Research Article Modeling the Spread of Tuberculosis in...

Hindawi Publishing CorporationComputational and Mathematical Methods in MedicineVolume 2013 Article ID 648291 19 pageshttpdxdoiorg1011552013648291

Research ArticleModeling the Spread of Tuberculosis inSemiclosed Communities

Mauricio Herrera1 Paul Bosch2 Manuel Naacutejera3 and Ximena Aguilera3

1 Facultad de Ingenierıa Universidad del Desarrollo Santiago 7620001 Chile2 Facultad de Ingenierıa Universidad Diego Portales Ejercito 441 Santiago 8370179 Chile3 CEPS Facultad de Medicina Clınica Alemana Universidad del Desarrollo Santiago 7710162 Chile

Correspondence should be addressed to Paul Bosch paulboschudpcl

Received 23 December 2012 Revised 11 March 2013 Accepted 11 March 2013

Academic Editor Nestor V Torres

Copyright copy 2013 Mauricio Herrera et al This is an open access article distributed under the Creative Commons AttributionLicense which permits unrestricted use distribution and reproduction in any medium provided the original work is properlycited

We address the problem of long-term dynamics of tuberculosis (TB) and latent tuberculosis (LTB) in semiclosed communitiesThese communities are congregate settings with the potential for sustained daily contact for weeks months and even years betweentheirmembers Basic examples of these communities are prisons but certain urbanrural communities some schools among otherscould possibly fit well into this definition These communities present a sort of ideal conditions for TB spread In order to describekey relevant dynamics of the disease in these communities we consider a five compartments SEIR model with five possible routestoward TB infection primary infection after a contact with infected and infectious individuals (fast TB) endogenous reactivationafter a period of latency (slow TB) relapse by natural causes after a cure exogenous reinfection of latently infected and exogenousreinfection of recovered individuals We discuss the possible existence of multiple endemic equilibrium states and the role that thetwo types of exogenous reinfections in the long-term dynamics of the disease could play

1 Introduction

Dynamics of tuberculosis (TB) spread has been the subject ofa considerable body of theoretical and mathematical workFor review see for example [1 2] and references thereinThe choice of a particular model is strongly connected to thequestions we want to answer and in the present work we willaddress the problem of long-term dynamics of tuberculosisand latent tuberculosis (LTB) in semiclosed communities

For semiclosed communities we mean not strictly closedcommunities with certain mobility of their members out orinto the community with a recruitment of newmembers anddeparture of others But essentially these communities arecongregate settings with the potential for sustained daily con-tact for weeks months and even years between communitymembers Basic examples of these communities are prisonsbut certain urbanrural communities schools among otherscould possibly fit well into this general definition Thesecommunities present a sort of ideal conditions for frequent

TB outbreaks enhanced TB transmission and acceleratedspread of the disease

The basic characteristics of such settings including thepossibility of high concentrations of infectious individualsand immunodeficient hosts improper precautions taken forprotection delay in diagnosis sustained contact with theindex case and inadequate ventilation andor overcrowdingmake them well suited for TB transmission creating thisway genuine high transmission pockets of TB inserted in thegeneral population [3 4]

In fact prisons are especially high burden communitiesin which incidence and prevalence of TB are very high andconsequently the frequency of infections and reinfectionsconsiderably increases in comparison with population atlarge see the works by Chiang and Riley [5] and by Baussanoet al [6]

Studying the dynamics of the TB spread in semiclosedcommunities is an interesting and significant topic by itselfhowever there is an important phenomenon due to which

2 Computational and Mathematical Methods in Medicine

the study of these types of communities is essential in thecontext of TB spread This phenomenon has been called theReservoir Effect [6 7] Indeed semiclosed communities suchas prisons represent a reservoir for disease transmission to thepopulation at large and should be a source of public concernFor example TB infection may spread into the generalpopulation through prison staff visitors and close contactsof released prisoners The transmission dynamics betweenprisoners and the general population [6] together withimmigration from developing countries with high prevalenceof TB [8 9] has been hypothesized to play a key role indriving overall population-level TB incidence prevalenceand mortality rates

In a recent work [4] the authors have even gone furtherin relation to this effect and have named these communitiesInstitutional Amplifiers of TB Propagation Some examples ofcommunities given by these authors are poor hospitals inwhich dozens of patients share poorly ventilated communalrooms crowded prison cell blocks and mining barracksamong others

The transmission and progression of TB infection hasbeen relatively well understood on a population scale Gener-ally it is assumed that once an individual is infected with TBhe or she is immune from further infection events Moreoverit was proposedwhat came to be known as the unitary conceptof pathogenesis [10] which states that TB always begins withprimary infection and subsequent episodes of active TB aredue to reactivation of dormant bacilli from this primaryinfection However a persistent evidence has recently beenshown (see [5] for a review) that the paths to TB infectionare not as linear as was suggested by the unitary conceptof pathogenesis The availability of individual strain-specificinfection histories (see eg [11ndash13]) has made it clear thatexogenous reinfection in people with previously documentedTB infection does occur The important question is whetherreinfection occurs commonly enough to have an effect on theoverall infection dynamics of the population [14]The relativeimportance of these pathways to the development of activedisease has significant implications for treatment and controlstrategies most notably in deciding whether latently infectedand treated individuals are at risk of reinfection [15]

Several authors [15ndash20] have declared that exogenousreinfection plays an important role in the disease progressionand that the inhalation of tubercle bacilli by persons whohave had a primary TB infection previously for more thanfive years represents an increasing risk to develop activeTB soon after reinfection A study from South Africa [21]has demonstrated that the rate of reinfection by TB aftersuccessful treatment could be higher than the rate of new TBinfections In this study the reinfection rate after successfultreatment was estimated at 22 per 100 person-years whichwas approximately seven times the crude incidence rate(313 per 100 000 population per year) and approximatelyfour times the age-adjusted incidence rate of new TB (515per 100 000 population per year) So ignoring exogenousreinfection when modeling TB spread in high-incidenceand high-prevalence community setting such as semiclosedcommunities has been seen to be inappropriate (Henao-Tamayo et al in [22] recently published a mouse model of TB

reinfection that could help to explain immunological aspectsof reinfection risk in high-incidence areas)

We will use an SEIR standard compartmental modelsee for example the works by Blower et al [23] and morerecently by Liao et al [24] with somemodifications explainedbellow that turn out to be quite useful in the study of theparticularities of TB spread at this type of communitiesThis model assumes that the population in the communityis homogeneous that it does not consider the heterogeneitiesin the social structure between community members and itis based on the so-calledmass action or fully mixing approxi-mation This means that individuals with whom a susceptibleindividual has contact are chosen at random from the wholecommunity It is also assumed that all individuals haveapproximately the same number of contacts in the sametime and that all contacts transmit the disease with the sameprobability

The model we use in this work takes into account thefollowing relevant facts in the context of semiclosed commu-nities

(1) The overcrowding in the community can increase(compared to what occurs in the population atlarge) the likelihood of exogenous reinfection dueto repeated contacts with active infected individualsThat is besides primary infection themodel considersthe possible reinfection of individuals with LTB (indi-viduals who are assumed to be asymptomatic andnoninfectious but capable of progressing to active TB)and recovered individuals (individuals who have beentreated for TB in the past and been declared cured)If latently infected or recovered individuals remain inthe community they could be infected again

(2) At present it is not completely clear whether inall cases previous infections with MycobacteriumTB with or without subsequent recovery offer someprotection that could be translated into a reducedsusceptibility to reinfection [5 21 22 25] So we willbe open at exploring different situations with regardto this fact in the model

(3) Poor nutrition immunodepression and other dis-eases increase the likelihood of accelerated progres-sion to active TB

We will see that considering exogenous reinfection todescribe TB spread produces a richer and more complexdynamics than the one observed in previous models (see eg[23 25 26]) In particular unlike the model published byFeng et al in [26] which uses a single parameter for exoge-nous reinfection our model uses two parameters related totwo possible reinfections (reinfection of latently infected andreinfection of recovered individuals)

2 Basic Epidemiology of TB Sources andProbability of Infection in SemiclosedCommunities

The risk of infection with Mycobacterium tuberculosis thebacterium causing TB depends mainly on two factors first

Computational and Mathematical Methods in Medicine 3

significant exposure to a source of infection and second theprobability of getting infection if there is exposure

TB is mostly transmitted through the air tubercle bacillithat depends on host and agent factors is distributed intiny liquid droplets that are produced when someone withclinical or active TB coughs sneezes spits or speaks allowinginfected individual to infect others In closed places thebacteria are expelled into a finite volume of air unless thereis ventilation see [27] In these conditions they may remainviable and suspended in the air for a prolonged period oftime But the number of bacilli excreted bymost personswithactive pulmonary TB is relatively small [16] so the probabilityof TB transmission per contact per unit of time is in generalquite low The risk of infection is very small during a singleencounter with an infectious individual [28] However theprobability of TB transmission can be enhanced by systematicand long exposure of susceptible individuals to particularinfectious individuals

The risk of TB transmission is particularly high in settingswith poorly ventilated areas (places with reduced air volumeper occupant with ventilation systems which recirculate theair or with poorly filtered air exchanges) andor closed areasin which people are in close and frequent contact Closedregime prisons are examples of these high-risk areas Ineffect the occurrence of TB in prisons for example is usuallyreported to be much higher than the average levels reportedfor the corresponding general population [6]

Although most exposed individuals develop an effec-tive immune response to the initial infection [17] there isanother factor that raises the chances of TB contagion thefact that TB is an opportunistic disease Indeed infectedindividuals with weakened immune systems are at significantrisk of developing clinical TB disease (active TB) High TBprevalence is therefore observed in individuals with HIVinfection poor nutritional status alcoholism drug abuseconcurrence of other pathology and psychological stressdecrease immune response levels These conditions occurfrequently in imprisoned peoples

TB is usually described as a slow disease because of itslong and variable period of latency and because of its shortand relatively narrow infectious period distribution Longperiods of latency (inactive TB or latent TB or LTB) implythat new cases of infection are not clinically noticeable andtherefore remain unobserved for a period of time Immuneresponse of susceptible individuals can restrict proliferationof the bacilli leading to what seems to be long-lastingpartial immunity against reinfection or a response capable ofstopping the progression from LTB to active TB

Exposed individuals may remain in the latent stage forlong and variable periods of time In fact it often happensthat the host dies without ever developing active TB Theprogression from latent to active TB is uncommon in thepopulation at large It is estimated that only about 5 to 10percent of LTB individuals develop clinical or active TB[16] but due to the above described extreme conditions atsemiclosed communities such as prisons persons lived inthese communities may be at risk of rapid progression fromLTB to active TB following recent infection or reactivation oflatent infection or reinfection see [6]

Some additional known epidemiological facts to be con-sidered for TB disease are the following

(1) Most of the secondary infections generated by aninfected individual do take place within the firstmonths following TB activation [29]

(2) In the work by Styblo [16] it was noted that nearly 60percent of the new cases arose during the first yearfollowing infection while the cumulative number ofcases generated over the first five years after infectionaccounted for nearly 95 percent of the total observedcases People ill with TB can infect up to 10ndash15 otherpeople through close contact over the course of a year[30]

(3) Case fatality among untreated pulmonary TB cases isaround 666 percent [30]

(4) Recovered individuals naturally or from treatmentmay develop active TB again a phenomenon knownas TB relapse (Recurrent cases (formerly relapsecases) have been treated for TB in the past andbeen declared successfully treated (curedtreatmentcompleted) at the end of their treatment regimenRecurrent cases include relapses due to the sameMycobacterium tuberculosis strain as for the previousepisode as well as new episodes of TB due to reinfec-tion)

(5) Individuals with LTB may progress to active TB dueto reexposure and reinfection The extent to whichlatent tuberculosis infection could reduce the riskof progressive disease following reinfection is notknown [31]

3 A Compartmental Model for the TB Spread

In order to describe key relevant dynamics in the study ofthe TB spread in semiclosed communities we consider fivecompartments SEIR model represented in Figure 1

The compartments are uninfected individuals (suscep-tible) the 119878 class the latent class 119864 that is individualswho are assumed to be asymptomatic and noninfectious butcapable of progressing to the clinical disease or active TBthe infectious class 119868 is subdivided into two subclasses (a)infected and infectious individuals 119868

119868and (b) infected and

noninfectious individuals 119868119873 and the 119877 class of recovered by

treatment self cure or quarantineEvery individual in the 119864 119868

119868 and 119868

119873classes is con-

sidered infected There are five possible routes toward TBinfection according to this model primary infection after acontact with infected and infectious individuals (fast TB)endogenous reactivation after a period of latency (slow TB)relapse by natural causes after a cure exogenous reinfectionof latently infected and exogenous reinfection of recoveredindividuals

The 119891 and 119902 are probability of developing infectious TB ifone develops fast and slow TB respectively 2119908 is the relapserate to active TB Uninfected individuals are recruited at therate Π and 120583 is the natural mortality rate Individuals withTB experience a death rate 120583

119879due to TB infection


119901119891120573119878119868119868

119902120575120573119864119868119868

120583119879119868119868 120583119868119868

120583119864120583119878 119902119864119888119868119868

1199032119868119868

120583119877120503 (1 minus 119901)120573119878119868

119877119868119864119878

120575(1 minus 119902)120573119864119868119868

120578120573119877119868119868

(1 minus 119902)119864

(1 minus 119891)119901120573119878119868119868

1199031119868119873

119888119868119873

120583119879119868119873 120583119868119873

119868119868

119868119873

119908119877

119908119877

Figure 1 Flow chart of TB compartmental model

After infection a fraction 119901 of individuals progresses todisease relatively soon (ie within the first two years) afterinfection the remaining fraction 1minus119901 of infected individualsbecome latently infected Of newly infected individuals whothus progress quickly to disease119891 represents the fraction thatdevelops infectious disease and 1 minus 119891 represents the fractionthat develops noninfectious disease

The 119864 class latently infected individuals does not shedbacilli and is not infective to others In some latently infectedindividuals the infection remains latent and it may persist forlife But in a small minority of latently infected individualsreactivation of the latent infection leads to the diseaseThe coefficients 119903

1and 119903

2denote the treatment rates for

infected and infectious individuals 119868119868class and infected and

noninfectious individuals 119868119873

class respectively The modeldoes not consider unsuccessful treatments

The parameter 120573 is the primary TB transmission rate thisparameter summarizes socioeconomic and environmentalfactors that affect primary TB transmission We assume thattransmission rates are determined by broad demographicand social contexts as well as by characteristics of boththe transmitter and recipient (ie the number viability andvirulence of the organisms within sputum droplet nucleiimmune status of the recipient etc)

TB transmission rate in case of reinfection might bedifferent than the transmission rate of primary infectionThequantities that take into account these differences in case ofreinfection of latently infected individuals and reinfectionof recovered individuals are given by the dimensionlessparameters 120575 and 120578 respectively The parameter 120575 is theproportion in TB transmission due to exogenous reinfectionof latently infected individuals and 120578 is the proportion inTB transmission due to exogenous reinfection of recoveredindividuals Thus 120575120573 is the exogenous reinfection rate oflatently infected and 120578120573 is the exogenous reinfection rate ofrecovered individuals

A conservative point of viewwill consider that biologicallyplausible values for the reinfection parameters 120575 and 120578 aregiven within the intervals 0 le 120575 le 1 0 le 120578 le 1In this case the parameters 120575 and 120578 can be interpretedas factors reducing the risk of reinfection of an individualwho has previously been infected and has acquired somedegree of protective immunity However studies on geneticpredisposition [22] or in communities with cases as thosereported in [21] have gathered some evidence that in certainsituations there may be some increased susceptibility toreinfection Therefore we are willing to explore in the nextsections other mathematical possibilities where the reinfec-tion parameters can take even less usual values 120575 gt 1 and120578 gt 1

However recurrent TB due to endogenous reactivation(relapse) and exogenous reinfection could be clinically indis-tinguishable [32] they are independent events For thisreason beside primary infection we will include in themodelthe possibility of endogenous reactivation and exogenousreinfection as different way toward infection So we have thefollowing

(1) TB due to the endogenous reactivation of primaryinfection (exacerbation of an old infection) is consid-ered in the model by the terms 119902]119864 and (1 minus 119902)]119864

(2) TB due to reactivation of primary infection inducedby exogenous reinfection is considered by the terms120575119902120573119864119868

119868and 120575(1 minus 119902)120573119864119868

119868

(3) Recurrent TB due to exogenous reinfection after acure or treatment is described by the term 120578120573119868

119868119877

Theparameters of themodel its descriptions and its unitsare given in Table 1


Table 1 Parameters of the model its descriptions and its units

Parameter Description Unit120573 Transmission rate 1yearΠ Recruitment rate 1year119888 Natural cure rate 1year] Progression rate from latent TB to active TB 1year120583 Natural mortality rate 1year120583119879 Mortality rate or fatality rate due to TB 1year

119908 Relapse rate 1year119902 Probability to develop TB (slow case) mdash119891 Probability to develop TB (fast case) mdash

119901Proportion of new infections that produceactive TB mdash

120575120573 Exogenous reinfection rate of latent 1year120578120573 Exogenous reinfection rate of recovered 1year1199031 Treatment rates for 119868

1198681year

1199032 Treatment rates for 119868

1198731year

All these considerations give us the following system ofequations

119889119878

119889119905= Π minus 120573119878119868

119868minus 120583119878

119889119864

119889119905= (1 minus 119901) 120573119878119868

119868+ 120578120573119877119868

119868minus (] + 120583) 119864 minus 120575120573119864119868

119868

119889119868119868

119889119905= 119891119901120573119878119868

119868+ 119902]119864 + 119908119877 minus (120583 + 120583

119879+ 119888 + 119903

1) 119868119868+ 120575119902120573119864119868

119868

119889119868119873

119889119905= (1 minus 119891) 119901120573119878119868

119868+ (1 minus 119902) ]119864

+ 119908119877 minus (120583 + 120583119879+ 119888 + 119903

2) 119868119873+ 120575 (1 minus 119902) 120573119868

119868119864

119889119877

119889119905= 119888 (119868119868+ 119868119873) minus (2119908 + 120583) 119877 minus 120578120573119877119868

119868+ 1199031119868119868+ 1199032119868119873

(1)

Adding all the equations in (1) together we have

119889119873

119889119905= minus120583119873 minus 120583

119879(119868119868+ 119868119873) + Π (2)

where119873 = 119878 + 119864 + 119868119868+ 119868119873+ 119877 represents the total number of

the population and the region

119863 = (119878 119864 119868119868 119868119873 119877) isin R

5

+ 119878 + 119864 + 119868

119868+ 119868119873+ 119877 le

Π

120583 (3)

is positively invariant of system (1)It is a common practice in epidemic research to introduce

the basic reproduction number 1198770 defined as the average

number of secondary infections produced by an infectedindividual in a completely susceptible population as themeasure for the epidemic thresholds if 119877

0gt 1 an epidemic

will arise

We have calculated 1198770for this model using the Next

Generation Method [35] and it is given by

1198770= 120573Π ((ℎ119891119901 + (1 minus 119901) ]119902) (119886119887 minus 119898119908)

+ 119898119908 (ℎ (1 minus 119891) 119901 + (1 minus 119901) ] (1 minus 119902)))

times (120583ℎ119886 (119886119887 minus 119892119908 minus 119898119908))minus1

(4)

where

119886 = 120583 + 120583119879+ 119888

119887 = 2119908 + 120583

ℎ = ] + 120583

119892 = 1199031+ 119888

119898 = 1199032+ 119888

(5)

31 Steady-State Solutions In order to find steady-statesolutions for (1) we have to solve the following system ofequations

0 = Π minus 120573119878119868119868minus 120583119878

0 = (1 minus 119901) 120573119878119868119868+ 120578120573119877119868

119868minus (] + 120583) 119864 minus 120575120573119864119868

119868

0 = 119891119901120573119878119868119868+ 119902]119864 + 119908119877 minus (120583 + 120583

119879+ 119888 + 119903

1) 119868119868+ 120575119902120573119864119868

119868

0 = (1 minus 119891) 119901120573119878119868119868+ (1 minus 119902) ]119864 + 119908119877

minus (120583 + 120583119879+ 119888 + 119903

2) 119868119873+ 120575 (1 minus 119902) 120573119868

119868119864

0 = 119888 (119868119868+ 119868119873) minus (2 119908 + 120583) 119877 minus 120578120573119877119868

119868+ 1199031119868119868+ 1199032119868119873

(6)

