and researches of the E-bola virus done detailedly. Now wed like to build a mathematical model aboutthe quantity of infected crowd based on these documents of the Ebola virus.Figure 1: West Africa ebola infections distributionTeam # 41545 Page 5 of 202 Model 1: A mathematical model of SIR2.1 AssumptionsLet the study object as the ideal crowd, the total number of populationremain at the fixed level of N. The phenomenon of immigration and emi-gration and other causes of death are not considered.Assuming that cured patients all have immunity in long term.2.2 A mathematical model of SIR without considering the la-tency2.2.1 Model Structurei(t) on the temporal growth rate about time is proportional to s(t)and the pro-portionality constant is X. The reduce speed of the number of patients and thetotal number of patients is proportional, and the constant of proportionality isV. The cured patients have immunity in long term, which represent the patientswouldt be infected twice after the medical treatment.s(t) + r(t) + i(t) = 1 (1)The susceptibles become the infections after contacting with infections effec-tively. Assume lS(t) represents susceptibles, which each patient contacts effec-tively per day. NI(t) infections can make lS(t)NI(t) susceptibles become thevirus lurks per day. Thus:dS(t)dt = lS(t)I(t) (2)The change of recuperators is equal to the incoming of infections, which isdR(t)dt = nI(t) (3)The change of infections is equal to the incoming susceptibles.dI(t)dt = lS(t)I(t) nI(t) (4)The proportion of patients and healthy people is S0, R0 separately(R0 = 0).From (1)and (4), we can get:S(t) = S0e Rs (5)Equations cant be solved analytically, we define a new variables = n/l. toobtain the root is i(t) = (s0 + i0 s + sln ss0 )Analyzing the change ofS(t), I(t), R(t). A In any case of initial conditionsfor S0, R0. The patient will disappear eventually, that is i = 0. B The ratio ofTeam # 41545 Page 6 of 20healthy people is s in the end, its the root of equation.(s0 +i0) s + sln ss0 = 0.C, If s0 1/s:i(t) increased at first, when s0 = 1/s,i(t) reaches the maximumvalue, then i(t) decreases and tends to 0, s (t) decreases to s monotonically. D,If s0 1/s ,i(t) decreases monotonically to 5, s (t) is monotonically decreaseto s. We found that if the peoples health level is higher, the daily contact ratewill be smaller; if the medical level is higher, there will be higher daily curerate. Therefore improving the health level and medical level are conducive tocontrol the spread of disease.We set Guinean as an example application of the model for analysis. Ac-cording to (5)and the data from table 2, we can get l = 3.72, m = 0.863. Thenwe can get the number of days of infectionS = 1882ch2(0.195t 6.02)and draw its figure:In figure 2, we anchored February 10. Form the figure we can see that thelargest number of days of infection arise in the first 32 weeks or 33 weeks. Inother words, the largest number of days of infection will reach a maximum inlate July of this year or early July. With curing actively and increasing the effortsof prevention, the number of people infected will be gradually reduced.We analyzed data according to the specific circumstances and assumptions,and draw a scatter plot of infections and deaths in Guinean(Figure 8). We chosethe data of infection and mortality in last 18 days:Then we get the fitted curve shown in figure 9 from original data.Assuming the sum of drug production and delivery time is td in Guinean.Different areas has different td. Taking into account the drug delivery processand there will be a new number of infections. Form the Fitting functionI = 232.94t 13.911t2 + 0.41905t3 + 824.35, t = td + 18Figure 2: The relations between the number of infections and weeksTeam # 41545 Page 7 of 20. We can predict the number of people infected with drug-delivery in Guineani (t) roughly.2.3 A mathematical model of SIR considering the latency2.3.1 Model StructureBecause of the incubation period of Ebola virus is 2 to 21 days, on average of 5to 12 days, thus we set up a mathematical model of SIR which take latency intoaccount. In this case, we suppose that in the Ebola virus propagation period,the total number of people in west africa are N billions constantly, for neitherdeath, nor consider migration. The crowd are divided into Susceptibleslatentcrowd of virus. Infections and Recuperators(including the dead and Survivors).The ratio of these four kinds of people in the total number of West Africa ares(t), e(t), i(t), r(t) at the moment t.Counting the number of people each patient contact each day is H, namelydaily contact rate. When infections and susceptibles contact effectively, it willmake susceptibles become latent crowd of virus. In a period of time, latentcrowd of virus will become infections, then the infections will be cured.After contacting with infections of Ebola effectively, the susceptibles be-come the latent crowd of virus. Assuming that the average daily number ofsuspectibles with which each patient contact effectively is l(t)S(t), NI(t) infec-tions