Solving system (6)with respect to 119868119868wehave the following

equation

119868119868(1198601198683

119868+ 1198611198682

119868+ 119862119868119868+ 119863) = 0 (7)

The coefficients of (7) are all expressed as functions of theparameters listed in Table 1 However these expressions aretoo long to be written here See Appendix A for explicit formsof the coefficients

311 Disease-Free Equilibrium For 119868119868= 0we get the disease-

free steady-state solution

1198750= (1198780 1198640 1198681198680 1198681198730

1198770) = (

Π

120583 0 0 0 0) (8)

Lemma 1 The disease-free steady-state solution 1198750is locally

asymptotic stable for 1198770lt 1


Proof The Jacobian matrix of system (6) evaluated in 1198750=

((Π120583) 0 0 0 0) is

119869 =

[[[[[[[[

[

minus120583 0 minus120573 Π

1205830 0

0 minus] minus 120583(1 minus 119901) 120573Π

1205830 0

0 119902]119891119901120573Π

120583minus 120583 minus 120583119879 minus 119888 minus 1199031 0 119908

0 (1 minus 119902) ](1 minus 119891) 119901120573Π

120583minus120583 minus 120583119879 minus 119888 minus 1199032 119908

0 0 1199031 + 119888 1199032 + 119888 minus2119908 minus 120583

]]]]]]]]

]

(9)

The characteristic equation for 119869 have the form

(120582 + 120583) (1205824+ 11988631205823+ 11988621205822+ 1198861120582 + 1198860) = 0 (10)

Given the polynomial

119875 (120582) = 1205824+ 11988631205823+ 11988621205822+ 1198861120582 + 1198860= 0 (11)

in the special case when 1198861 1198862 1198863gt 0 3 roots of the polyno-

mial 119875(120582) have negative real part and if

(i) 1198860= 0 the 4th root or largest eigenvalue is zero

(ii) 1198860gt 0 all eigenvalues are negative

(iii) 1198860lt 0 the largest eigenvalue has positive real part

Thus the stability of disease-free steady-state solution isdetermined solely by the sign of the constant term 119886

0of the

polynomial 119875(120582) [36]The coefficients 119886

0 1198861 1198862 1198863are all decreasing functions

of 120573 and they are linear functions with respect to theparameter 120573 so they all take the general form 119886

119894(120573) = minus119860

119894120573+

119861119894with 119894 = 0 1 2 3 and 119860

119894 119861119894gt 0 We can define 120573

119894= 119861119894119860119894

It is easy to see that for 120573 lt 120573119894we have 119886

119894(120573) gt 0

For example

1198863(120573) = minus

119891119901120573 Π

120583+ ] + 3120583 + 120583

119879+ 119888 + 119903

2+ 2119908

1205733=

(] + 3120583 + 120583119879+ 119888 + 119903

2+ 2119908) 120583

119891119901Π

(12)

By straightforward calculations and reminding that thecoefficients 119886

119894are decreasing functions of 120573 we found that

1198862(1205733) lt 0 997904rArr 120573

2lt 1205733

1198861(1205732) lt 0 997904rArr 120573

1lt 1205732

1198860(1205731) lt 0 997904rArr 120573

0lt 1205731

(13)

This way 1205730lt 1205731lt 1205732lt 1205733and if we take 120573 lt 120573

0 all the

coefficients 119886119894are positive But for 119877

0lt 1 we can see that the

condition 120573 lt 1205730is fulfilled Indeed the constant term 119886

0of

the polynomial 119875(120582) can be written as

1198860(120573) = minus119860

0120573 + 1198610= 1198610(1 minus

1198600120573

1198610

) = 1198610(1 minus 119877

0) (14)

Using this form for the coefficient 1198860we can see that if

1198770lt 1 then 119886

0(120573) gt 0 so 120573 lt 120573

0

Remark 2 For 1198770gt 1 we have 119886

0lt 0 and the disease-free

steady-state solution is unstable Indeed if 1205821 1205822 1205823 and 120582

4

are the roots of polynomial 119875(120582) = 0 we have 1205821120582212058231205824=

1198860lt 0 and it is impossible that the four roots have negative

real part

312 Endemic Equilibria From (7) we get that nontrivialsolutions are possible if

1198601198683

119868+ 1198611198682

119868+ 119862119868119868+ 119863 = 0 (15)

The explicit expressions for the coefficients 119860 and119863 are

119860 = 1205733120575120590 (120583 + 120583

119879) (120583 + 120583

119879+ 119888 + (1 minus 119902) 119903

1+ 1199021199032) gt 0 (16)

119863 = ℎ ([119886119887 + 1199081199032+ 1199031(119908 + 120583)] (120583 + 120583119905)

+ 120583119898 (1199031+ 119886)) (1 minus 119877

0)

(17)

where parameters 119886 119887 ℎ and119898 are defined as in (5)The coefficients 119861 and 119862 can be written in the following

general form

119861 = 1205732119891119861(120573)

119862 = 120573119891119862(120573)

(18)

where119891119861(120573) = minus119861

1120573 + 1198612

119891119862(120573) = minus119862

1120573 + 119862

2

(19)

The coefficients 119861119894=12

and 119862119894=12

are all positive anddepend on the parameters given in Table 1 See Appendix Afor the explicit form of these coefficients

Changes in the signs of the coefficient 119861 and 119862 asfunction of transmission rate 120573 can be explained using theabove defined functions 119891

119861(120573) and 119891

119862(120573) respectively The

functions 119891119861(120573) and 119891

119862(120573) both are linear and decreasing

functions of 120573Consider the polynomial function

119875 (119909) = 1198601199093+ 1198611199092+ 119862119909 + 119863 (20)

From (17) we can see that for 1198770

gt 1 the coefficient 119863is negative so we have 119875(0) = 119863 lt 0 On the other handbecause the coefficient 119860 is always positive there must bea value 119909

lowast such that for 119909 gt 119909lowast it holds that 119875(119909) gt 0

Since function 119875(119909) is continuous this implies the existenceof solution for the equation 119875(119909) = 0

To determine howmany possible endemic states arise weconsider the derivative 1198751015840(119909) = 3119860119909

2+ 2119861119909+119862 and then we

analyse the following cases

(1) If Δ = 1198612minus 3119860119862 le 0 119875

1015840(119909) ge 0 for all 119909

then 119875(119909) is monotonically increasing function andwe have a unique solution that is a unique endemicequilibrium

(2) If Δ ge 0 we have solutions of the equation 1198751015840(119909) = 0

given by

11990921

=minus119861 plusmn radic1198612 minus 3119860119862

3119860

(21)


and 1198751015840(119909) ge 0 for all 119909 ge 119909

2and 119909 le 119909

1 So we need to

consider the positions of the roots 1199091and 119909

2in the real line

We have the following possible cases(i) If 119862 le 0 then for both cases 119861 ge 0 and 119861 lt 0 we

have 1199091lt 0 119909

2gt 0 and 119875

1015840(119909) gt 0 for all 119909 ge 119909

2ge 0

Given that 119875(0) = 119863 lt 0 this implies the existence ofa unique endemic equilibrium

(ii) If 119861 ge 0 and 119862 ge 0 then both roots 1199091and 119909

2are

negative and 1198751015840(119909) gt 0 for all 119909 ge 0

(iii) If 119861 lt 0 and 119862 gt 0 then both roots 1199091and 119909

2

are positive and we have the possibility of multipleendemic equilibria This is a necessary condition butnot sufficient It must be fulfilled also that 119875(119909

1) ge 0

Let 120573119861be the value of 120573 such that 119891

119861(120573119861) = 0 and 120573

119862the

value of120573 such that119891119862(120573119862) = 0Moreover let120573

1198770be the value

for which the basic reproduction number 1198770is equal to one

(the value of 120573 such that coefficient119863 becomes zero)

Lemma 3 If the condition 1205731198770

lt 120573119862lt 120573119861is met then system

(1) has a unique endemic equilibrium for all 120573 gt 1205731198770

(Table 3)

Proof Using similar arguments to those used in the proof ofLemma 1 we have given the condition 120573

1198770lt 120573119862lt 120573119861 that

for all values of 120573 such that 120573 lt 1205731198770 all polynomial coeffi-

cients are positive therefore all solutions of the polynomialare negative and there is no endemic equilibrium (positiveepidemiologically meaningful solution)

For 1205731198770

lt 120573 lt 120573119862the coefficients 119861 and 119862 are both

positive while the coefficient119863 is negative therefore appearsonly one positive solution of the polynomial (the greatestone) so we have a unique endemic equilibrium

For 120573119862

lt 120573 lt 120573119861 the coefficient 119862 is negative and 119861 is

positive According to the cases studied above we have in thissituation a unique endemic equilibrium

Finally for 120573 gt 120573119861

the coefficients 119861 and 119862 areboth negative and according to the study of cases givenabove we also have a unique positive solution or endemicequilibrium

Let us first consider biologically plausible values for thereinfection parameters 120578 and 120575 that is values within theintervals 0 le 120575 le 1 0 le 120578 le 1 This means that thelikelihood of both variants of reinfections is no greater thanthe likelihood of primary TB So we are considering herepartial immunity after a primary TB infection

Lemma 4 For biologically plausible values (120575 120578) isin [0 1] times

[0 1] system (1) fulfils the condition 1205731198770

lt 120573119862lt 120573119861

Proof Using straightforward but cumbersome calculations(we use a symbolic software for this task) we were able toprove that if we consider all parameters positive (as it is thecase) and taking into account biologically plausible values(120575 120578) isin [0 1] times [0 1] then 119891

119861(120573119862) gt 0 and 119863(120573

119861) gt 0

and it is easy to see that these inequalities are equivalent to1205731198770

lt 120573119862lt 120573119861

We have proven that the condition 1205731198770

lt 120573119862

lt 120573119861

implies that the system can only realize two epidemiologically

Table 2 Qualitative behaviour for system (1) as a function of thedisease transmission rate 120573 when the condition 120573

1198770lt 120573119862

lt 120573119861is

fulfilled

Interval Coefficients Type ofequilibrium

120573 lt 1205731198770

119860 gt 0 119861 gt 0 119862 gt 0119863 gt 0Disease-freeequilibrium

1205731198770

lt 120573 lt 120573119862 119860 gt 0 119861 gt 0 119862 gt 0119863 lt 0

Unique endemicequilibrium

120573119862lt 120573 lt 120573

119861 119860 gt 0 119861 gt 0 119862 lt 0119863 lt 0Unique endemicequilibrium

120573 gt 120573119861 119860 gt 0 119861 lt 0 119862 lt 0119863 lt 0


800

600

400

200

0

minus200

minus400

119909

1198770 gt 1

1205731198770120573119862 120573119861

00002 00004 00006 00008 0001120573

Figure 2 Bifurcation diagram (solution 119909 of polynomial (20) versus120573) for the condition 120573

1198770lt 120573119862

lt 120573119861 1205731198770

is the bifurcation valueThe blue branch in the graph is a stable endemic equilibrium whichappears for 119877

0gt 1

meaningful (nonnegative) equilibrium states Indeed if weconsider the disease transmission rate 120573 as a bifurcationparameter for (1) then we can see that the system experiencesa transcritical bifurcation at 120573 = 120573

1198770 that is when 119877

0= 1

(see Figure 2) If the condition 1205731198770

lt 120573119862

lt 120573119861is met the

system has a single steady-state solution corresponding tozero prevalence and elimination of the TB epidemic for 120573 lt

1205731198770 that is119877

0lt 1 and two equilibrium states corresponding

to endemic TB and zero prevalence when 120573 gt 1205731198770 that is

1198770

gt 1 Moreover according to Lemma 4 this condition isfulfilled in the biologically plausible domain for exogenousreinfection parameters (120575 120578) isin [0 1] times [0 1] This case issummarized in Table 2

From Table 2 we can see that although the signs of thepolynomial coefficients may change other new biologicallymeaningful solutions (nonnegative solutions) do not arisein this case The system can only display the presence oftwo equilibrium states disease-free or a unique endemicequilibrium


Table 3 Qualitative behaviour for system (1) as function of thedisease transmission rate 120573 when the condition 120573

119861lt 120573119862

lt 1205731198770

isfulfilled Here Δ

1is the discriminant of the cubic polynomial (20)


120573 lt 120573119861 119860 gt 0 119861 gt 0 119862 gt 0119863 gt 0

Disease-freeequilibrium

120573119861lt 120573 lt 120573

119862 119860 gt 0 119861 lt 0 119862 gt 0119863 gt 0

Two equilibria(Δ1lt 0) or none(Δ1gt 0)

120573119862lt 120573 lt 120573

1198770119860 gt 0 119861 lt 0 119862 lt 0119863 gt 0


1205731198770

lt 120573 119860 gt 0 119861 lt 0 119862 lt 0119863 lt 0Unique endemicequilibrium

The basic reproduction number 1198770in this case explains

well the appearance of the transcritical bifurcation that iswhen a unique endemic state arises and the disease-freeequilibrium becomes unstable (see blue line in Figure 2)

However the change in signs of the polynomial coef-ficients modifies the qualitative type of the equilibria Thisfact is shown in Figures 5 and 7 illustrating the existenceof focus or node type steady-sate solutions These differenttypes of equilibria as we will see in the next section cannotbe explained using solely the reproduction number 119877

0

In the next section we will explore numerically the para-metric space of system (1) looking for different qualitativedynamics of TB epidemics We will discuss in more detailhow dynamics depends on the parameters given in Table 1especially on the transmission rate 120573 which will be used asbifurcation parameter for the model

Let us consider here briefly two examples of parametricregimes for the model in order to illustrate the possibility toencounter a more complex dynamics which cannot be solelyexplained by changes in the value of the basic reproductionnumber 119877

0

Example I Suppose 120573 = 1205731198770 this implies that 119877

0= 1 and

119863 = 0 therefore we have the equation

119875 (119909) = 1198601199093+ 1198611199092+ 119862119909

= 119909 (1198601199092+ 1198611199092+ 119862) = 0

(22)

It is easy to see that besides zero solution if 119861 lt 0 119862 gt 0

and 1198612minus 4119860119862 gt 0 (22) has two positive solutions 119909

1and 119909

2

So we have in this case three nonnegative equilibria for thesystem

The condition 119861 lt 0 for 120573 = 1205731198770

means 119891119861(1205731198770) lt 0 and

this in turn implies that 120573119861

lt 1205731198770 On the other hand the

condition 119862 gt 0 implies 119891119862(1205731198770) gt 0 and therefore 120573

1198770lt 120573119862

Gathering both inequalities we can conclude that if 120573119861lt

1205731198770

lt 120573119862 then the system has the possibility of multiple

equilibriaSince the coefficients 119860 and 119861 are both continuous

functions of 120573 we can always find a 120598 neighbourhood of 1205731198770

|120573 minus 1205731198770| lt 120598 such that the signs of these coefficients are

preserved Although in this case we do not have the solution

times10minus3

119875(119909)

01

005

minus005

minus01

minus200 0 200 400 600119909

Figure 3 Polynomial 119875(119909) for different values of 120573 with thecondition 120573

119861lt 1205731198770

lt 120573119862 The graphs were obtained for values of

120575 = 30 and 120578 = 22The dashed black line indicates the case120573 = 1205731198770

The figure shows the existence of multiple equilibria

119909 = 0 we eventually could still have two positive solutionsand consequently multiple equilibrium states see the greenline in Figure 3Example II Suppose we take numerical values for the param-eters in Table 1 such that the condition 120573

119861lt 120573119862

lt 1205731198770

isfulfilled

If 120573 lt 120573119861 then all coefficients of the polynomial (20) are

positive and there is not nonnegative solutions In this casethe system has only a disease-free equilibrium

For 120573119861

lt 120573 lt 120573119862and 120573

119862lt 120573 lt 120573

1198770the signs of the

coefficients of the polynomial are 119860 gt 0 119861 lt 0 119862 gt 0 and119863 gt 0 119860 gt 0 119861 lt 0 119862 lt 0119863 gt 0 respectively

In both cases the polynomial has two possibilities

(a) three real solutions one negative and two positivesolutions for Δ

1lt 0

(b) one negative and two complex conjugate solutions forΔ1gt 0

Here Δ1is the discriminant for the polynomial (20)

In the (a) case we have the possibility of multiple endemicstates for system (1) This case is illustrated in numericalsimulations in the next section by Figures 8 and 9

We should note that the value 120573 = 120573119861is not a bifurcation

value for the parameter 120573If 120573 = 120573

119861 then 119860 gt 0 119861 = 0 119862 gt 0 and 119863 gt 0 In this

case we have

Δ1=

1

4

1198632

1198602+

1

27

1198623

1198603gt 0 (23)

ThediscriminantΔ1is a continuous function of120573 for this

reason this sign will be preserved in a 120598 neighbourhood of 120573119861

We should be able to find a bifurcation value solvingnumerically the equation

Δ1(120573lowast) = 0 (24)


600

500

400

300

200

100

0

minus100

minus200

minus300

119909

1198770 gt 1

1205731198770120573119862120573119861 120573lowast

000005 00001 000015 00002120573

Figure 4 Bifurcation diagram for the condition 120573119861

lt 120573119862

lt 1205731198770

120573lowast is the bifurcation value The blue branch in the graph is a stable

endemic equilibrium which appears even for 1198770lt 1

where 120573lowast can be bounded by the interval 120573119861lt 120573lowastlt 1205731198770

(seeFigure 4)

4 Numerical Simulations

In this section we will show some numerical simulationswith the compartmental model (1) This model has fourteenparameters that have been gathered in Table 1 In order tomake the numerical exploration of the model more manage-able we will adopt the following strategy

(i) First instead of fourteen parameters we will reducethe parametric space using four independent param-eters 120573

1198770 120573 120575 and 120578 The parameters 120573 120575 and 120578 are

the transmission rate of primary infection exogenousreinfection rate of latently infected and exogenousreinfection rate of recovered individuals respectively1205731198770

is the value of 120573 such that basic reproductionnumber 119877

0is equal to one (or the value of 120573 such that

coefficient 119863 in the polynomial (20) becomes zero)On the other hand 120573

1198770depends on parameters given

in the list Λ = Π 119888 119891 120583 ] 119901 119902 119908 120583119905 1199031 1199032 This

means that if we keep all the parameters fixed in thelistΛ then 120573

1198770is also fixed In simulations we will use

1205731198770

instead of using basic reproduction number 1198770

(ii) Second we will fix parameters in the list Λ accordingto the values reported in the literature In Table 4 areshown numerical values that will be used in some ofthe simulations besides the corresponding referencesfrom where these values were taken Mostly thesenumerical values are related to data obtained fromthe population at large and in the next simulationswe will change some of them for considering theconditions of extremely high incidenceprevalence of

Table 4 Numerical values for the parameters in the list Λ Someof the given numerical values for the model parameters are mainlyrelated to the spread of TB in the population at large and are basicallytaken as reference Other values assuming for the parametersdifferent than those given in this table will be clearly indicated inthe text

Parameter Description ValueΠ Recruitment rate 200 (assumed)119888 Natural cure rate 0058 [23 33 34]

]Progression rate from latent TB toactive TB 00256 [33 34]

120583 Natural mortality rate 00222 [2]120583119879 Mortality rate due to TB 0139 [2 33]

119908 Relapse rate 0005 [2 33 34]119902 Probability to develop TB (slow case) 085 [2 33]119891 Probability to develop TB (fast case) 070 [2 33]

119901Proportion of new infections thatproduce active TB 005 [2 33 34]


119868050 (assumed)


119873020 (assumed)

TB in semiclosed communities In any case thesechanges will be clearly indicated in the text

(iii) Third for any pairs of values 120575 and 120578 we can compute120573119861and 120573

119862 that is the values of 120573 such that 119861 = 0

and 119862 = 0 respectively in the polynomial (20) Sowe have that the exploration of parametric space isreduced at this point to the study of the parameters1205731198770 120573119861 120573119862 and 120573 According to the chosen values for

120575 120578 and 1205731198770 we have six possible orderings for the

parameters 1205731198770 120573119861 and 120573

119862(see Appendix B)

The dynamic behavior of system (1) will depend ofthese orderings In particular from Table 5 it is easyto see that if 120573 le min(120573

1198770 120573119861 120573119862) then the system

has a unique equilibrium point which represents adisease-free state and if 120573 ge max(120573

1198770 120573119861 120573119862) then

the system has a unique endemic equilibrium besidesan unstable disease-free equilibrium

(iv) Fourth and finally we will change the value of 120573which is considered a bifurcation parameter for sys-tem (1) taking into account the previous mentionedordering to find different qualitative dynamics

It is especially interesting to explore the consequencesof modifications in the values of the reinfection parameterswithout changing the values in the list Λ because in this casethe threshold 120573

1198770remains unchanged Thus we can study in

a better way the influence of the reinfection in the dynamicsof the TB spread

The values given for the reinfection parameters 120575 and 120578

in the next simulations could be extreme trying to capturethis way the special conditions of high burden semiclosedcommunitiesExample I (Case 120573

1198770lt 120573119862

lt 120573119861 120575 = 09 120578 = 001) Let us

consider here the case when the condition 1205731198770

lt 120573119862lt 120573119861is


metWe know from the previous section that this condition ismet under biologically plausible values (120575 120578) isin [0 1] times [0 1]

According to Lemmas 3 and 4 in this case the behaviourof the system is characterized by the evolution towardsdisease-free equilibrium if 120573 lt 120573

1198770and the existence of

a unique endemic equilibrium for 120573 gt 1205731198770 Changes in

the parameters of the list Λ alter the numerical value of thethreshold 120573

1198770but do not change this behaviour

First we consider the following numerical values for theseparameters 120575 = 09 120578 = 001 and 120573 = 000052 We alsofix the list of parameters Λ according to the numerical valuesgiven in Table 4

The basic reproduction number for these numerical val-ues gives119877

0= 3585422172The initial conditions considered

were

119878 (0) = 4980 119864 (0) = 0 119868119868 (0) = 20

119868119873 (0) = 0 119877 (0) = 0

(25)

We also have the following values

1205731198770

= 00001450317354

120573119861= 001087387065

120573119862= 00002715343808

(26)

These values clearly meet the condition 1205731198770

lt 120573119862

lt 120573119861

and according to Lemma 3 the system must have in this casea unique endemic equilibrium for all 120573 gt 120573

1198770

Figure 5 shows that under the above described situationthe system will converge to an endemic equilibrium given bythe focus type stationary steady solution

119878infin

= 1616 119877infin

= 4080 119868119873infin

= 103

119868119868infin

= 195 119864infin

= 1150

(27)

By straightforward calculations we can show that thisfocus is stable and nomatter what initial conditions are takenfor the system the solutions always evolve to this endemicstate

Figure 6 shows the trajectories of the system for multipleinitial conditions in a three-dimensional phase space inwhich the horizontal axes are susceptible 119878 and recovered 119877

individuals while the vertical axis is the prevalence 119868119868+119868119873+119864

Example II (Case 1205731198770

lt 120573119862lt 120573119861 120575 = 00 120578 = 09) For our

next numerical simulation we consider the following valuesfor the used parameters 120575 = 001 120578 = 09 120573 = 000052 andas before the list of parametersΛ is fixed according to Table 4

The basic reproduction number for these parameters asbefore gives the same value 119877

0= 3585422172 The used

initial conditions were

119878 (0) = 4980 119864 (0) = 0 119868119868 (0) = 20

119868119873 (0) = 0 119877 (0) = 0

(28)

119905

7000

6000

5000

4000

3000

2000

1000

00 50 100 150 200

119878

119877

119868119873119864

119868119868 + 119868119873119868119868119873

119868119868 + 119868119873 + 119864 + 119877

Figure 5 Numerical simulation for 1198770

= 3585422172 120575 = 09120578 = 001 and 120573 = 000052 The system goes toward a focus typestable stationary equilibrium

4000

4000 5000

3000

3000

2000

2000

1000

10004000

6000

20000

119868 119868+119868 119873+119864

119878 119877 4000 500

3000000000000000000000000000000

3000

200202000020200202002020200000000200202020020020200200202020202002002000202002000020202020200020202000220000200202000020202222000222020200220000202022022220022002202222220022222 02 02222 0000000000000000000000000000000000000000000000000000000000000000

2000

100110010010010010010010010100100100100010010010010010010010010000010100100100100100100100100110000001001000000110010110010010010010001000001110100010000010000000001010000011000001110000000001001001000000000000001000010000111000101 000000000000000000000000000000000000000000000000000000000000000000000000000000000000000

10004000

6000

20000

119868119868119868119868119868119868119868119868119868119868119868119868119868119868119868119868119868119868119868119868119868119868119868119868119868119868119868119868119868119868119868119868119868119868119868119868119868119868119868119868 119868119868119868119868119868119868119868119868119868119868119868119868119868119868++++++++++++++++++++++++++119868 119873119868+119864

119878 119877

Figure 6 Phase space representation of the evolution of the systemtoward a stable focus type equilibrium In this representation wereused multiple initial conditions and the following values 119877

0=

3585422172 120575 = 09 120578 = 001 and 120573 = 000052


1205731198770

= 00001450317354

120573119861= 001226355348

120573119862= 00003132229272

(29)

These values meet the condition 1205731198770

lt 120573119861lt 120573119862 and as in

the previous simulation the system evolves toward a uniqueendemic equilibrium but this time the dynamical propertiesof the equilibrium have changed

In fact Figure 7 shows the evolution of the system towarda stable node type endemic equilibrium

119878infin

= 1938 119877infin

= 974 119868119873infin

= 60

119868119868infin

= 156 119864infin

= 4530

(30)


119905

7000

6000

5000

4000

3000

2000

1000

00 50 100 150 200

119878

119877

119868119873119864

119868119868 + 119868119873119868119868119873

119868119868 + 119868119873 + 119864 + 119877

Figure 7 Numerical simulation for 1198770= 3585422172 120575 = 001

120578 = 09 and 120573 = 000052 In this case the system converges to astable node type equilibrium

In our model considering biologically plausible domainfor exogenous reinfection parameters (120575 120578) isin [0 1] times [0 1]the condition 120573

1198770lt 120573119862

lt 120573119861is fulfilled Under this

condition we have a unique endemic equilibrium for120573 gt 1205731198770

The emergence by a transcritical bifurcation of this endemicstate is properly explained by the basic reproduction number1198770 However changes in the reinfection parameters 120575 120578

can modify the qualitative nature of the dynamics of thedisease in addition to changing the numbers of individualsin the different compartments of the model in the endemicequilibrium state without having any change in the value ofthe basic reproduction number 119877

0 which in this case fails to

describe these variations in the dynamics of the diseaseExample III (Case 120573

119861lt 120573119862

lt 1205731198770 120575 = 30 120578 = 25) There

is now evidence that rates of secondary tuberculosis in highendemic communities (for example semiclosed communi-ties) in patients with LTB orand already treated for primarydisease are actually higher than in people presenting withprimary infection [21 22] Taking this into considerationwe consider now the following numerical values for theparameters 120573 = 000014 120575 = 30 120578 = 25 In thiscase the basic reproduction number takes the value 119877

0=

09653059690 Additionally we have

1205731198770

= 00001450317354

120573119861= 00001066568066

120573119862= 00001225687204

(31)

For these parameter we have that the condition120573119861lt 120573119862lt

1205731198770

is fulfilled and the system has the possibility of multipleequilibria In fact we have in this case the following stationarypoints 119875 = (119878 119877 119868

119894 119868119899 119864)

1198751= (9009 0 0 0 0)

1198752= (8507 182 9 5 2166)

1198753= (3221 1406 285 103 1566)

(32)

800

700

600

500

400

300

200

100

07006005004003002001000

119868 119868

119905


and 120578 = 25The systemcan evolve to twodifferent equilibria 119868119868infin

= 0

(red lines) or 119868119868infin

= 285 (dark green lines) according to differentinitial conditions

11987531198751198751198751198751198751198751198751198751198751198751198751198751198751198751198751198751198751198751198751198751198751198751198751198751198751198751198751198751198751198751198751198751198751198751198751198751198751198751198751198751198751198751198751198751198751198751198751198751198751198751198751198751198751198751198751198751198751198751198751198751198751198751198751198751198751198751198751198751198751198751198751198751198753119875119875

16001200

800400

900080007000

600050004000300020000

1000

2000

3000

119868 119868+119868 119873+119864

1198781198771198752

1198751


and 120578 = 25 Phase space representation of the system with multipleequilibrium points

1198751is a stable disease-free equilibrium point (stable node) 119875

3

is a stable endemic equilibrium (stable focus) and 1198752is an

unstable equilibrium point (saddle point)Figure 8 shows the convergence to 119868

119868infin= 0 or to 119868

119868infin=

285 according to with different initial conditionsIn Figure 9 is shown another representation (phase space)

of the evolution of the system toward 1198751or to 119875

3according

to different initial conditions The representation is a three-dimensional phase space in which the horizontal axes are


susceptible 119878 and recovered 119877 individuals while the verticalaxis is the prevalence 119868

119868+ 119868119873+ 119864

For the previously numerical values the system expe-riences a backward bifurcation [37] at the value 120573

lowast=

00001261648723with 120573119861lt 120573lowastlt 1205731198770 For 120573 gt 120573

lowast the system

possesses two stable equilibrium points and one unstable (seeFigure 4)Example IV (Case 120573

119861lt 1205731198770

lt 120573119862 120575 = 30 120578 = 25) Consider

now a more extreme situation with 120578 = 25 120575 = 30 and119901 = 07 (the other parameters kept the same values given inTable 4) In this case the condition 120573

119861lt 1205731198770

lt 120573119862is fulfilled

This example is shown in order to illustratemore complexand rich dynamics that might admit system (1) which ismathematically possible and could in principle be a modelcase for an extreme hypothetical situation in a semiclosedhigh burden community For these parameters we have

1205731198770

= 00001679568390

120573119862= 00001729256777

120573119861= 00001489092005

(33)

which clearly satisfy the condition 120573119861lt 1205731198770

lt 120573119862 Therefore

as was explained in the previous section the system has thepossibility of multiple equilibria

In fact for the bifurcation value 1205731= 00001673533706

of the disease transmission rate which satisfies the condition120573119861

lt 1205731lt 1205731198770 the system acquires two positive equilibria

apart from the disease-free equilibriumWhen 120573 = 120573

1198770appear three positive equilibrium points

and the disease-free equillibrium becomes unstable For 1205732=

00001688612368 with 1205731198770

lt 1205732lt 120573119862the system admits a

unique and stable endemic equilibrium (see Figure 10)We take now the value 120573 = 00001675 which satisfies the

condition 1205731lt 120573 lt 120573

1198770

With these numerical values the basic reproductionnumber is119877

0= 09972800211 lt 1 and therefore the disease-

free equilibrium is stableWe have in this case the following stationary points 119875 =

(119878 119877 119868119894 119868119899 119864)

1198750= (5148 0 0 0 0)

1198751= (3372 1041 122 60 482)

1198752= (2828 1283 190 88 651)

(34)

1198750is the stable disease-free equillibrium point (stable node)

1198751is an unstable equilibrium point (saddle point) and 119875

2is

a stable endemic equilibrium (stable focus) Figure 11 showsthe convergence to 119868

119868infin= 0 or to 119868

119868infin= 190 according to the

initial conditionIn Figure 12 is shown another representation (phase

space) of the evolution of the system toward 1198750or to 119875

2

according to the initial conditionsLet us take now the value 120573 = 00001683 which satisfies

the condition 1205731198770

lt 120573 lt 1205732 In this case the basic

reproduction number has the value 1198770= 1002043150 We

still have that the condition 120573119861

lt 1205731198770


1205731198770

1205731

1205732

120573119862120573119861

000015 000016 000017 000018120573

600

500

400

300

200

100

0

119909

Figure 10 Bifurcation diagram (solution 119909 of polynomial (20)versus 120573) for the condition 120573

119861lt 1205731198770

lt 120573119862 The system experiences

multiple bifurcations at 1205731 1205731198770 and 120573

2

500

400

300

200

100

0

119868 119868

0 500 1000 1500 2000119905



= 0

or 119868119868infin

= 190 according to the initial condition

and the system in this case has four equilibrium points 119875 =

(119878 119877 119868119894 119868119899 119864)

1198750= (5148 0 0 0 0)

1198751= (5042 76 5 3 20)

1198752= (3971 734 69 36 298)

1198753= (2491 1413 246 109 750)

(35)


160014001200800600400200

500

2000

0

1000

1500

1000

2000

3000

4000

5000

119878119877

11987521198751

1198750 160014001200800600444400420000002000200

5005005000

000

1000

1500

1000

20020020000000

300003003003003000000000000

400404004004004000040040040000000 119878119877

1198752119875119875

1198751119875119875

119868 119868+119868 119873+119877



1198750is the unstable disease-free equillibrium point (saddle

point ) 1198751is a stable endemic equilibrium point (node) 119875

2

is an unstable equilibrium (saddle point) and 1198753is a stable

endemic equilibrium point (focus)Figure 13 shows the phase space representation of this

caseFor further numerical analysis we set all the parameters

in the list Λ according to the numerical values given inTable 4 leaving free the parameters 120573 120578 and 120575 related to theprimary transmission rate and reinfection rates of the disease

We will explore the parametric space of system (1) andrelate it to the signs of the coefficients of the polynomial (20)

In Figure 14 we consider values of 120573 such that 1198770

gt

1 We can observe from this figure that as the primarytransmission rate of the disease 120573 increases and with it thebasic reproduction number 119877

0 the system under biological

plausible condition represented in the figure by the square(120575 120578) isin [0 1] times [0 1] evolves such that initially (for lowervalues of 120573) coefficients 119861 and 119862 are both positive then 119861

remains positive and 119862 becomes negative and finally bothcoefficients become negative

This change in the coefficients signs as the transmissionrate 120573 increases agrees with the results summarized in Table 2when the condition 120573

1198770lt 120573119862lt 120573119861is fulfilled

Next in order to explore another mathematical possibili-ties we will modify some numerical values for the parametersin the list Λ in a more extreme manner taking a hypotheticalregime with Λ

lowast= 120583 = 003885 120583

119905= 001520 119901 = 08

] = 00266 119891 = 08 119902 = 085 119908 = 0005 119888 = 04 1199031= 05

1199032= 02In Figure 15 besides signs of 119861 and 119862 we consider also the

signs of the discriminant Δ of the quadratic equation 1198751015840(119909) =

0 where 119875(119909) is the polynomial (20)From Figure 15 in particular we can see that the domain

with 119861 lt 0 119862 gt 0 Δ gt 0 and 119875(1199091) gt 0 represented in

160014001200

800600400200

500

2000

0

1000

1500

1000

2000

3000

4000

5000

119868 119868+119868 119873+

119878119877

119877

1198752

1198751

1198753

1198750 160014001200

8006004004040200200200200200200

500

000

1000

1500

1000

2002002002002 000000

3000030033000000000000

4004004004004000040000400 00000

119868 119868119868+119868 119873119868+

119878119877

119877

1198752119875119875

1198751119875119875

1198753119875119875

0

Figure 13 Numerical simulation for1198770= 1002043150 120575 = 30 and

120578 = 25 The system can evolve to two different equilibria 1198751(stable

node) or 1198753(stable focus) according to the initial condition 119875

0and

1198752are unstable equilibria

red allows the possibility of multiple endemic equilibria forsystem (1) However despite the extreme numerical valuesfor the parameters taking in Λ

lowast this domain is still farfrom the domain of biologically plausible values for 120575 and120578 represented in the figures by the square (120575 120578) isin [0 1]times

[0 1]As the transmission rate of the disease 120573 increases and

with it the basic reproduction number 1198770 this red domain

of the parametric space becomes increasingly smaller untilfinally it disappears In fact the red domain is only significantwhen the basic reproduction number 119877

0is near one

In Figure 16 we show some numerical simulation wherebasic reproduction number 119877

0is less than one The red

domain indicates the possibility of multiple endemic equilib-ria for the system even for119877

0lt 1We can see that this domain

is far from the domain of biologically plausible values for 120575and 120578 represented in the figure by the square (120575 120578) isin [0 1] times

[0 1] As the transmission rate of the disease 120573 decreases andwith it the number 119877

0 this parameter domain moves away

from the square In all represented cases 119861 gt 0 and 119862 gt 0

inside the square (120575 120578) isin [0 1] times [0 1]

5 Discussion and Conclusions

In order to consider high incidence and prevalence of TB andLTB in semiclosed communities we have used in this worka compartmental SEIR model with five possible pathwaysto TB disease The extra nonlinear terms considered in themodel lead to a more complex mathematical treatment incomparisonwith previously usedmodels (see eg [23 25ndash2932 35 36]) But the special form of some coefficients obtainedfrom the analysis of standard SEIR models with constanttransmission rate allowed us to move forward with someanalytical results that were confirmed later by numericalsimulations


5

5

4

4

3

3

2

2

1

10

0

120578

119861 gt 0 119862 gt 0

120575

119861 lt 0 119862 lt 0

119861 gt 0 119862 lt 0

(a) 1198770 = 1

5

5

4

4

3

3

2

2

1

10

0

120578

120575

(b) 1198770 = 1689504264

5

5

4

4

3

3

2

2

1

10

0

120578

120575

(c) 1198770 = 2379008527

5

5

4

4

3

3

2

2

1

10

0

120578

120575

(d) 1198770 = 7895042636

Figure 14 Signs of coefficients 119861 and 119862 as functions of exogenous reinfection rate of latent 120575 and exogenous reinfection rate of recovered 120578

for 1198770⩾ 1 The parameter 120573 has the values (a) 120573 = 120573

1198770= 00001450317354 (b) 120573 = 00002450317354 (c) 120573 = 00003450317354 and (d)

120573 = 0001145031735

In this paper we follow a newmethodology for numericalexploration of this kind ofmodels which allowedus to handlethe high dimensionality of its parameter spaces and thusstudy its different dynamic behaviors We found that thetransmission rate 120573 can be used as a bifurcation parameterand that the system undergoes qualitative changes in thedynamics when this parameter varies In this context theanalysis of the parametric space is reduced to the study ofvariations of the parameters 120573

1198770 120573119861 120573119862 and 120573 We divided

the parametric space into six possible arrangements for theseparameters which in turn determine all possible differentqualitative behaviours of the system dynamics

From model (1) we can see that reinfection requireslatently infected recovered and actively infectious individ-uals The basic reproduction number 119877

0has to do solely

with infections produced when an infectious individual isintroduced into an uninfected population but reinfectiondoes not alter the dynamics of an uninfected population Sothe number 119877

0does not completely describe the dynamics of

the model when the reinfection is incorporated as was notedbefore by Feng et al in [26] Unlike the model published bythese authors which uses a single parameter for exogenousreinfection in our model we use two parameters related totwo possible pathways of reinfection (reinfection of latentlyinfected and reinfection of recovered individuals)This is areason why our model shows a more complex and richerdynamics

We have showed through theoretical analysis inSection 3 and numerical simulations in Section 4 that if weaccept as valid the plausible assumption that exposure to


5

5

4

4

3

3

2

2

1

10

0

120578

120575

Δ gt 0

119891(1199091) gt 0

119861 gt 0 119862 lt 0

119861 lt 0 119862 lt 0

119861 lt 0 119862 gt 0

119861 lt 0119862 gt 0

119861 gt 0 119862 gt 0

(a) 1198770 = 1

5

5

4

4

3

3

2

2

1

10

0

120578

120575

(b) 1198770 = 1004390340

5

5

4

4

3

3

2

2

1

10

0

120578

120575

(c) 1198770 = 1087806817

5

5

4

4

3

3

2

2

1

10

0

120578

120575

(d) 1198770 = 2317102260

Figure 15 Signs of coefficients 119861 119862 and discriminant Δ = 1198612minus 3119860119862 as functions of exogenous reinfection rate of latent 120575 and exogenous

reinfection rate of recovered 120578 for 1198770⩾ 1 The parameter 120573 has the values (a) 120573 = 120573

1198770= 00002277727471 (b) 120573 = 00002287727471 (c)

120573 = 00002477727471 and (d) 120573 = 00005277727471

mycobacterium induces an immune response which ispartially protective against reinfection then the systemfor semiclosed communities (1) reproduces well commonobserved trends in TB epidemiology that are similar to whathappens in population at large which is basically that for1198770lt 1 there is only one disease-free status while for 119877

0gt 1

there exists a unique endemic state with nonzero prevalenceFor 119877

0= 1 occurs a transcritical bifurcation from which

emerges an endemic stable stateMoreover according to Lemmas 3 and 4 any values of

reinfection parameters in this parametric regime (120575 120578) isin

[0 1]times[0 1]would lead to the same qualitative dynamics andwill not affect this already classical behavior in SEIR modelsIn this case only one of the aforementioned arrangements

(1205731198770

lt 120573119862

lt 120573119861) emerges as valid under this biologically

plausible conditionSince the two parameters related to exogenous reinfection

of latently infected and recovered individuals do not affect thevalue of the number 119877