can make l(t)S(t)NI(t) susceptibles become the latent crowd of virus aFigure 3: case-death rateTeam # 41545 Page 8 of 20Figure 4: Fitting figureday on average. So we get a formula:NdS(t)dt = l(t)S(t)NI(t)The changes of latent crowd of virus is equal to the number of susceptiblesminus the newly increased number of infections. Which is:dE(t)dt = l(t)S(t)NI(t) a(t)E(t)Atmeans the daily infection rate of the incubation period. The changes of recu-perators in per unit time is equal to the reduce of infections,that isdR(t)dt = n(t)I(t)V(t) means the daily exit rate. The Changes of infections is equal to the numberof people into the latent crowd of viruswhich isdI(t)dt = a(t)E(t) n(t)I(t)s(t) + i(t) + r(t) = 1. At the initial time, the ratio of Susceptiblesinfections and recuperators ares0(s0 0),i0(i0 0)r0 = 0.2.3.2 ConclusionsThe number of people in latency cant be confirmed. So it can be a part ofsusceptible, thus the solving process is the same as Ignore the latency.Team # 41545 Page 9 of 202.4 Values2.4.1 strengthThe SIR model assumes that the cured patients have immunity in longterm, namely the patients wont be infected twice after the medical treat-ment, which reflects our aspiration for Ebola virus.Considering the latency makes the model more realistic.2.4.2 WeaknessThe number of people in incubation period are unable to confirm, themodel cant describe the process of infectious disease precisely.In reality, the infectious disease virus may mutate through evolution, whichcan cause more infections. Some Datas show that the Ebola virus has aterm of incubation up to 21days, the model did not consider the stage.3 Model 2:BA scale-free network modelIn reality, people select different crowd to contact in separate area accordingto their own preferences. Resulting from the effect of personality, social statusand other subjective and objective factors. The communication range of eachperson is distinct, thus the routes and risks of the Ebola virus are also distinctfrom man to man. So we choose the BA scale-free network model to simulatethe interpersonal network in real life.Two important characteristics of actual network:1. Growth characteristic, namely the size of the network is not fixed, but inexpanding.2. Preferential connection characteristic, namely the different nodes preferthe nodes with high degree to contact. This phenomenon is referred to as the”Matthew Effect”, or ”the rich get richer”.3.1 AssumptionsAssuming that M represents population, M0 as the initial population number,node I for specific individuals, Ki is the level of node. For the diverse peoplewho are led into the network, the degree is also the personal choice and thebasis of the contact in the population.3.2 Model StructureBased on the Growth characteristic and Preferential connection characteristic,the evolution rules of BA scale-free network model are as follows:Team # 41545 Page 10 of 20First step:Based on the Growth characteristic.Starting from m0 nodes in anetwork, a node are introduced every time, connect it to m(m m0) differentexisting nodes.Second step: Based on the Preferential connection characteristic,when thenew nodes choose the exsiting nodes to contact, the probability of the new nodeconnect the node i depends on the degree ki of the node i.that isi= ki j kj(6)After t stepsthis algorithm produces a network including N = t + m0 nodes,mtsides. There are two nodes in the initial network.According to the Preferentialconnection characteristic,the new node which is introduced contact the exsitingtwo nodes in the network.2. Degree distributionThere are three main methods for theoretical research of the degree distri-bution in BA scale-free network: Continuous field theory, the master equationmethod and rate equation method. The asymptotic results of the three methodsare the same. The second method is equivalent to the third method. Function ofthe degree distribution in BA scale-free network by using the master equationmethod isp(k) = 2m(m + 1)k(k + 1)(k + 2) 2m2k 3 (7)3.3 ConclusionsSo we can draw a conclusionFunction of the degree distribution in BA scale-free network can be described by the power-law degree function of the powerindex of 3 approximately.3.4 Strength and Weakness3.4.1 StrengthBA scale-free network model considers the real network to be two mechanisms,including growth characteristic and preferential connection characteristic. itssimple and clear. It reflects the simple extraction essence from the complexphenomenon. Its a effective way to solve problem.3.4.2 WeaknessBA scale-free network model has limitation, the value of the power-law indexcan only be fixed for 3.but,in reality, the power-law index of different complexnetworks is not exactly the same.Team # 41545 Page 11 of 204 Model3:TheSIRstochasticmodeloftheadaptivenetwork of virusWe use the transmission dynamics to simulate the spread of the disease incrowd, when a node changes state, the network topology wont change, asshown in the following figures:However, this model doesnt consider the change of in