0 even under the plausible assumption

of partial immunity variation of reinfection parameterscan make that for the same value of the number 119877

0 the

quality of dynamics and the number of affected by diseaseindividuals (incidence and prevalence) drastically changeFor example Figures 5 and 7 show two types of dynamicsthat is convergences to different stationary points a focusand a node for the same basic reproduction number119877

0 Some

evidence of this variability in tuberculosis epidemiologydue to dynamic balance between primary infection and


5

5

4

4

3

3

2

2

1

10

0

120578

120575

Δ gt 0

119891(1199091) gt 0

119861 gt 0 119862 lt 0

119861 lt 0 119862 lt 0

119861 lt 0 119862 gt 0

119861 lt 0119862 gt 0

119861 gt 0 119862 gt 0

(a) 1198770 = 09560965911

5

5

4

4

3

3

2

2

1

10

0

120578

120575

(b) 1198770 = 08902414781

5

5

4

4

3

3

2

2

1

10

0

120578

120575

(c) 1198770 = 07804829561

5

5

4

4

3

3

2

2

1

10

0

120578

120575

(d) 1198770 = 05609659131


reinfection rate of recovered 120578 for 1198770⪕ 1 The parameter 120573 has the values (a) 120573 = 00002177727471 (b) 120573 = 00002027727471 (c) 120573 =

00001777727471 and (d) 120573 = 00001277727471

reinfection has been presented in several works (see eg[26 38])

Taking less plausible assumption but already evidencedin several works [5 21 22 26] of an increased susceptibilityto reinfection over primary infection in some cases leadsus to a further study of model (1) For 120575 gt 1 and 120578 gt

1 system (1) experiences a rich and complex dynamicswith successive and different kind of bifurcations as thetransmission rate 120573 changesThese cases incorporate possiblemultiple endemic states regardless of whether the values forthe basic reproduction number 119877

0were less than or greater

than 1 So these behaviors cannot be explained using onlythis number It is in this context that the use of the diseasetransmission rate 120573 as bifurcation parameter instead of 119877

0

acquires real usefulness

Some important implications of the simulations with 120575 gt

1 and 120578 gt 1 lie in the fact that many of the measures takento stop and control an epidemics are designed to reduce thevalue of the basic reproduction number 119877

0such that disease-

free status for1198770lt 1 is achievedHowever in this parametric

regime reinfection might cause the system to fall into a stateunable to eliminate endemic disease although it fulfills that1198770

lt 1 Thus semiclosed communities with this kind ofregime will become in genuine high transmission pockets ofTB inserted in the general population [4] Indeed semiclosedcommunities such as prisons might become in a reservoir fordisease transmission to the population at large and should bea source of public concern [4 6 7]

The theoretical approach and numerical simulations pre-sented in this paper for the study of the impact of reinfection


Table 5 Different possible orderings for 1205731198770 120573119861 and 120573

119862 In every case 119860 gt 0 Δ

1is the cubic discriminant of the equation 119875(119909) = 0 Δ is the

discriminant of the quadratic equation 1198751015840(119909) = 0 where 119875(119909) is the polynomial (20)

Interval Coefficients Equilibria1205731198770

lt 120573119862lt 120573119861

120573 lt 1205731198770

119861 gt 0 119862 gt 0119863 gt 0 Disease-free equilibrium1205731198770

lt 120573 lt 120573119862

119861 gt 0 119862 gt 0119863 lt 0 Unique endemic equilibrium120573119862lt 120573 lt 120573

119861119861 gt 0 119862 lt 0119863 lt 0 Unique endemic equilibrium

120573119861lt 120573 119861 lt 0 119862 lt 0119863 lt 0 Unique endemic equilibrium

120573119862lt 1205731198770

lt 120573119861

120573 lt 120573119862

119861 gt 0 119862 gt 0119863 gt 0 Disease-free equilibrium120573119862lt 120573 lt 120573

1198770119861 gt 0 119862 lt 0119863 gt 0 Two equilibria if 119875(119909

2) le 0 or Δ

1lt 0 none if 119875(119909

2) ge 0 or Δ

1gt 0

1205731198770

lt 120573 lt 120573119861

119861 gt 0 119862 lt 0119863 lt 0 One equilibrium for Δ1lt 0 or Δ

1gt 0


120573119862lt 120573119861lt 1205731198770

120573 lt 120573119862



2) le 0 or Δ

1lt 0 none if 119875(119909

2) ge 0 or Δ

1gt 0

120573119861lt 120573 lt 120573

1198770119861 lt 0 119862 lt 0119863 gt 0 Two equilibria (Δ

1lt 0) or none (Δ

1gt 0)

1205731198770

lt 120573 119861 lt 0 119862 lt 0119863 lt 0 Unique endemic equilibrium1205731198770

lt 120573119861lt 120573119862

120573 lt 1205731198770


lt 120573 lt 120573119861


119862119861 lt 0 119862 gt 0119863 lt 0 One equilibrium (Δ

1gt 0) three equilibria (Δ

1lt 0)


120573119861lt 1205731198770

lt 120573119862

120573 lt 120573119861


1198770119861 lt 0 119862 gt 0119863 gt 0 Two equilibria (Δ

1lt 0) or none (Δ

1gt 0)

1205731198770

lt 120573 lt 120573119862

119861 lt 0 119862 gt 0119863 lt 0 One equilibrium (Δ1gt 0) three equilibria (Δ

1lt 0)


120573119861lt 120573119862lt 1205731198770

120573 lt 120573119861



1lt 0) or none (Δ

1gt 0)

120573119862lt 120573 lt 120573


1lt 0) or none (Δ

1gt 0)

1205731198770

lt 120573 119861 lt 0 119862 lt 0119863 lt 0 Unique endemic equilibrium

on TB dynamics in semiclosed communities could haveimportant implications at multiple levels including vaccinedesign control programdesign epidemiology of tuberculosisin regions where the risk of reexposure is high and forsystems-based computer models which to date assume thatprimary infection will confer at least some degree of (stable)memory immunity to a secondary infection but that in factalso have to consider less plausible assumptions about anincreased susceptibility to reinfection

Appendices

A Explicit Form of Coefficients

Introducing the notations

119886 = 120583 + 120583119879+ 119888

119887 = 2119908 + 120583

ℎ = ] + 120583

119892 = 1199031+ 119888

119898 = 1199032+ 119888

120579 = 120583 + 120583119905

120598 = 1 minus 119901

1198711= 120579 + 119892 (1 minus 119902) + 119898119902

1198712= ]1199021199032+ ℎ119886 + 119903

1] (1 minus 119902) + 119903

1120583

1198701= 1199032119908 + 119887119886 + 119903

1(119908 + 120583)

1198702= ]1199021199032+ ℎ119886 + 119903

1] (1 minus 119902) + 119903

1120583

(A1)

We have

1198611= (120579 (119891119901 + (1 minus 119901) 119902) + 119898119902)Π120575120590

1198612= 120590 (120583119871

1120579 120575 + 119871

2120579 + 119898120583 (119886 + 119903

1))

+ (1199032119908120579 + 119903

1(119908 + 120583) 120579

+ 119887119886120579 + 119898120583 (119886 + 1199031)) 120575


1198621= Π (120575 ((119891119901 + 120598119902) (119887120583119905 + (119898 + 119887) 120583) + 119898119908)

+ 120590 ((]119902120598 + 119891ℎ119901) 120579 + 119898 (120583119891119901 + ] 119902)))

1198622= 120583 (119870

1120579 + 119898120583 (119886 + 119903

1)) 120575 + 120583 (119870

2120579 + 119898120583 (119886 + 119903

1)) 120590

+ ℎ (1198701120579 + 119898120583 (119886 + 119903

1))

(A2)

The coefficients 119861 and 119862 can be written in the followinggeneral form

119861 = 1205732119891119861(120573)

119862 = 120573119891119862(120573)

(A3)

where119891119861(120573) = minus119861

1120573 + 1198612

119891119862(120573) = minus119862

1120573 + 119862

2

119860 = 1205733120575120590 (120583 + 120583

119879) (120583 + 120583

119879+ 119888 + (1 minus 119902) 119903

1+ 1199021199032) gt 0

119863 = ℎ ([119886119887 + 1199081199032+ 1199031(119908 + 120583)] (120583 + 120583119905)

+ 120583119898 (1199031+ 119886)) (1 minus 119877

0)

(A4)

B Summary for Possible Orderings for theDifferent Transmission Rate Parameters

Table 5 shows different orderings for the parameters 1205731198770 120573119861

and 120573119862and their implications concerning the possible equi-

librium or stationary points of system (1) This table iscomplemented with the analysis given in Section 312

Acknowledgments

This work was supported by Concurso Nacional de Proyectosde Investigacion en Salud FONIS projectCode SA11I2073The authors wish to thank Claudia Gonzalez MacarenaHirmas Patricia Gonzalez Juan Carlos Hormazabal IrisDelgado and the students Matias Meyer and Ilani Kauffmanfor their valuable collaboration to the research

References

[1] C Castillo-Chavez andB Song ldquoDynamicalmodels of tubercu-losis and applicationsrdquoMathematical Biosciences and Engineer-ing vol 1 pp 361ndash404 2004

[2] C Ozcaglar A Shabbeer S L Vandenberg B Yener andK P Bennett ldquoEpidemiological models of Mycobacteriumtuberculosis complex infectionsrdquoMathematical Biosciences vol236 pp 77ndash96 2012

[3] J Raffalli K A Sepkowitz and D Armstrong ldquoCommunity-based outbreaks of tuberculosisrdquo Archives of Internal Medicinevol 156 no 10 pp 1053ndash1060 1996

[4] S Basu D Stuckle and M McKee ldquoAddressing institu-tional amplifiers in the dynamics and control of tuberculosisepidemicsrdquo The American Journal of Tropical Medicine andHygiene vol 84 no 1 pp 30ndash37 2011

[5] C Y Chiang and L W Riley ldquoExogenous reinfection intuberculosisrdquo The Lancet Infectious Diseases vol 5 no 10 pp629ndash636 2005

[6] I Baussano B G Williams P Nunn M Beggiato U Fedeliand F Scano ldquoTuberculosis incidence in prisons a systematicreviewrdquo PLoS Medicine vol 7 no 12 Article ID e1000381 2010

[7] FONIS Project Code SA11I2073 Ongoing research in deter-minants of TB transmission in the prison population and itsimpact as a reservoir for the general population of ChileCEPS Facultad deMedicina Clinica Alemana Universidad delDesarrollo

[8] Z W Jia G Y Tang Z Jin et al ldquoModeling the impact ofimmigration on the epidemiology of tuberculosisrdquo TheoreticalPopulation Biology vol 73 no 3 pp 437ndash448 2008

[9] Y Zhou K Khan Z Feng and J Wu ldquoProjection of tubercu-losis incidence with increasing immigration trendsrdquo Journal ofTheoretical Biology vol 254 no 2 pp 215ndash228 2008

[10] W W Stead ldquoPathogenesis of a first episode of chronic pul-monary tuberculosis in man recrudescence of residuals of theprimary infection or exogenous reinfectionrdquo American Reviewof Respiratory Disease vol 95 no 5 pp 729ndash745 1967

[11] R Sahadevan S Narayanan C N Paramasivan R Prabhakarand P R Narayanan ldquoRestriction fragment length polymor-phism typing of clinical isolates of Mycobacterium tuberculosisfrom patients with pulmonary tuberculosis inMadras India byuse of direct-repeat proberdquo Journal of Clinical Microbiology vol33 no 11 pp 3037ndash3039 1995

[12] L K Fitzpatrick A Okwera R Mugerwa R Ridzon J Ellnerand I Onorato ldquoAn investigation of suspected exogenousreinfection in tuberculosis patients in Kampala Ugandardquo Inter-national Journal of Tuberculosis and Lung Disease vol 6 no 6pp 550ndash552 2002

[13] A Kruuner L Pehme S Ghebremichael T Koivula S EHoffner andMMikelsaar ldquoUse of molecular techniques to dis-tinguish between treatment failure and exogenous reinfectionwith Mycobacterium tuberculosisrdquo Clinical Infectious Diseasesvol 35 no 2 pp 146ndash155 2002

[14] P D van Helden ldquoMolecular epidemiology of TB challengingdogmas and asking new questionsrdquo IUBMB Life vol 53 no 4-5pp 219ndash223 2002

[15] P E M Fine and P M Small ldquoExogenous reinfection intuberculosisrdquoTheNew England Journal ofMedicine vol 341 no16 pp 1226ndash1227 1999

[16] K Styblo ldquoEpidemiology of tuberculosisrdquo Selected Papers 24Royal Netherlands Tuberculosis Association The Hague TheNetherlands 1991

[17] P G Smith and A R Moss ldquoEpidemiology of tuberculosisrdquo inTuberculosis Pathogenesis Protection and Control B R BloomEd ASM Press Washington DC USA 1994

[18] J A Romeyn ldquoExogenous reinfection in tuberculosisrdquo Ameri-can Review of Respiratory Disease vol 101 no 6 pp 923ndash9271970

[19] A van Rie R Warren M Richardson et al ldquoExogenous rein-fection as a cause of recurrent tuberculosis after curativetreatmentrdquo The New England Journal of Medicine vol 341 no16 pp 1174ndash1179 1999

[20] A de Boer R van Soolingen and M Borgdorff ldquoRecurrenttuberculosis due to exogenous reinfectionrdquo The New EnglandJournal of Medicine vol 342 no 14 pp 1050ndash1051 2000

[21] S Verver R M Warren N Beyers et al ldquoRate of reinfectiontuberculosis after successful treatment is higher than rate of new


tuberculosisrdquo American Journal of Respiratory and Critical CareMedicine vol 171 no 12 pp 1430ndash1435 2005

[22] M Henao-Tamayo A Obregn-Henao D J Ordway S ShangC G Duncan and I M Orme ldquoAmouse model of tuberculosisreinfectionrdquo Tuberculosis vol 92 no 3 pp 211ndash217 2012

[23] S M Blower A R McLean T C Porco et al ldquoThe intrin-sic transmission dynamics of tuberculosis epidemicsrdquo NatureMedicine vol 1 no 8 pp 815ndash821 1995

[24] C M Liao Y H Cheng Y J Lin et al ldquoA probabilistic trans-mission and population dynamic model to assess tuberculosisinfection riskrdquo Risk Analysis vol 32 no 8 pp 1420ndash1432 2012

[25] J P Aparicio and C Castillo-Chavez ldquoMathematical modellingof tuberculosis epidemicsrdquoMathematical Biosciences and Engi-neering vol 6 no 2 pp 209ndash237 2009

[26] Z Feng C Castillo-Chavez and A F Capurro ldquoA model fortuberculosis with exogenous reinfectionrdquo Theoretical Popula-tion Biology vol 57 no 3 pp 235ndash247 2000

[27] E A Nardell J Keegan S A Cheney and S C Etkind ldquoAir-borne Infection theoretical limits of protection achievable bybuilding ventilationrdquo American Review of Respiratory Diseasevol 144 no 2 pp 302ndash306 1991

[28] T Baxter ldquoLow infectivity of tuberculosisrdquoThe Lancet vol 342no 8867 article 371 1993

[29] D S Barnes The Making of a Social Disease Tuberculosis inthe Nineteenth-Century France University of California PressBerkeley Calif USA 1995

[30] WHO TB fact sheets 2012 httpwwwwhointmediacentrefactsheetsfs104en

[31] J R Andrews F Noubary R P Walensky R Cerda E LosinaandC R Horsburgh ldquoRisk of progression to active tuberculosisfollowing reinfection withMycobacterium tuberculosisrdquoClinicalInfectious Diseases vol 54 no 6 pp 784ndash791 2012

[32] M L Lambert E Hasker A van Deun D Roberfroid MBoelaert and P van der Stuyft ldquoRecurrence in tuberculosisrelapse or reinfectionrdquoTheLancet Infectious Diseases vol 3 no5 pp 282ndash287 2003

[33] M A Sanchez and S M Blower ldquoUncertainty and sensitivityanalysis of the basic reproductive rate tuberculosis as anexamplerdquoAmerican Journal of Epidemiology vol 145 no 12 pp1127ndash1137 1997

[34] J Legrand A Sanchez F Le Pont L Camacho and B LarouzeldquoModeling the impact of tuberculosis control strategies inhighly endemic overcrowded prisonsrdquo PLoS ONE vol 3 no 5Article ID e2100 2008

[35] O Diekmann J A Heesterbeek and J A Metz ldquoOn thedefinition and the computation of the basic reproductionratio R

0in models for infectious diseases in heterogeneous

populationsrdquo Journal of Mathematical Biology vol 35 pp 503ndash522 1990

[36] J M Heffernan R J Smith and L M Wahl ldquoPerspectiveson the basic reproductive ratiordquo Journal of the Royal SocietyInterface vol 2 pp 281ndash293 2005

[37] J Dushoff W Huang and C Castillo-Chavez ldquoBackwardsbifurcations and catastrophe in simple models of fatal diseasesrdquoJournal of Mathematical Biology vol 36 no 3 pp 227ndash2481998

[38] M G M Gomes A O Franco M C Gomes and G F MedleyldquoThe reinfection threshold promotes variability in tuberculosisepidemiology and vaccine efficacyrdquo Proceedings of the RoyalSociety B vol 271 no 1539 pp 617ndash623 2004

Submit your manuscripts athttpwwwhindawicom

Stem CellsInternational

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014


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of


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EndocrinologyInternational Journal of



Disease Markers


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The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

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Parkinsonrsquos Disease

Evidence-Based Complementary and Alternative Medicine

Volume 2014Hindawi Publishing Corporationhttpwwwhindawicom


the study of these types of communities is essential in thecontext of TB spread This phenomenon has been called theReservoir Effect [6 7] Indeed semiclosed communities suchas prisons represent a reservoir for disease transmission to thepopulation at large and should be a source of public concernFor example TB infection may spread into the generalpopulation through prison staff visitors and close contactsof released prisoners The transmission dynamics betweenprisoners and the general population [6] together withimmigration from developing countries with high prevalenceof TB [8 9] has been hypothesized to play a key role indriving overall population-level TB incidence prevalenceand mortality rates

In a recent work [4] the authors have even gone furtherin relation to this effect and have named these communitiesInstitutional Amplifiers of TB Propagation Some examples ofcommunities given by these authors are poor hospitals inwhich dozens of patients share poorly ventilated communalrooms crowded prison cell blocks and mining barracksamong others

The transmission and progression of TB infection hasbeen relatively well understood on a population scale Gener-ally it is assumed that once an individual is infected with TBhe or she is immune from further infection events Moreoverit was proposedwhat came to be known as the unitary conceptof pathogenesis [10] which states that TB always begins withprimary infection and subsequent episodes of active TB aredue to reactivation of dormant bacilli from this primaryinfection However a persistent evidence has recently beenshown (see [5] for a review) that the paths to TB infectionare not as linear as was suggested by the unitary conceptof pathogenesis The availability of individual strain-specificinfection histories (see eg [11ndash13]) has made it clear thatexogenous reinfection in people with previously documentedTB infection does occur The important question is whetherreinfection occurs commonly enough to have an effect on theoverall infection dynamics of the population [14]The relativeimportance of these pathways to the development of activedisease has significant implications for treatment and controlstrategies most notably in deciding whether latently infectedand treated individuals are at risk of reinfection [15]

Several authors [15ndash20] have declared that exogenousreinfection plays an important role in the disease progressionand that the inhalation of tubercle bacilli by persons whohave had a primary TB infection previously for more thanfive years represents an increasing risk to develop activeTB soon after reinfection A study from South Africa [21]has demonstrated that the rate of reinfection by TB aftersuccessful treatment could be higher than the rate of new TBinfections In this study the reinfection rate after successfultreatment was estimated at 22 per 100 person-years whichwas approximately seven times the crude incidence rate(313 per 100 000 population per year) and approximatelyfour times the age-adjusted incidence rate of new TB (515per 100 000 population per year) So ignoring exogenousreinfection when modeling TB spread in high-incidenceand high-prevalence community setting such as semiclosedcommunities has been seen to be inappropriate (Henao-Tamayo et al in [22] recently published a mouse model of TB

reinfection that could help to explain immunological aspectsof reinfection risk in high-incidence areas)

We will use an SEIR standard compartmental modelsee for example the works by Blower et al [23] and morerecently by Liao et al [24] with somemodifications explainedbellow that turn out to be quite useful in the study of theparticularities of TB spread at this type of communitiesThis model assumes that the population in the communityis homogeneous that it does not consider the heterogeneitiesin the social structure between community members and itis based on the so-calledmass action or fully mixing approxi-mation This means that individuals with whom a susceptibleindividual has contact are chosen at random from the wholecommunity It is also assumed that all individuals haveapproximately the same number of contacts in the sametime and that all contacts transmit the disease with the sameprobability

The model we use in this work takes into account thefollowing relevant facts in the context of semiclosed commu-nities

(1) The overcrowding in the community can increase(compared to what occurs in the population atlarge) the likelihood of exogenous reinfection dueto repeated contacts with active infected individualsThat is besides primary infection themodel considersthe possible reinfection of individuals with LTB (indi-viduals who are assumed to be asymptomatic andnoninfectious but capable of progressing to active TB)and recovered individuals (individuals who have beentreated for TB in the past and been declared cured)If latently infected or recovered individuals remain inthe community they could be infected again

(2) At present it is not completely clear whether inall cases previous infections with MycobacteriumTB with or without subsequent recovery offer someprotection that could be translated into a reducedsusceptibility to reinfection [5 21 22 25] So we willbe open at exploring different situations with regardto this fact in the model

(3) Poor nutrition immunodepression and other dis-eases increase the likelihood of accelerated progres-sion to active TB

We will see that considering exogenous reinfection todescribe TB spread produces a richer and more complexdynamics than the one observed in previous models (see eg[23 25 26]) In particular unlike the model published byFeng et al in [26] which uses a single parameter for exoge-nous reinfection our model uses two parameters related totwo possible reinfections (reinfection of latently infected andreinfection of recovered individuals)

2 Basic Epidemiology of TB Sources andProbability of Infection in SemiclosedCommunities

The risk of infection with Mycobacterium tuberculosis thebacterium causing TB depends mainly on two factors first




















119868 and 119868






119901119891120573119878119868119868

119902120575120573119864119868119868

120583119879119868119868 120583119868119868

120583119864120583119878 119902119864119888119868119868

1199032119868119868

120583119877120503 (1 minus 119901)120573119878119868

119877119868119864119878

120575(1 minus 119902)120573119864119868119868

120578120573119877119868119868

(1 minus 119902)119864

(1 minus 119891)119901120573119878119868119868

1199031119868119873

119888119868119873

120583119879119868119873 120583119868119873

119868119868

119868119873

119908119877

119908119877




1and 119903











119868and 120575(1 minus 119902)120573119864119868

119868


119868119877








1198681year


1198731year


119889119878

119889119905= Π minus 120573119878119868

119868minus 120583119878

119889119864

119889119905= (1 minus 119901) 120573119878119868

119868+ 120578120573119877119868

119868minus (] + 120583) 119864 minus 120575120573119864119868

119868

119889119868119868

119889119905= 119891119901120573119878119868

119868+ 119902]119864 + 119908119877 minus (120583 + 120583

119879+ 119888 + 119903

1) 119868119868+ 120575119902120573119864119868

119868

119889119868119873

119889119905= (1 minus 119891) 119901120573119878119868

119868+ (1 minus 119902) ]119864

+ 119908119877 minus (120583 + 120583119879+ 119888 + 119903

2) 119868119873+ 120575 (1 minus 119902) 120573119868

119868119864

119889119877

119889119905= 119888 (119868119868+ 119868119873) minus (2119908 + 120583) 119877 minus 120578120573119877119868

119868+ 1199031119868119868+ 1199032119868119873

(1)


119889119873

119889119905= minus120583119873 minus 120583

119879(119868119868+ 119868119873) + Π (2)



119863 = (119878 119864 119868119868 119868119873 119877) isin R

5

+ 119878 + 119864 + 119868

119868+ 119868119873+ 119877 le

Π

120583 (3)




0gt 1 an epidemic

will arise



1198770= 120573Π ((ℎ119891119901 + (1 minus 119901) ]119902) (119886119887 minus 119898119908)



(4)

where

119886 = 120583 + 120583119879+ 119888

119887 = 2119908 + 120583

ℎ = ] + 120583

119892 = 1199031+ 119888

119898 = 1199032+ 119888

(5)


0 = Π minus 120573119878119868119868minus 120583119878

0 = (1 minus 119901) 120573119878119868119868+ 120578120573119877119868

119868minus (] + 120583) 119864 minus 120575120573119864119868

119868

0 = 119891119901120573119878119868119868+ 119902]119864 + 119908119877 minus (120583 + 120583

119879+ 119888 + 119903

1) 119868119868+ 120575119902120573119864119868

119868

0 = (1 minus 119891) 119901120573119878119868119868+ (1 minus 119902) ]119864 + 119908119877

minus (120583 + 120583119879+ 119888 + 119903

2) 119868119873+ 120575 (1 minus 119902) 120573119868

119868119864

0 = 119888 (119868119868+ 119868119873) minus (2 119908 + 120583) 119877 minus 120578120573119877119868

119868+ 1199031119868119868+ 1199032119868119873

(6)


equation

119868119868(1198601198683

119868+ 1198611198682

119868+ 119862119868119868+ 119863) = 0 (7)




1198750= (1198780 1198640 1198681198680 1198681198730

1198770) = (

Π

120583 0 0 0 0) (8)





((Π120583) 0 0 0 0) is

119869 =

[[[[[[[[

[


1205830 0


1205830 0

0 119902]119891119901120573Π


0 (1 minus 119902) ](1 minus 119891) 119901120573Π


0 0 1199031 + 119888 1199032 + 119888 minus2119908 minus 120583

]]]]]]]]

]

(9)


(120582 + 120583) (1205824+ 11988631205823+ 11988621205822+ 1198861120582 + 1198860) = 0 (10)


119875 (120582) = 1205824+ 11988631205823+ 11988621205822+ 1198861120582 + 1198860= 0 (11)







0of the




119894(120573) = minus119860

119894120573+

119861119894with 119894 = 0 1 2 3 and 119860

119894 119861119894gt 0 We can define 120573

119894= 119861119894119860119894


119894(120573) gt 0

For example

1198863(120573) = minus

119891119901120573 Π

120583+ ] + 3120583 + 120583

119879+ 119888 + 119903

2+ 2119908

1205733=

(] + 3120583 + 120583119879+ 119888 + 119903

2+ 2119908) 120583

119891119901Π

(12)



1198862(1205733) lt 0 997904rArr 120573

2lt 1205733

1198861(1205732) lt 0 997904rArr 120573

1lt 1205732

1198860(1205731) lt 0 997904rArr 120573

0lt 1205731

(13)


0 all the




0of


1198860(120573) = minus119860

0120573 + 1198610= 1198610(1 minus

1198600120573

1198610

) = 1198610(1 minus 119877

0) (14)


1198770lt 1 then 119886

0(120573) gt 0 so 120573 lt 120573

0




4



real part


1198601198683

119868+ 1198611198682

119868+ 119862119868119868+ 119863 = 0 (15)


119860 = 1205733120575120590 (120583 + 120583

119879) (120583 + 120583

119879+ 119888 + (1 minus 119902) 119903

1+ 1199021199032) gt 0 (16)

119863 = ℎ ([119886119887 + 1199081199032+ 1199031(119908 + 120583)] (120583 + 120583119905)

+ 120583119898 (1199031+ 119886)) (1 minus 119877

0)

(17)


general form

119861 = 1205732119891119861(120573)

119862 = 120573119891119862(120573)

(18)

where119891119861(120573) = minus119861

1120573 + 1198612

119891119862(120573) = minus119862

1120573 + 119862

2

(19)


and 119862119894=12



119861(120573) and 119891


functions 119891119861(120573) and 119891



119875 (119909) = 1198601199093+ 1198611199092+ 119862119909 + 119863 (20)






2+ 2119861119909+119862 and then we


(1) If Δ = 1198612minus 3119860119862 le 0 119875

1015840(119909) ge 0 for all 119909



given by

11990921


3119860

(21)


and 1198751015840(119909) ge 0 for all 119909 ge 119909

2and 119909 le 119909

1 So we need to


2in the real line


have 1199091lt 0 119909

2gt 0 and 119875

1015840(119909) gt 0 for all 119909 ge 119909

2ge 0



2are



2


1) ge 0


119861(120573119861) = 0 and 120573

119862the


1198770be the value






(Table 3)


1198770lt 120573119862lt 120573119861 that



For 1205731198770



For 120573119862



Finally for 120573 gt 120573119861





lt 120573119862lt 120573119861


119861(120573119862) gt 0 and 119863(120573

119861) gt 0


lt 120573119862lt 120573119861


lt 120573119862

lt 120573119861



1198770lt 120573119862

lt 120573119861is

fulfilled


120573 lt 1205731198770


1205731198770

lt 120573 lt 120573119862 119860 gt 0 119861 gt 0 119862 gt 0119863 lt 0


120573119862lt 120573 lt 120573


120573 gt 120573119861 119860 gt 0 119861 lt 0 119862 lt 0119863 lt 0


800

600

400

200

0

minus200

minus400

119909

1198770 gt 1

1205731198770120573119862 120573119861

00002 00004 00006 00008 0001120573


1198770lt 120573119862

lt 120573119861 1205731198770


0gt 1


1198770 that is when 119877

0= 1


lt 120573119862



1205731198770 that is119877



1198770





119861lt 120573119862

lt 1205731198770

isfulfilled Here Δ



120573 lt 120573119861 119860 gt 0 119861 gt 0 119862 gt 0119863 gt 0


120573119861lt 120573 lt 120573

119862 119860 gt 0 119861 lt 0 119862 gt 0119863 gt 0


120573119862lt 120573 lt 120573

1198770119860 gt 0 119861 lt 0 119862 lt 0119863 gt 0


1205731198770





0



0


0= 1 and


119875 (119909) = 1198601199093+ 1198611199092+ 119862119909

= 119909 (1198601199092+ 1198611199092+ 119862) = 0

(22)



1and 119909

2



means 119891119861(1205731198770) lt 0 and




1198770lt 120573119862


1205731198770






times10minus3

119875(119909)

01

005

minus005

minus01

minus200 0 200 400 600119909


119861lt 1205731198770





119861lt 120573119862

lt 1205731198770

isfulfilled



For 120573119861

lt 120573 lt 120573119862and 120573

119862lt 120573 lt 120573





1lt 0







case we have

Δ1=

1

4

1198632

1198602+

1

27

1198623

1198603gt 0 (23)




Δ1(120573lowast) = 0 (24)


600

500

400

300

200

100

0

minus100

minus200

minus300

119909

1198770 gt 1

1205731198770120573119862120573119861 120573lowast

000005 00001 000015 00002120573


lt 120573119862

lt 1205731198770




(seeFigure 4)













1205731198770










119868050 (assumed)


119873020 (assumed)






parameters 1205731198770 120573119861 and 120573



1198770 120573119861 120573119862) then the system


1198770 120573119861 120573119862) then








1198770lt 120573119862

lt 120573119861 120575 = 09 120578 = 001) Let us


lt 120573119862lt 120573119861is











were

119878 (0) = 4980 119864 (0) = 0 119868119868 (0) = 20

119868119873 (0) = 0 119877 (0) = 0

(25)


1205731198770

= 00001450317354

120573119861= 001087387065

120573119862= 00002715343808

(26)


lt 120573119862

lt 120573119861


1198770


119878infin

= 1616 119877infin

= 4080 119868119873infin

= 103

119868119868infin

= 195 119864infin

= 1150

(27)





lt 120573119862lt 120573119861 120575 = 00 120578 = 09) For our



0= 3585422172 The used


119878 (0) = 4980 119864 (0) = 0 119868119868 (0) = 20

119868119873 (0) = 0 119877 (0) = 0

(28)

119905

7000

6000

5000

4000

3000

2000

1000

00 50 100 150 200

119878

119877

119868119873119864

119868119868 + 119868119873119868119868119873

119868119868 + 119868119873 + 119864 + 119877



4000

4000 5000

3000

3000

2000

2000

1000

10004000

6000

20000

119868 119868+119868 119873+119864

119878 119877 4000 500

3000000000000000000000000000000

3000

200202000020200202002020200000000200202020020020200200202020202002002000202002000020202020200020202000220000200202000020202222000222020200220000202022022220022002202222220022222 02 02222 0000000000000000000000000000000000000000000000000000000000000000

2000

100110010010010010010010010100100100100010010010010010010010010000010100100100100100100100100110000001001000000110010110010010010010001000001110100010000010000000001010000011000001110000000001001001000000000000001000010000111000101 000000000000000000000000000000000000000000000000000000000000000000000000000000000000000

10004000

6000

20000

119868119868119868119868119868119868119868119868119868119868119868119868119868119868119868119868119868119868119868119868119868119868119868119868119868119868119868119868119868119868119868119868119868119868119868119868119868119868119868119868 119868119868119868119868119868119868119868119868119868119868119868119868119868119868++++++++++++++++++++++++++119868 119873119868+119864

119878 119877


0=

3585422172 120575 = 09 120578 = 001 and 120573 = 000052


1205731198770

= 00001450317354

120573119861= 001226355348

120573119862= 00003132229272

(29)


lt 120573119861lt 120573119862 and as in



119878infin

= 1938 119877infin

= 974 119868119873infin

= 60

119868119868infin

= 156 119864infin

= 4530

(30)


119905

7000

6000

5000

4000

3000

2000

1000

00 50 100 150 200

119878

119877

119868119873119864

119868119868 + 119868119873119868119868119873

119868119868 + 119868119873 + 119864 + 119877




1198770lt 120573119862







119861lt 120573119862

lt 1205731198770 120575 = 30 120578 = 25) There


0=


1205731198770

= 00001450317354

120573119861= 00001066568066

120573119862= 00001225687204

(31)


1205731198770


119894 119868119899 119864)

1198751= (9009 0 0 0 0)

1198752= (8507 182 9 5 2166)

1198753= (3221 1406 285 103 1566)

(32)

800

700

600

500

400

300

200

100

07006005004003002001000

119868 119868

119905



= 0



11987531198751198751198751198751198751198751198751198751198751198751198751198751198751198751198751198751198751198751198751198751198751198751198751198751198751198751198751198751198751198751198751198751198751198751198751198751198751198751198751198751198751198751198751198751198751198751198751198751198751198751198751198751198751198751198751198751198751198751198751198751198751198751198751198751198751198751198751198751198751198751198751198751198753119875119875

16001200

800400

900080007000

600050004000300020000

1000

2000

3000

119868 119868+119868 119873+119864

1198781198771198752

1198751




3



119868infin= 0 or to 119868

119868infin=



3according




119868+ 119868119873+ 119864


lowast=


lowast the system


119861lt 1205731198770

lt 120573119862 120575 = 30 120578 = 25) Consider


119861lt 1205731198770



1205731198770

= 00001679568390

120573119862= 00001729256777

120573119861= 00001489092005

(33)










00001688612368 with 1205731198770



condition 1205731lt 120573 lt 120573

1198770




(119878 119877 119868119894 119868119899 119864)

1198750= (5148 0 0 0 0)

1198751= (3372 1041 122 60 482)

1198752= (2828 1283 190 88 651)

(34)



2is


119868infin= 0 or to 119868




2






lt 1205731198770


1205731198770

1205731

1205732

120573119862120573119861

000015 000016 000017 000018120573

600

500

400

300

200

100

0

119909


119861lt 1205731198770



2

500

400

300

200

100

0

119868 119868

0 500 1000 1500 2000119905



= 0

or 119868119868infin



(119878 119877 119868119894 119868119899 119864)

1198750= (5148 0 0 0 0)

1198751= (5042 76 5 3 20)

1198752= (3971 734 69 36 298)

1198753= (2491 1413 246 109 750)

(35)


160014001200800600400200

500

2000

0

1000

1500

1000

2000

3000

4000

5000

119878119877

11987521198751

1198750 160014001200800600444400420000002000200

5005005000

000

1000

1500

1000

20020020000000

300003003003003000000000000

400404004004004000040040040000000 119878119877

1198752119875119875

1198751119875119875

119868 119868+119868 119873+119877





2







gt








lowast= 120583 = 003885 120583

119905= 001520 119901 = 08

] = 00266 119891 = 08 119902 = 085 119908 = 0005 119888 = 04 1199031= 05





160014001200

800600400200

500

2000

0

1000

1500

1000

2000

3000

4000

5000

119868 119868+119868 119873+

119878119877

119877

1198752

1198751

1198753

1198750 160014001200

8006004004040200200200200200200

500

000

1000

1500

1000

2002002002002 000000

3000030033000000000000

4004004004004000040000400 00000

119868 119868119868+119868 119873119868+

119878119877

119877

1198752119875119875

1198751119875119875

1198753119875119875

0




0and







0is near one













5

5

4

4

3

3

2

2

1

10

0

120578

119861 gt 0 119862 gt 0

120575

119861 lt 0 119862 lt 0

119861 gt 0 119862 lt 0

(a) 1198770 = 1

5

5

4

4

3

3

2

2

1

10

0

120578

120575

(b) 1198770 = 1689504264

5

5

4

4

3

3

2

2

1

10

0

120578

120575

(c) 1198770 = 2379008527

5

5

4

4

3

3

2

2

1

10

0

120578

120575

(d) 1198770 = 7895042636



1198770= 00001450317354 (b) 120573 = 00002450317354 (c) 120573 = 00003450317354 and (d)

120573 = 0001145031735


1198770 120573119861 120573119862 and 120573 We divided



0has to do solely






5

5

4

4

3

3

2

2

1

10

0

120578

120575

Δ gt 0

119891(1199091) gt 0

119861 gt 0 119862 lt 0

119861 lt 0 119862 lt 0

119861 lt 0 119862 gt 0

119861 lt 0119862 gt 0

119861 gt 0 119862 gt 0

(a) 1198770 = 1

5

5

4

4

3

3

2

2

1

10

0

120578

120575

(b) 1198770 = 1004390340

5

5

4

4

3

3

2

2

1

10

0

120578

120575

(c) 1198770 = 1087806817

5

5

4

4

3

3

2

2

1

10

0

120578

120575

(d) 1198770 = 2317102260



1198770= 00002277727471 (b) 120573 = 00002287727471 (c)

120573 = 00002477727471 and (d) 120573 = 00005277727471


0gt 1






(1205731198770

lt 120573119862






0 the


0 Some



5

5

4

4

3

3

2

2

1

10

0

120578

120575

Δ gt 0

119891(1199091) gt 0

119861 gt 0 119862 lt 0

119861 lt 0 119862 lt 0

119861 lt 0 119862 gt 0

119861 lt 0119862 gt 0

119861 gt 0 119862 gt 0

(a) 1198770 = 09560965911

5

5

4

4

3

3

2

2

1

10

0

120578

120575

(b) 1198770 = 08902414781

5

5

4

4

3

3

2

2

1

10

0

120578

120575

(c) 1198770 = 07804829561

5

5

4

4

3

3

2

2

1

10

0

120578

120575

(d) 1198770 = 05609659131



00001777727471 and (d) 120573 = 00001277727471






0




0such that disease-











lt 120573119862lt 120573119861

120573 lt 1205731198770


lt 120573 lt 120573119862




120573119862lt 1205731198770

lt 120573119861

120573 lt 120573119862



2) le 0 or Δ

1lt 0 none if 119875(119909

2) ge 0 or Δ

1gt 0

1205731198770

lt 120573 lt 120573119861


1gt 0


120573119862lt 120573119861lt 1205731198770

120573 lt 120573119862



2) le 0 or Δ

1lt 0 none if 119875(119909

2) ge 0 or Δ

1gt 0

120573119861lt 120573 lt 120573


1lt 0) or none (Δ

1gt 0)

1205731198770


lt 120573119861lt 120573119862

120573 lt 1205731198770


lt 120573 lt 120573119861




1lt 0)


120573119861lt 1205731198770

lt 120573119862

120573 lt 120573119861



1lt 0) or none (Δ

1gt 0)

1205731198770

lt 120573 lt 120573119862


1lt 0)


120573119861lt 120573119862lt 1205731198770

120573 lt 120573119861



1lt 0) or none (Δ

1gt 0)

120573119862lt 120573 lt 120573


1lt 0) or none (Δ

1gt 0)

1205731198770



Appendices



119886 = 120583 + 120583119879+ 119888

119887 = 2119908 + 120583

ℎ = ] + 120583

119892 = 1199031+ 119888

119898 = 1199032+ 119888

120579 = 120583 + 120583119905

120598 = 1 minus 119901

1198711= 120579 + 119892 (1 minus 119902) + 119898119902

1198712= ]1199021199032+ ℎ119886 + 119903

1] (1 minus 119902) + 119903

1120583

1198701= 1199032119908 + 119887119886 + 119903

1(119908 + 120583)

1198702= ]1199021199032+ ℎ119886 + 119903

1] (1 minus 119902) + 119903

1120583

(A1)

We have

1198611= (120579 (119891119901 + (1 minus 119901) 119902) + 119898119902)Π120575120590

1198612= 120590 (120583119871

1120579 120575 + 119871

2120579 + 119898120583 (119886 + 119903

1))

+ (1199032119908120579 + 119903

1(119908 + 120583) 120579

+ 119887119886120579 + 119898120583 (119886 + 1199031)) 120575


1198621= Π (120575 ((119891119901 + 120598119902) (119887120583119905 + (119898 + 119887) 120583) + 119898119908)

+ 120590 ((]119902120598 + 119891ℎ119901) 120579 + 119898 (120583119891119901 + ] 119902)))

1198622= 120583 (119870

1120579 + 119898120583 (119886 + 119903

1)) 120575 + 120583 (119870

2120579 + 119898120583 (119886 + 119903

1)) 120590

+ ℎ (1198701120579 + 119898120583 (119886 + 119903

1))

(A2)


119861 = 1205732119891119861(120573)

119862 = 120573119891119862(120573)

(A3)

where119891119861(120573) = minus119861

1120573 + 1198612

119891119862(120573) = minus119862

1120573 + 119862

2

119860 = 1205733120575120590 (120583 + 120583

119879) (120583 + 120583

119879+ 119888 + (1 minus 119902) 119903

1+ 1199021199032) gt 0

119863 = ℎ ([119886119887 + 1199081199032+ 1199031(119908 + 120583)] (120583 + 120583119905)

+ 120583119898 (1199031+ 119886)) (1 minus 119877

0)

(A4)





Acknowledgments


References
















































of






Disease Markers



OncologyJournal of





PPAR Research



Journal of

ObesityJournal of



































119868 and 119868






119901119891120573119878119868119868

119902120575120573119864119868119868

120583119879119868119868 120583119868119868

120583119864120583119878 119902119864119888119868119868

1199032119868119868

120583119877120503 (1 minus 119901)120573119878119868

119877119868119864119878

120575(1 minus 119902)120573119864119868119868

120578120573119877119868119868

(1 minus 119902)119864

(1 minus 119891)119901120573119878119868119868

1199031119868119873

119888119868119873

120583119879119868119873 120583119868119873

119868119868

119868119873

119908119877

119908119877




1and 119903











119868and 120575(1 minus 119902)120573119864119868

119868


119868119877








1198681year


1198731year


119889119878

119889119905= Π minus 120573119878119868

119868minus 120583119878

119889119864

119889119905= (1 minus 119901) 120573119878119868

119868+ 120578120573119877119868

119868minus (] + 120583) 119864 minus 120575120573119864119868

119868

119889119868119868

119889119905= 119891119901120573119878119868

119868+ 119902]119864 + 119908119877 minus (120583 + 120583

119879+ 119888 + 119903

1) 119868119868+ 120575119902120573119864119868

119868

119889119868119873

119889119905= (1 minus 119891) 119901120573119878119868

119868+ (1 minus 119902) ]119864

+ 119908119877 minus (120583 + 120583119879+ 119888 + 119903

2) 119868119873+ 120575 (1 minus 119902) 120573119868

119868119864

119889119877

119889119905= 119888 (119868119868+ 119868119873) minus (2119908 + 120583) 119877 minus 120578120573119877119868

119868+ 1199031119868119868+ 1199032119868119873

(1)


119889119873

119889119905= minus120583119873 minus 120583

119879(119868119868+ 119868119873) + Π (2)



119863 = (119878 119864 119868119868 119868119873 119877) isin R

5

+ 119878 + 119864 + 119868

119868+ 119868119873+ 119877 le

Π

120583 (3)




0gt 1 an epidemic

will arise



1198770= 120573Π ((ℎ119891119901 + (1 minus 119901) ]119902) (119886119887 minus 119898119908)



(4)

where

119886 = 120583 + 120583119879+ 119888

119887 = 2119908 + 120583

ℎ = ] + 120583

119892 = 1199031+ 119888

119898 = 1199032+ 119888

(5)


0 = Π minus 120573119878119868119868minus 120583119878

0 = (1 minus 119901) 120573119878119868119868+ 120578120573119877119868

119868minus (] + 120583) 119864 minus 120575120573119864119868

119868

0 = 119891119901120573119878119868119868+ 119902]119864 + 119908119877 minus (120583 + 120583

119879+ 119888 + 119903

1) 119868119868+ 120575119902120573119864119868

119868

0 = (1 minus 119891) 119901120573119878119868119868+ (1 minus 119902) ]119864 + 119908119877

minus (120583 + 120583119879+ 119888 + 119903

2) 119868119873+ 120575 (1 minus 119902) 120573119868

119868119864

0 = 119888 (119868119868+ 119868119873) minus (2 119908 + 120583) 119877 minus 120578120573119877119868

119868+ 1199031119868119868+ 1199032119868119873

(6)


equation

119868119868(1198601198683

119868+ 1198611198682

119868+ 119862119868119868+ 119863) = 0 (7)




1198750= (1198780 1198640 1198681198680 1198681198730

1198770) = (

Π

120583 0 0 0 0) (8)





((Π120583) 0 0 0 0) is

119869 =

[[[[[[[[

[


1205830 0


1205830 0

0 119902]119891119901120573Π


0 (1 minus 119902) ](1 minus 119891) 119901120573Π


0 0 1199031 + 119888 1199032 + 119888 minus2119908 minus 120583

]]]]]]]]

]

(9)


(120582 + 120583) (1205824+ 11988631205823+ 11988621205822+ 1198861120582 + 1198860) = 0 (10)


119875 (120582) = 1205824+ 11988631205823+ 11988621205822+ 1198861120582 + 1198860= 0 (11)







0of the




119894(120573) = minus119860

119894120573+

119861119894with 119894 = 0 1 2 3 and 119860

119894 119861119894gt 0 We can define 120573

119894= 119861119894119860119894


119894(120573) gt 0

For example

1198863(120573) = minus

119891119901120573 Π

120583+ ] + 3120583 + 120583

119879+ 119888 + 119903

2+ 2119908

1205733=

(] + 3120583 + 120583119879+ 119888 + 119903

2+ 2119908) 120583

119891119901Π

(12)



1198862(1205733) lt 0 997904rArr 120573

2lt 1205733

1198861(1205732) lt 0 997904rArr 120573

1lt 1205732

1198860(1205731) lt 0 997904rArr 120573

0lt 1205731

(13)


0 all the




0of


1198860(120573) = minus119860

0120573 + 1198610= 1198610(1 minus

1198600120573

1198610

) = 1198610(1 minus 119877

0) (14)


1198770lt 1 then 119886

0(120573) gt 0 so 120573 lt 120573

0




4



real part


1198601198683

119868+ 1198611198682

119868+ 119862119868119868+ 119863 = 0 (15)


119860 = 1205733120575120590 (120583 + 120583

119879) (120583 + 120583

119879+ 119888 + (1 minus 119902) 119903

1+ 1199021199032) gt 0 (16)

119863 = ℎ ([119886119887 + 1199081199032+ 1199031(119908 + 120583)] (120583 + 120583119905)

+ 120583119898 (1199031+ 119886)) (1 minus 119877

0)

(17)


general form

119861 = 1205732119891119861(120573)

119862 = 120573119891119862(120573)

(18)

where119891119861(120573) = minus119861

1120573 + 1198612

119891119862(120573) = minus119862

1120573 + 119862

2

(19)


and 119862119894=12



119861(120573) and 119891


functions 119891119861(120573) and 119891



119875 (119909) = 1198601199093+ 1198611199092+ 119862119909 + 119863 (20)






2+ 2119861119909+119862 and then we


(1) If Δ = 1198612minus 3119860119862 le 0 119875

1015840(119909) ge 0 for all 119909



given by

11990921


3119860

(21)


and 1198751015840(119909) ge 0 for all 119909 ge 119909

2and 119909 le 119909

1 So we need to


2in the real line


have 1199091lt 0 119909

2gt 0 and 119875

1015840(119909) gt 0 for all 119909 ge 119909

2ge 0



2are



2


1) ge 0


119861(120573119861) = 0 and 120573

119862the


1198770be the value






(Table 3)


1198770lt 120573119862lt 120573119861 that



For 1205731198770



For 120573119862



Finally for 120573 gt 120573119861





lt 120573119862lt 120573119861


119861(120573119862) gt 0 and 119863(120573

119861) gt 0


lt 120573119862lt 120573119861


lt 120573119862

lt 120573119861



1198770lt 120573119862

lt 120573119861is

fulfilled


120573 lt 1205731198770


1205731198770

lt 120573 lt 120573119862 119860 gt 0 119861 gt 0 119862 gt 0119863 lt 0


120573119862lt 120573 lt 120573


120573 gt 120573119861 119860 gt 0 119861 lt 0 119862 lt 0119863 lt 0


800

600

400

200

0

minus200

minus400

119909

1198770 gt 1

1205731198770120573119862 120573119861

00002 00004 00006 00008 0001120573


1198770lt 120573119862

lt 120573119861 1205731198770


0gt 1


1198770 that is when 119877

0= 1


lt 120573119862



1205731198770 that is119877



1198770





119861lt 120573119862

lt 1205731198770

isfulfilled Here Δ



120573 lt 120573119861 119860 gt 0 119861 gt 0 119862 gt 0119863 gt 0


120573119861lt 120573 lt 120573

119862 119860 gt 0 119861 lt 0 119862 gt 0119863 gt 0


120573119862lt 120573 lt 120573

1198770119860 gt 0 119861 lt 0 119862 lt 0119863 gt 0


1205731198770





0



0


0= 1 and


119875 (119909) = 1198601199093+ 1198611199092+ 119862119909

= 119909 (1198601199092+ 1198611199092+ 119862) = 0

(22)



1and 119909

2



means 119891119861(1205731198770) lt 0 and




1198770lt 120573119862


1205731198770






times10minus3

119875(119909)

01

005

minus005

minus01

minus200 0 200 400 600119909


119861lt 1205731198770





119861lt 120573119862

lt 1205731198770

isfulfilled



For 120573119861

lt 120573 lt 120573119862and 120573

119862lt 120573 lt 120573





1lt 0







case we have

Δ1=

1

4

1198632

1198602+

1

27

1198623

1198603gt 0 (23)




Δ1(120573lowast) = 0 (24)


600

500

400

300

200

100

0

minus100

minus200

minus300

119909

1198770 gt 1

1205731198770120573119862120573119861 120573lowast

000005 00001 000015 00002120573


lt 120573119862

lt 1205731198770




(seeFigure 4)













1205731198770










119868050 (assumed)


119873020 (assumed)






parameters 1205731198770 120573119861 and 120573



1198770 120573119861 120573119862) then the system


1198770 120573119861 120573119862) then








1198770lt 120573119862

lt 120573119861 120575 = 09 120578 = 001) Let us


lt 120573119862lt 120573119861is











were

119878 (0) = 4980 119864 (0) = 0 119868119868 (0) = 20

119868119873 (0) = 0 119877 (0) = 0

(25)


1205731198770

= 00001450317354

120573119861= 001087387065

120573119862= 00002715343808

(26)


lt 120573119862

lt 120573119861


1198770


119878infin

= 1616 119877infin

= 4080 119868119873infin

= 103

119868119868infin

= 195 119864infin

= 1150

(27)





lt 120573119862lt 120573119861 120575 = 00 120578 = 09) For our



0= 3585422172 The used


119878 (0) = 4980 119864 (0) = 0 119868119868 (0) = 20

119868119873 (0) = 0 119877 (0) = 0

(28)

119905

7000

6000

5000

4000

3000

2000

1000

00 50 100 150 200

119878

119877

119868119873119864

119868119868 + 119868119873119868119868119873

119868119868 + 119868119873 + 119864 + 119877



4000

4000 5000

3000

3000

2000

2000

1000

10004000

6000

20000

119868 119868+119868 119873+119864

119878 119877 4000 500

3000000000000000000000000000000

3000

200202000020200202002020200000000200202020020020200200202020202002002000202002000020202020200020202000220000200202000020202222000222020200220000202022022220022002202222220022222 02 02222 0000000000000000000000000000000000000000000000000000000000000000

2000

100110010010010010010010010100100100100010010010010010010010010000010100100100100100100100100110000001001000000110010110010010010010001000001110100010000010000000001010000011000001110000000001001001000000000000001000010000111000101 000000000000000000000000000000000000000000000000000000000000000000000000000000000000000

10004000

6000

20000

119868119868119868119868119868119868119868119868119868119868119868119868119868119868119868119868119868119868119868119868119868119868119868119868119868119868119868119868119868119868119868119868119868119868119868119868119868119868119868119868 119868119868119868119868119868119868119868119868119868119868119868119868119868119868++++++++++++++++++++++++++119868 119873119868+119864

119878 119877


0=

3585422172 120575 = 09 120578 = 001 and 120573 = 000052


1205731198770

= 00001450317354

120573119861= 001226355348

120573119862= 00003132229272

(29)


lt 120573119861lt 120573119862 and as in



119878infin

= 1938 119877infin

= 974 119868119873infin

= 60

119868119868infin

= 156 119864infin

= 4530

(30)


119905

7000

6000

5000

4000

3000

2000

1000

00 50 100 150 200

119878

119877

119868119873119864

119868119868 + 119868119873119868119868119873

119868119868 + 119868119873 + 119864 + 119877




1198770lt 120573119862







119861lt 120573119862

lt 1205731198770 120575 = 30 120578 = 25) There


0=


1205731198770

= 00001450317354

120573119861= 00001066568066

120573119862= 00001225687204

(31)


1205731198770


119894 119868119899 119864)

1198751= (9009 0 0 0 0)

1198752= (8507 182 9 5 2166)

1198753= (3221 1406 285 103 1566)

(32)

800

700

600

500

400

300

200

100

07006005004003002001000

119868 119868

119905



= 0



11987531198751198751198751198751198751198751198751198751198751198751198751198751198751198751198751198751198751198751198751198751198751198751198751198751198751198751198751198751198751198751198751198751198751198751198751198751198751198751198751198751198751198751198751198751198751198751198751198751198751198751198751198751198751198751198751198751198751198751198751198751198751198751198751198751198751198751198751198751198751198751198751198751198753119875119875

16001200

800400

900080007000

600050004000300020000

1000

2000

3000

119868 119868+119868 119873+119864

1198781198771198752

1198751




3



119868infin= 0 or to 119868

119868infin=



3according




119868+ 119868119873+ 119864


lowast=


lowast the system


119861lt 1205731198770

lt 120573119862 120575 = 30 120578 = 25) Consider


119861lt 1205731198770



1205731198770

= 00001679568390

120573119862= 00001729256777

120573119861= 00001489092005

(33)










00001688612368 with 1205731198770



condition 1205731lt 120573 lt 120573

1198770




(119878 119877 119868119894 119868119899 119864)

1198750= (5148 0 0 0 0)

1198751= (3372 1041 122 60 482)

1198752= (2828 1283 190 88 651)

(34)



2is


119868infin= 0 or to 119868




2






lt 1205731198770


1205731198770

1205731

1205732

120573119862120573119861

000015 000016 000017 000018120573

600

500

400

300

200

100

0

119909


119861lt 1205731198770



2

500

400

300

200

100

0

119868 119868

0 500 1000 1500 2000119905



= 0

or 119868119868infin



(119878 119877 119868119894 119868119899 119864)

1198750= (5148 0 0 0 0)

1198751= (5042 76 5 3 20)

1198752= (3971 734 69 36 298)

1198753= (2491 1413 246 109 750)

(35)


160014001200800600400200

500

2000

0

1000

1500

1000

2000

3000

4000

5000

119878119877

11987521198751

1198750 160014001200800600444400420000002000200

5005005000

000

1000

1500

1000

20020020000000

300003003003003000000000000

400404004004004000040040040000000 119878119877

1198752119875119875

1198751119875119875

119868 119868+119868 119873+119877





2







gt








lowast= 120583 = 003885 120583

119905= 001520 119901 = 08

] = 00266 119891 = 08 119902 = 085 119908 = 0005 119888 = 04 1199031= 05





160014001200

800600400200

500

2000

0

1000

1500

1000

2000

3000

4000

5000

119868 119868+119868 119873+

119878119877

119877

1198752

1198751

1198753

1198750 160014001200

8006004004040200200200200200200

500

000

1000

1500

1000

2002002002002 000000

3000030033000000000000

4004004004004000040000400 00000

119868 119868119868+119868 119873119868+

119878119877

119877

1198752119875119875

1198751119875119875

1198753119875119875

0




0and







0is near one













5

5

4

4

3

3

2

2

1

10

0

120578

119861 gt 0 119862 gt 0

120575

119861 lt 0 119862 lt 0

119861 gt 0 119862 lt 0

(a) 1198770 = 1

5

5

4

4

3

3

2

2

1

10

0

120578

120575

(b) 1198770 = 1689504264

5

5

4

4

3

3

2

2

1

10

0

120578

120575

(c) 1198770 = 2379008527

5

5

4

4

3

3

2

2

1

10

0

120578

120575

(d) 1198770 = 7895042636



1198770= 00001450317354 (b) 120573 = 00002450317354 (c) 120573 = 00003450317354 and (d)

120573 = 0001145031735


1198770 120573119861 120573119862 and 120573 We divided



0has to do solely






5

5

4

4

3

3

2

2

1

10

0

120578

120575

Δ gt 0

119891(1199091) gt 0

119861 gt 0 119862 lt 0

119861 lt 0 119862 lt 0

119861 lt 0 119862 gt 0

119861 lt 0119862 gt 0

119861 gt 0 119862 gt 0

(a) 1198770 = 1

5

5

4

4

3

3

2

2

1

10

0

120578

120575

(b) 1198770 = 1004390340

5

5

4

4

3

3

2

2

1

10

0

120578

120575

(c) 1198770 = 1087806817

5

5

4

4

3

3

2

2

1

10

0

120578

120575

(d) 1198770 = 2317102260



1198770= 00002277727471 (b) 120573 = 00002287727471 (c)

120573 = 00002477727471 and (d) 120573 = 00005277727471


0gt 1






(1205731198770

lt 120573119862






0 the


0 Some



5

5

4

4

3

3

2

2

1

10

0

120578

120575

Δ gt 0

119891(1199091) gt 0

119861 gt 0 119862 lt 0

119861 lt 0 119862 lt 0

119861 lt 0 119862 gt 0

119861 lt 0119862 gt 0

119861 gt 0 119862 gt 0

(a) 1198770 = 09560965911

5

5

4

4

3

3

2

2

1

10

0

120578

120575

(b) 1198770 = 08902414781

5

5

4

4

3

3

2

2

1

10

0

120578

120575

(c) 1198770 = 07804829561

5

5

4

4

3

3

2

2

1

10

0

120578

120575

(d) 1198770 = 05609659131



00001777727471 and (d) 120573 = 00001277727471






0




0such that disease-











lt 120573119862lt 120573119861

120573 lt 1205731198770


lt 120573 lt 120573119862




120573119862lt 1205731198770

lt 120573119861

120573 lt 120573119862



2) le 0 or Δ

1lt 0 none if 119875(119909

2) ge 0 or Δ

1gt 0

1205731198770

lt 120573 lt 120573119861


1gt 0


120573119862lt 120573119861lt 1205731198770

120573 lt 120573119862



2) le 0 or Δ

1lt 0 none if 119875(119909

2) ge 0 or Δ

1gt 0

120573119861lt 120573 lt 120573


1lt 0) or none (Δ

1gt 0)

1205731198770


lt 120573119861lt 120573119862

120573 lt 1205731198770


lt 120573 lt 120573119861




1lt 0)


120573119861lt 1205731198770

lt 120573119862

120573 lt 120573119861



1lt 0) or none (Δ

1gt 0)

1205731198770

lt 120573 lt 120573119862


1lt 0)


120573119861lt 120573119862lt 1205731198770

120573 lt 120573119861



1lt 0) or none (Δ

1gt 0)

120573119862lt 120573 lt 120573


1lt 0) or none (Δ

1gt 0)

1205731198770



Appendices



119886 = 120583 + 120583119879+ 119888

119887 = 2119908 + 120583

ℎ = ] + 120583

119892 = 1199031+ 119888

119898 = 1199032+ 119888

120579 = 120583 + 120583119905

120598 = 1 minus 119901

1198711= 120579 + 119892 (1 minus 119902) + 119898119902

1198712= ]1199021199032+ ℎ119886 + 119903

1] (1 minus 119902) + 119903

1120583

1198701= 1199032119908 + 119887119886 + 119903

1(119908 + 120583)

1198702= ]1199021199032+ ℎ119886 + 119903

1] (1 minus 119902) + 119903

1120583

(A1)

We have

1198611= (120579 (119891119901 + (1 minus 119901) 119902) + 119898119902)Π120575120590

1198612= 120590 (120583119871

1120579 120575 + 119871

2120579 + 119898120583 (119886 + 119903

1))

+ (1199032119908120579 + 119903

1(119908 + 120583) 120579

+ 119887119886120579 + 119898120583 (119886 + 1199031)) 120575


1198621= Π (120575 ((119891119901 + 120598119902) (119887120583119905 + (119898 + 119887) 120583) + 119898119908)

+ 120590 ((]119902120598 + 119891ℎ119901) 120579 + 119898 (120583119891119901 + ] 119902)))

1198622= 120583 (119870

1120579 + 119898120583 (119886 + 119903

1)) 120575 + 120583 (119870

2120579 + 119898120583 (119886 + 119903

1)) 120590

+ ℎ (1198701120579 + 119898120583 (119886 + 119903

1))

(A2)


119861 = 1205732119891119861(120573)

119862 = 120573119891119862(120573)

(A3)

where119891119861(120573) = minus119861

1120573 + 1198612

119891119862(120573) = minus119862

1120573 + 119862

2

119860 = 1205733120575120590 (120583 + 120583

119879) (120583 + 120583

119879+ 119888 + (1 minus 119902) 119903

1+ 1199021199032) gt 0

119863 = ℎ ([119886119887 + 1199081199032+ 1199031(119908 + 120583)] (120583 + 120583119905)

+ 120583119898 (1199031+ 119886)) (1 minus 119877

0)

(A4)





Acknowledgments


References
















































of






Disease Markers



OncologyJournal of





PPAR Research



Journal of

ObesityJournal of

















119901119891120573119878119868119868

119902120575120573119864119868119868

120583119879119868119868 120583119868119868

120583119864120583119878 119902119864119888119868119868

1199032119868119868

120583119877120503 (1 minus 119901)120573119878119868

119877119868119864119878

120575(1 minus 119902)120573119864119868119868

120578120573119877119868119868

(1 minus 119902)119864

(1 minus 119891)119901120573119878119868119868

1199031119868119873

119888119868119873

120583119879119868119873 120583119868119873

119868119868

119868119873

119908119877

119908119877




1and 119903











119868and 120575(1 minus 119902)120573119864119868

119868


119868119877








1198681year


1198731year


119889119878

119889119905= Π minus 120573119878119868

119868minus 120583119878

119889119864

119889119905= (1 minus 119901) 120573119878119868

119868+ 120578120573119877119868

119868minus (] + 120583) 119864 minus 120575120573119864119868

119868

119889119868119868

119889119905= 119891119901120573119878119868

119868+ 119902]119864 + 119908119877 minus (120583 + 120583

119879+ 119888 + 119903

1) 119868119868+ 120575119902120573119864119868

119868

119889119868119873

119889119905= (1 minus 119891) 119901120573119878119868

119868+ (1 minus 119902) ]119864

+ 119908119877 minus (120583 + 120583119879+ 119888 + 119903

2) 119868119873+ 120575 (1 minus 119902) 120573119868

119868119864

119889119877

119889119905= 119888 (119868119868+ 119868119873) minus (2119908 + 120583) 119877 minus 120578120573119877119868

119868+ 1199031119868119868+ 1199032119868119873

(1)


119889119873

119889119905= minus120583119873 minus 120583

119879(119868119868+ 119868119873) + Π (2)



119863 = (119878 119864 119868119868 119868119873 119877) isin R

5

+ 119878 + 119864 + 119868

119868+ 119868119873+ 119877 le

Π

120583 (3)




0gt 1 an epidemic

will arise



1198770= 120573Π ((ℎ119891119901 + (1 minus 119901) ]119902) (119886119887 minus 119898119908)



(4)

where

119886 = 120583 + 120583119879+ 119888

119887 = 2119908 + 120583

ℎ = ] + 120583

119892 = 1199031+ 119888

119898 = 1199032+ 119888

(5)


0 = Π minus 120573119878119868119868minus 120583119878

0 = (1 minus 119901) 120573119878119868119868+ 120578120573119877119868

119868minus (] + 120583) 119864 minus 120575120573119864119868

119868

0 = 119891119901120573119878119868119868+ 119902]119864 + 119908119877 minus (120583 + 120583

119879+ 119888 + 119903

1) 119868119868+ 120575119902120573119864119868

119868

0 = (1 minus 119891) 119901120573119878119868119868+ (1 minus 119902) ]119864 + 119908119877

minus (120583 + 120583119879+ 119888 + 119903

2) 119868119873+ 120575 (1 minus 119902) 120573119868

119868119864

0 = 119888 (119868119868+ 119868119873) minus (2 119908 + 120583) 119877 minus 120578120573119877119868

119868+ 1199031119868119868+ 1199032119868119873

(6)


equation

119868119868(1198601198683

119868+ 1198611198682

119868+ 119862119868119868+ 119863) = 0 (7)




1198750= (1198780 1198640 1198681198680 1198681198730

1198770) = (

Π

120583 0 0 0 0) (8)





((Π120583) 0 0 0 0) is

119869 =

[[[[[[[[

[


1205830 0


1205830 0

0 119902]119891119901120573Π


0 (1 minus 119902) ](1 minus 119891) 119901120573Π


0 0 1199031 + 119888 1199032 + 119888 minus2119908 minus 120583

]]]]]]]]

]

(9)


(120582 + 120583) (1205824+ 11988631205823+ 11988621205822+ 1198861120582 + 1198860) = 0 (10)


119875 (120582) = 1205824+ 11988631205823+ 11988621205822+ 1198861120582 + 1198860= 0 (11)







0of the




119894(120573) = minus119860

119894120573+

119861119894with 119894 = 0 1 2 3 and 119860

119894 119861119894gt 0 We can define 120573

119894= 119861119894119860119894


119894(120573) gt 0

For example

1198863(120573) = minus

119891119901120573 Π

120583+ ] + 3120583 + 120583

119879+ 119888 + 119903

2+ 2119908

1205733=

(] + 3120583 + 120583119879+ 119888 + 119903

2+ 2119908) 120583

119891119901Π

(12)



1198862(1205733) lt 0 997904rArr 120573

2lt 1205733

1198861(1205732) lt 0 997904rArr 120573

1lt 1205732

1198860(1205731) lt 0 997904rArr 120573

0lt 1205731

(13)


0 all the




0of


1198860(120573) = minus119860

0120573 + 1198610= 1198610(1 minus

1198600120573

1198610

) = 1198610(1 minus 119877

0) (14)


1198770lt 1 then 119886

0(120573) gt 0 so 120573 lt 120573

0




4



real part


1198601198683

119868+ 1198611198682

119868+ 119862119868119868+ 119863 = 0 (15)


119860 = 1205733120575120590 (120583 + 120583

119879) (120583 + 120583

119879+ 119888 + (1 minus 119902) 119903

1+ 1199021199032) gt 0 (16)

119863 = ℎ ([119886119887 + 1199081199032+ 1199031(119908 + 120583)] (120583 + 120583119905)

+ 120583119898 (1199031+ 119886)) (1 minus 119877

0)

(17)


general form

119861 = 1205732119891119861(120573)

119862 = 120573119891119862(120573)

(18)

where119891119861(120573) = minus119861

1120573 + 1198612

119891119862(120573) = minus119862

1120573 + 119862

2

(19)


and 119862119894=12



119861(120573) and 119891


functions 119891119861(120573) and 119891



119875 (119909) = 1198601199093+ 1198611199092+ 119862119909 + 119863 (20)






2+ 2119861119909+119862 and then we


(1) If Δ = 1198612minus 3119860119862 le 0 119875

1015840(119909) ge 0 for all 119909



given by

11990921


3119860

(21)


and 1198751015840(119909) ge 0 for all 119909 ge 119909

2and 119909 le 119909

1 So we need to


2in the real line


have 1199091lt 0 119909

2gt 0 and 119875

1015840(119909) gt 0 for all 119909 ge 119909

2ge 0



2are



2


1) ge 0


119861(120573119861) = 0 and 120573

119862the


1198770be the value






(Table 3)


1198770lt 120573119862lt 120573119861 that



For 1205731198770



For 120573119862



Finally for 120573 gt 120573119861





lt 120573119862lt 120573119861


119861(120573119862) gt 0 and 119863(120573

119861) gt 0


lt 120573119862lt 120573119861


lt 120573119862

lt 120573119861



1198770lt 120573119862

lt 120573119861is

fulfilled


120573 lt 1205731198770


1205731198770

lt 120573 lt 120573119862 119860 gt 0 119861 gt 0 119862 gt 0119863 lt 0


120573119862lt 120573 lt 120573


120573 gt 120573119861 119860 gt 0 119861 lt 0 119862 lt 0119863 lt 0


800

600

400

200

0

minus200

minus400

119909

1198770 gt 1

1205731198770120573119862 120573119861

00002 00004 00006 00008 0001120573


1198770lt 120573119862

lt 120573119861 1205731198770


0gt 1


1198770 that is when 119877

0= 1


lt 120573119862



1205731198770 that is119877



1198770





119861lt 120573119862

lt 1205731198770

isfulfilled Here Δ



120573 lt 120573119861 119860 gt 0 119861 gt 0 119862 gt 0119863 gt 0


120573119861lt 120573 lt 120573

119862 119860 gt 0 119861 lt 0 119862 gt 0119863 gt 0


120573119862lt 120573 lt 120573

1198770119860 gt 0 119861 lt 0 119862 lt 0119863 gt 0


1205731198770





0



0


0= 1 and


119875 (119909) = 1198601199093+ 1198611199092+ 119862119909

= 119909 (1198601199092+ 1198611199092+ 119862) = 0

(22)



1and 119909

2



means 119891119861(1205731198770) lt 0 and




1198770lt 120573119862


1205731198770






times10minus3

119875(119909)

01

005

minus005

minus01

minus200 0 200 400 600119909


119861lt 1205731198770





119861lt 120573119862

lt 1205731198770

isfulfilled



For 120573119861

lt 120573 lt 120573119862and 120573

119862lt 120573 lt 120573





1lt 0







case we have

Δ1=

1

4

1198632

1198602+

1

27

1198623

1198603gt 0 (23)




Δ1(120573lowast) = 0 (24)


600

500

400

300

200

100

0

minus100

minus200

minus300

119909

1198770 gt 1

1205731198770120573119862120573119861 120573lowast

000005 00001 000015 00002120573


lt 120573119862

lt 1205731198770




(seeFigure 4)













1205731198770










119868050 (assumed)


119873020 (assumed)






parameters 1205731198770 120573119861 and 120573



1198770 120573119861 120573119862) then the system


1198770 120573119861 120573119862) then








1198770lt 120573119862

lt 120573119861 120575 = 09 120578 = 001) Let us


lt 120573119862lt 120573119861is











were

119878 (0) = 4980 119864 (0) = 0 119868119868 (0) = 20

119868119873 (0) = 0 119877 (0) = 0

(25)


1205731198770

= 00001450317354

120573119861= 001087387065

120573119862= 00002715343808

(26)


lt 120573119862

lt 120573119861


1198770


119878infin

= 1616 119877infin

= 4080 119868119873infin

= 103

119868119868infin

= 195 119864infin

= 1150

(27)





lt 120573119862lt 120573119861 120575 = 00 120578 = 09) For our



0= 3585422172 The used


119878 (0) = 4980 119864 (0) = 0 119868119868 (0) = 20

119868119873 (0) = 0 119877 (0) = 0

(28)

119905

7000

6000

5000

4000

3000

2000

1000

00 50 100 150 200

119878

119877

119868119873119864

119868119868 + 119868119873119868119868119873

119868119868 + 119868119873 + 119864 + 119877



4000

4000 5000

3000

3000

2000

2000

1000

10004000

6000

20000

119868 119868+119868 119873+119864

119878 119877 4000 500

3000000000000000000000000000000

3000

200202000020200202002020200000000200202020020020200200202020202002002000202002000020202020200020202000220000200202000020202222000222020200220000202022022220022002202222220022222 02 02222 0000000000000000000000000000000000000000000000000000000000000000

2000

100110010010010010010010010100100100100010010010010010010010010000010100100100100100100100100110000001001000000110010110010010010010001000001110100010000010000000001010000011000001110000000001001001000000000000001000010000111000101 000000000000000000000000000000000000000000000000000000000000000000000000000000000000000

10004000

6000

20000

119868119868119868119868119868119868119868119868119868119868119868119868119868119868119868119868119868119868119868119868119868119868119868119868119868119868119868119868119868119868119868119868119868119868119868119868119868119868119868119868 119868119868119868119868119868119868119868119868119868119868119868119868119868119868++++++++++++++++++++++++++119868 119873119868+119864

119878 119877


0=

3585422172 120575 = 09 120578 = 001 and 120573 = 000052


1205731198770

= 00001450317354

120573119861= 001226355348

120573119862= 00003132229272

(29)


lt 120573119861lt 120573119862 and as in



119878infin

= 1938 119877infin

= 974 119868119873infin

= 60

119868119868infin

= 156 119864infin

= 4530

(30)


119905

7000

6000

5000

4000

3000

2000

1000

00 50 100 150 200

119878

119877

119868119873119864

119868119868 + 119868119873119868119868119873

119868119868 + 119868119873 + 119864 + 119877




1198770lt 120573119862







119861lt 120573119862

lt 1205731198770 120575 = 30 120578 = 25) There


0=


1205731198770

= 00001450317354

120573119861= 00001066568066

120573119862= 00001225687204

(31)


1205731198770


119894 119868119899 119864)

1198751= (9009 0 0 0 0)

1198752= (8507 182 9 5 2166)

1198753= (3221 1406 285 103 1566)

(32)

800

700

600

500

400

300

200

100

07006005004003002001000

119868 119868

119905



= 0



11987531198751198751198751198751198751198751198751198751198751198751198751198751198751198751198751198751198751198751198751198751198751198751198751198751198751198751198751198751198751198751198751198751198751198751198751198751198751198751198751198751198751198751198751198751198751198751198751198751198751198751198751198751198751198751198751198751198751198751198751198751198751198751198751198751198751198751198751198751198751198751198751198751198753119875119875

16001200

800400

900080007000

600050004000300020000

1000

2000

3000

119868 119868+119868 119873+119864

1198781198771198752

1198751




3



119868infin= 0 or to 119868

119868infin=



3according




119868+ 119868119873+ 119864


lowast=


lowast the system


119861lt 1205731198770

lt 120573119862 120575 = 30 120578 = 25) Consider


119861lt 1205731198770



1205731198770

= 00001679568390

120573119862= 00001729256777

120573119861= 00001489092005

(33)










00001688612368 with 1205731198770



condition 1205731lt 120573 lt 120573

1198770




(119878 119877 119868119894 119868119899 119864)

1198750= (5148 0 0 0 0)

1198751= (3372 1041 122 60 482)

1198752= (2828 1283 190 88 651)

(34)



2is


119868infin= 0 or to 119868




2






lt 1205731198770


1205731198770

1205731

1205732

120573119862120573119861

000015 000016 000017 000018120573

600

500

400

300

200

100

0

119909


119861lt 1205731198770



2

500

400

300

200

100

0

119868 119868

0 500 1000 1500 2000119905



= 0

or 119868119868infin



(119878 119877 119868119894 119868119899 119864)

1198750= (5148 0 0 0 0)

1198751= (5042 76 5 3 20)

1198752= (3971 734 69 36 298)

1198753= (2491 1413 246 109 750)

(35)


160014001200800600400200

500

2000

0

1000

1500

1000

2000

3000

4000

5000

119878119877

11987521198751

1198750 160014001200800600444400420000002000200

5005005000

000

1000

1500

1000

20020020000000

300003003003003000000000000

400404004004004000040040040000000 119878119877

1198752119875119875

1198751119875119875

119868 119868+119868 119873+119877





2







gt








lowast= 120583 = 003885 120583

119905= 001520 119901 = 08

] = 00266 119891 = 08 119902 = 085 119908 = 0005 119888 = 04 1199031= 05





160014001200

800600400200

500

2000

0

1000

1500

1000

2000

3000

4000

5000

119868 119868+119868 119873+

119878119877

119877

1198752

1198751

1198753

1198750 160014001200

8006004004040200200200200200200

500

000

1000

1500

1000

2002002002002 000000

3000030033000000000000

4004004004004000040000400 00000

119868 119868119868+119868 119873119868+

119878119877

119877

1198752119875119875

1198751119875119875

1198753119875119875

0




0and







0is near one













5

5

4

4

3

3

2

2

1

10

0

120578

119861 gt 0 119862 gt 0

120575

119861 lt 0 119862 lt 0

119861 gt 0 119862 lt 0

(a) 1198770 = 1

5

5

4

4

3

3

2

2

1

10

0

120578

120575

(b) 1198770 = 1689504264

5

5

4

4

3

3

2

2

1

10

0

120578

120575

(c) 1198770 = 2379008527

5

5

4

4

3

3

2

2

1

10

0

120578

120575

(d) 1198770 = 7895042636



1198770= 00001450317354 (b) 120573 = 00002450317354 (c) 120573 = 00003450317354 and (d)

120573 = 0001145031735


1198770 120573119861 120573119862 and 120573 We divided



0has to do solely






5

5

4

4

3

3

2

2

1

10

0

120578

120575

Δ gt 0

119891(1199091) gt 0

119861 gt 0 119862 lt 0

119861 lt 0 119862 lt 0

119861 lt 0 119862 gt 0

119861 lt 0119862 gt 0

119861 gt 0 119862 gt 0

(a) 1198770 = 1

5

5

4

4

3

3

2

2

1

10

0

120578

120575

(b) 1198770 = 1004390340

5

5

4

4

3

3

2

2

1

10

0

120578

120575

(c) 1198770 = 1087806817

5

5

4

4

3

3

2

2

1

10

0

120578

120575

(d) 1198770 = 2317102260



1198770= 00002277727471 (b) 120573 = 00002287727471 (c)

120573 = 00002477727471 and (d) 120573 = 00005277727471


0gt 1






(1205731198770

lt 120573119862






0 the


0 Some



5

5

4

4

3

3

2

2

1

10

0

120578

120575

Δ gt 0

119891(1199091) gt 0

119861 gt 0 119862 lt 0

119861 lt 0 119862 lt 0

119861 lt 0 119862 gt 0

119861 lt 0119862 gt 0

119861 gt 0 119862 gt 0

(a) 1198770 = 09560965911

5

5

4

4

3

3

2

2

1

10

0

120578

120575

(b) 1198770 = 08902414781

5

5

4

4

3

3

2

2

1

10

0

120578

120575

(c) 1198770 = 07804829561

5

5

4

4

3

3

2

2

1

10

0

120578

120575

(d) 1198770 = 05609659131



00001777727471 and (d) 120573 = 00001277727471






0




0such that disease-











lt 120573119862lt 120573119861

120573 lt 1205731198770


lt 120573 lt 120573119862




120573119862lt 1205731198770

lt 120573119861

120573 lt 120573119862



2) le 0 or Δ

1lt 0 none if 119875(119909

2) ge 0 or Δ

1gt 0

1205731198770

lt 120573 lt 120573119861


1gt 0


120573119862lt 120573119861lt 1205731198770

120573 lt 120573119862



2) le 0 or Δ

1lt 0 none if 119875(119909

2) ge 0 or Δ

1gt 0

120573119861lt 120573 lt 120573


1lt 0) or none (Δ

1gt 0)

1205731198770


lt 120573119861lt 120573119862

120573 lt 1205731198770


lt 120573 lt 120573119861




1lt 0)


120573119861lt 1205731198770

lt 120573119862

120573 lt 120573119861



1lt 0) or none (Δ

1gt 0)

1205731198770

lt 120573 lt 120573119862


1lt 0)


120573119861lt 120573119862lt 1205731198770

120573 lt 120573119861



1lt 0) or none (Δ

1gt 0)

120573119862lt 120573 lt 120573


1lt 0) or none (Δ

1gt 0)

1205731198770



Appendices



119886 = 120583 + 120583119879+ 119888

119887 = 2119908 + 120583

ℎ = ] + 120583

119892 = 1199031+ 119888

119898 = 1199032+ 119888

120579 = 120583 + 120583119905

120598 = 1 minus 119901

1198711= 120579 + 119892 (1 minus 119902) + 119898119902

1198712= ]1199021199032+ ℎ119886 + 119903

1] (1 minus 119902) + 119903

1120583

1198701= 1199032119908 + 119887119886 + 119903

1(119908 + 120583)

1198702= ]1199021199032+ ℎ119886 + 119903

1] (1 minus 119902) + 119903

1120583

(A1)

We have

1198611= (120579 (119891119901 + (1 minus 119901) 119902) + 119898119902)Π120575120590

1198612= 120590 (120583119871

1120579 120575 + 119871

2120579 + 119898120583 (119886 + 119903

1))

+ (1199032119908120579 + 119903

1(119908 + 120583) 120579

+ 119887119886120579 + 119898120583 (119886 + 1199031)) 120575


1198621= Π (120575 ((119891119901 + 120598119902) (119887120583119905 + (119898 + 119887) 120583) + 119898119908)

+ 120590 ((]119902120598 + 119891ℎ119901) 120579 + 119898 (120583119891119901 + ] 119902)))

1198622= 120583 (119870

1120579 + 119898120583 (119886 + 119903

1)) 120575 + 120583 (119870

2120579 + 119898120583 (119886 + 119903

1)) 120590

+ ℎ (1198701120579 + 119898120583 (119886 + 119903

1))

(A2)


119861 = 1205732119891119861(120573)

119862 = 120573119891119862(120573)

(A3)

where119891119861(120573) = minus119861

1120573 + 1198612

119891119862(120573) = minus119862

1120573 + 119862

2

119860 = 1205733120575120590 (120583 + 120583

119879) (120583 + 120583

119879+ 119888 + (1 minus 119902) 119903

1+ 1199021199032) gt 0

119863 = ℎ ([119886119887 + 1199081199032+ 1199031(119908 + 120583)] (120583 + 120583119905)

+ 120583119898 (1199031+ 119886)) (1 minus 119877

0)

(A4)





Acknowledgments


References
















































of






Disease Markers



OncologyJournal of





PPAR Research



Journal of

ObesityJournal of






















1198681year


1198731year


119889119878

119889119905= Π minus 120573119878119868

119868minus 120583119878

119889119864

119889119905= (1 minus 119901) 120573119878119868

119868+ 120578120573119877119868

119868minus (] + 120583) 119864 minus 120575120573119864119868

119868

119889119868119868

119889119905= 119891119901120573119878119868

119868+ 119902]119864 + 119908119877 minus (120583 + 120583

119879+ 119888 + 119903

1) 119868119868+ 120575119902120573119864119868

119868

119889119868119873

119889119905= (1 minus 119891) 119901120573119878119868

119868+ (1 minus 119902) ]119864

+ 119908119877 minus (120583 + 120583119879+ 119888 + 119903

2) 119868119873+ 120575 (1 minus 119902) 120573119868

119868119864

119889119877

119889119905= 119888 (119868119868+ 119868119873) minus (2119908 + 120583) 119877 minus 120578120573119877119868

119868+ 1199031119868119868+ 1199032119868119873

(1)


119889119873

119889119905= minus120583119873 minus 120583

119879(119868119868+ 119868119873) + Π (2)



119863 = (119878 119864 119868119868 119868119873 119877) isin R

5

+ 119878 + 119864 + 119868

119868+ 119868119873+ 119877 le

Π

120583 (3)




0gt 1 an epidemic

will arise



1198770= 120573Π ((ℎ119891119901 + (1 minus 119901) ]119902) (119886119887 minus 119898119908)



(4)

where

119886 = 120583 + 120583119879+ 119888

119887 = 2119908 + 120583

ℎ = ] + 120583

119892 = 1199031+ 119888

119898 = 1199032+ 119888

(5)


0 = Π minus 120573119878119868119868minus 120583119878

0 = (1 minus 119901) 120573119878119868119868+ 120578120573119877119868

119868minus (] + 120583) 119864 minus 120575120573119864119868

119868

0 = 119891119901120573119878119868119868+ 119902]119864 + 119908119877 minus (120583 + 120583

119879+ 119888 + 119903

1) 119868119868+ 120575119902120573119864119868

119868

0 = (1 minus 119891) 119901120573119878119868119868+ (1 minus 119902) ]119864 + 119908119877

minus (120583 + 120583119879+ 119888 + 119903

2) 119868119873+ 120575 (1 minus 119902) 120573119868

119868119864

0 = 119888 (119868119868+ 119868119873) minus (2 119908 + 120583) 119877 minus 120578120573119877119868

119868+ 1199031119868119868+ 1199032119868119873

(6)


equation

119868119868(1198601198683

119868+ 1198611198682

119868+ 119862119868119868+ 119863) = 0 (7)




1198750= (1198780 1198640 1198681198680 1198681198730

1198770) = (

Π

120583 0 0 0 0) (8)





((Π120583) 0 0 0 0) is

119869 =

[[[[[[[[

[


1205830 0


1205830 0

0 119902]119891119901120573Π


0 (1 minus 119902) ](1 minus 119891) 119901120573Π


0 0 1199031 + 119888 1199032 + 119888 minus2119908 minus 120583

]]]]]]]]

]

(9)


(120582 + 120583) (1205824+ 11988631205823+ 11988621205822+ 1198861120582 + 1198860) = 0 (10)


119875 (120582) = 1205824+ 11988631205823+ 11988621205822+ 1198861120582 + 1198860= 0 (11)







0of the




119894(120573) = minus119860

119894120573+

119861119894with 119894 = 0 1 2 3 and 119860

119894 119861119894gt 0 We can define 120573

119894= 119861119894119860119894


119894(120573) gt 0

For example

1198863(120573) = minus

119891119901120573 Π

120583+ ] + 3120583 + 120583

119879+ 119888 + 119903

2+ 2119908

1205733=

(] + 3120583 + 120583119879+ 119888 + 119903

2+ 2119908) 120583

119891119901Π

(12)



1198862(1205733) lt 0 997904rArr 120573

2lt 1205733

1198861(1205732) lt 0 997904rArr 120573

1lt 1205732

1198860(1205731) lt 0 997904rArr 120573

0lt 1205731

(13)


0 all the




0of


1198860(120573) = minus119860

0120573 + 1198610= 1198610(1 minus

1198600120573

1198610

) = 1198610(1 minus 119877

0) (14)


1198770lt 1 then 119886

0(120573) gt 0 so 120573 lt 120573

0




4



real part


1198601198683

119868+ 1198611198682

119868+ 119862119868119868+ 119863 = 0 (15)


119860 = 1205733120575120590 (120583 + 120583

119879) (120583 + 120583

119879+ 119888 + (1 minus 119902) 119903

1+ 1199021199032) gt 0 (16)

119863 = ℎ ([119886119887 + 1199081199032+ 1199031(119908 + 120583)] (120583 + 120583119905)

+ 120583119898 (1199031+ 119886)) (1 minus 119877

0)

(17)


general form

119861 = 1205732119891119861(120573)

119862 = 120573119891119862(120573)

(18)

where119891119861(120573) = minus119861

1120573 + 1198612

119891119862(120573) = minus119862

1120573 + 119862

2

(19)


and 119862119894=12



119861(120573) and 119891


functions 119891119861(120573) and 119891



119875 (119909) = 1198601199093+ 1198611199092+ 119862119909 + 119863 (20)






2+ 2119861119909+119862 and then we


(1) If Δ = 1198612minus 3119860119862 le 0 119875

1015840(119909) ge 0 for all 119909



given by

11990921


3119860

(21)


and 1198751015840(119909) ge 0 for all 119909 ge 119909

2and 119909 le 119909

1 So we need to


2in the real line


have 1199091lt 0 119909

2gt 0 and 119875

1015840(119909) gt 0 for all 119909 ge 119909

2ge 0



2are



2


1) ge 0


119861(120573119861) = 0 and 120573

119862the


1198770be the value






(Table 3)


1198770lt 120573119862lt 120573119861 that



For 1205731198770



For 120573119862



Finally for 120573 gt 120573119861





lt 120573119862lt 120573119861


119861(120573119862) gt 0 and 119863(120573

119861) gt 0


lt 120573119862lt 120573119861


lt 120573119862

lt 120573119861



1198770lt 120573119862

lt 120573119861is

fulfilled


120573 lt 1205731198770


1205731198770

lt 120573 lt 120573119862 119860 gt 0 119861 gt 0 119862 gt 0119863 lt 0


120573119862lt 120573 lt 120573


120573 gt 120573119861 119860 gt 0 119861 lt 0 119862 lt 0119863 lt 0


800

600

400

200

0

minus200

minus400

119909

1198770 gt 1

1205731198770120573119862 120573119861

00002 00004 00006 00008 0001120573


1198770lt 120573119862

lt 120573119861 1205731198770


0gt 1


1198770 that is when 119877

0= 1


lt 120573119862



1205731198770 that is119877



1198770





119861lt 120573119862

lt 1205731198770

isfulfilled Here Δ



120573 lt 120573119861 119860 gt 0 119861 gt 0 119862 gt 0119863 gt 0


120573119861lt 120573 lt 120573

119862 119860 gt 0 119861 lt 0 119862 gt 0119863 gt 0


120573119862lt 120573 lt 120573

1198770119860 gt 0 119861 lt 0 119862 lt 0119863 gt 0


1205731198770





0



0


0= 1 and


119875 (119909) = 1198601199093+ 1198611199092+ 119862119909

= 119909 (1198601199092+ 1198611199092+ 119862) = 0

(22)



1and 119909

2



means 119891119861(1205731198770) lt 0 and




1198770lt 120573119862


1205731198770






times10minus3

119875(119909)

01

005

minus005

minus01

minus200 0 200 400 600119909


119861lt 1205731198770





119861lt 120573119862

lt 1205731198770

isfulfilled



For 120573119861

lt 120573 lt 120573119862and 120573

119862lt 120573 lt 120573





1lt 0







case we have

Δ1=

1

4

1198632

1198602+

1

27

1198623

1198603gt 0 (23)




Δ1(120573lowast) = 0 (24)


600

500

400

300

200

100

0

minus100

minus200

minus300

119909

1198770 gt 1

1205731198770120573119862120573119861 120573lowast

000005 00001 000015 00002120573


lt 120573119862

lt 1205731198770




(seeFigure 4)













1205731198770










119868050 (assumed)


119873020 (assumed)






parameters 1205731198770 120573119861 and 120573



1198770 120573119861 120573119862) then the system


1198770 120573119861 120573119862) then








1198770lt 120573119862

lt 120573119861 120575 = 09 120578 = 001) Let us


lt 120573119862lt 120573119861is











were

119878 (0) = 4980 119864 (0) = 0 119868119868 (0) = 20

119868119873 (0) = 0 119877 (0) = 0

(25)


1205731198770

= 00001450317354

120573119861= 001087387065

120573119862= 00002715343808

(26)


lt 120573119862

lt 120573119861


1198770


119878infin

= 1616 119877infin

= 4080 119868119873infin

= 103

119868119868infin

= 195 119864infin

= 1150

(27)





lt 120573119862lt 120573119861 120575 = 00 120578 = 09) For our



0= 3585422172 The used


119878 (0) = 4980 119864 (0) = 0 119868119868 (0) = 20

119868119873 (0) = 0 119877 (0) = 0

(28)

119905

7000

6000

5000

4000

3000

2000

1000

00 50 100 150 200

119878

119877

119868119873119864

119868119868 + 119868119873119868119868119873

119868119868 + 119868119873 + 119864 + 119877



4000

4000 5000

3000

3000

2000

2000

1000

10004000

6000

20000

119868 119868+119868 119873+119864

119878 119877 4000 500

3000000000000000000000000000000

3000

200202000020200202002020200000000200202020020020200200202020202002002000202002000020202020200020202000220000200202000020202222000222020200220000202022022220022002202222220022222 02 02222 0000000000000000000000000000000000000000000000000000000000000000

2000

100110010010010010010010010100100100100010010010010010010010010000010100100100100100100100100110000001001000000110010110010010010010001000001110100010000010000000001010000011000001110000000001001001000000000000001000010000111000101 000000000000000000000000000000000000000000000000000000000000000000000000000000000000000

10004000

6000

20000

119868119868119868119868119868119868119868119868119868119868119868119868119868119868119868119868119868119868119868119868119868119868119868119868119868119868119868119868119868119868119868119868119868119868119868119868119868119868119868119868 119868119868119868119868119868119868119868119868119868119868119868119868119868119868++++++++++++++++++++++++++119868 119873119868+119864

119878 119877


0=

3585422172 120575 = 09 120578 = 001 and 120573 = 000052


1205731198770

= 00001450317354

120573119861= 001226355348

120573119862= 00003132229272

(29)


lt 120573119861lt 120573119862 and as in



119878infin

= 1938 119877infin

= 974 119868119873infin

= 60

119868119868infin

= 156 119864infin

= 4530

(30)


119905

7000

6000

5000

4000

3000

2000

1000

00 50 100 150 200

119878

119877

119868119873119864

119868119868 + 119868119873119868119868119873

119868119868 + 119868119873 + 119864 + 119877




1198770lt 120573119862







119861lt 120573119862

lt 1205731198770 120575 = 30 120578 = 25) There


0=


1205731198770

= 00001450317354

120573119861= 00001066568066

120573119862= 00001225687204

(31)


1205731198770


119894 119868119899 119864)

1198751= (9009 0 0 0 0)

1198752= (8507 182 9 5 2166)

1198753= (3221 1406 285 103 1566)

(32)

800

700

600

500

400

300

200

100

07006005004003002001000

119868 119868

119905



= 0



11987531198751198751198751198751198751198751198751198751198751198751198751198751198751198751198751198751198751198751198751198751198751198751198751198751198751198751198751198751198751198751198751198751198751198751198751198751198751198751198751198751198751198751198751198751198751198751198751198751198751198751198751198751198751198751198751198751198751198751198751198751198751198751198751198751198751198751198751198751198751198751198751198751198753119875119875

16001200

800400

900080007000

600050004000300020000

1000

2000

3000

119868 119868+119868 119873+119864

1198781198771198752

1198751




3



119868infin= 0 or to 119868

119868infin=



3according




119868+ 119868119873+ 119864


lowast=


lowast the system


119861lt 1205731198770

lt 120573119862 120575 = 30 120578 = 25) Consider


119861lt 1205731198770



1205731198770

= 00001679568390

120573119862= 00001729256777

120573119861= 00001489092005

(33)










00001688612368 with 1205731198770



condition 1205731lt 120573 lt 120573

1198770




(119878 119877 119868119894 119868119899 119864)

1198750= (5148 0 0 0 0)

1198751= (3372 1041 122 60 482)

1198752= (2828 1283 190 88 651)

(34)



2is


119868infin= 0 or to 119868




2






lt 1205731198770


1205731198770

1205731

1205732

120573119862120573119861

000015 000016 000017 000018120573

600

500

400

300

200

100

0

119909


119861lt 1205731198770



2

500

400

300

200

100

0

119868 119868

0 500 1000 1500 2000119905



= 0

or 119868119868infin



(119878 119877 119868119894 119868119899 119864)

1198750= (5148 0 0 0 0)

1198751= (5042 76 5 3 20)

1198752= (3971 734 69 36 298)

1198753= (2491 1413 246 109 750)

(35)


160014001200800600400200

500

2000

0

1000

1500

1000

2000

3000

4000

5000

119878119877

11987521198751

1198750 160014001200800600444400420000002000200

5005005000

000

1000

1500

1000

20020020000000

300003003003003000000000000

400404004004004000040040040000000 119878119877

1198752119875119875

1198751119875119875

119868 119868+119868 119873+119877





2







gt








lowast= 120583 = 003885 120583

119905= 001520 119901 = 08

] = 00266 119891 = 08 119902 = 085 119908 = 0005 119888 = 04 1199031= 05





160014001200

800600400200

500

2000

0

1000

1500

1000

2000

3000

4000

5000

119868 119868+119868 119873+

119878119877

119877

1198752

1198751

1198753

1198750 160014001200

8006004004040200200200200200200

500

000

1000

1500

1000

2002002002002 000000

3000030033000000000000

4004004004004000040000400 00000

119868 119868119868+119868 119873119868+

119878119877

119877

1198752119875119875

1198751119875119875

1198753119875119875

0




0and







0is near one













5

5

4

4

3

3

2

2

1

10

0

120578

119861 gt 0 119862 gt 0

120575

119861 lt 0 119862 lt 0

119861 gt 0 119862 lt 0

(a) 1198770 = 1

5

5

4

4

3

3

2

2

1

10

0

120578

120575

(b) 1198770 = 1689504264

5

5

4

4

3

3

2

2

1

10

0

120578

120575

(c) 1198770 = 2379008527

5

5

4

4

3

3

2

2

1

10

0

120578

120575

(d) 1198770 = 7895042636



1198770= 00001450317354 (b) 120573 = 00002450317354 (c) 120573 = 00003450317354 and (d)

120573 = 0001145031735


1198770 120573119861 120573119862 and 120573 We divided



0has to do solely






5

5

4

4

3

3

2

2

1

10

0

120578

120575

Δ gt 0

119891(1199091) gt 0

119861 gt 0 119862 lt 0

119861 lt 0 119862 lt 0

119861 lt 0 119862 gt 0

119861 lt 0119862 gt 0

119861 gt 0 119862 gt 0

(a) 1198770 = 1

5

5

4

4

3

3

2

2

1

10

0

120578

120575

(b) 1198770 = 1004390340

5

5

4

4

3

3

2

2

1

10

0

120578

120575

(c) 1198770 = 1087806817

5

5

4

4

3

3

2

2

1

10

0

120578

120575

(d) 1198770 = 2317102260



1198770= 00002277727471 (b) 120573 = 00002287727471 (c)

120573 = 00002477727471 and (d) 120573 = 00005277727471


0gt 1






(1205731198770

lt 120573119862






0 the


0 Some



5

5

4

4

3

3

2

2

1

10

0

120578

120575

Δ gt 0

119891(1199091) gt 0

119861 gt 0 119862 lt 0

119861 lt 0 119862 lt 0

119861 lt 0 119862 gt 0

119861 lt 0119862 gt 0

119861 gt 0 119862 gt 0

(a) 1198770 = 09560965911

5

5

4

4

3

3

2

2

1

10

0

120578

120575

(b) 1198770 = 08902414781

5

5

4

4

3

3

2

2

1

10

0

120578

120575

(c) 1198770 = 07804829561

5

5

4

4

3

3

2

2

1

10

0

120578

120575

(d) 1198770 = 05609659131



00001777727471 and (d) 120573 = 00001277727471






0




0such that disease-











lt 120573119862lt 120573119861

120573 lt 1205731198770


lt 120573 lt 120573119862




120573119862lt 1205731198770

lt 120573119861

120573 lt 120573119862



2) le 0 or Δ

1lt 0 none if 119875(119909

2) ge 0 or Δ

1gt 0

1205731198770

lt 120573 lt 120573119861


1gt 0


120573119862lt 120573119861lt 1205731198770

120573 lt 120573119862



2) le 0 or Δ

1lt 0 none if 119875(119909

2) ge 0 or Δ

1gt 0

120573119861lt 120573 lt 120573


1lt 0) or none (Δ

1gt 0)

1205731198770


lt 120573119861lt 120573119862

120573 lt 1205731198770


lt 120573 lt 120573119861




1lt 0)


120573119861lt 1205731198770

lt 120573119862

120573 lt 120573119861



1lt 0) or none (Δ

1gt 0)

1205731198770

lt 120573 lt 120573119862


1lt 0)


120573119861lt 120573119862lt 1205731198770

120573 lt 120573119861



1lt 0) or none (Δ

1gt 0)

120573119862lt 120573 lt 120573


1lt 0) or none (Δ

1gt 0)

1205731198770



Appendices



119886 = 120583 + 120583119879+ 119888

119887 = 2119908 + 120583

ℎ = ] + 120583

119892 = 1199031+ 119888

119898 = 1199032+ 119888

120579 = 120583 + 120583119905

120598 = 1 minus 119901

1198711= 120579 + 119892 (1 minus 119902) + 119898119902

1198712= ]1199021199032+ ℎ119886 + 119903

1] (1 minus 119902) + 119903

1120583

1198701= 1199032119908 + 119887119886 + 119903

1(119908 + 120583)

1198702= ]1199021199032+ ℎ119886 + 119903

1] (1 minus 119902) + 119903

1120583

(A1)

We have

1198611= (120579 (119891119901 + (1 minus 119901) 119902) + 119898119902)Π120575120590

1198612= 120590 (120583119871

1120579 120575 + 119871

2120579 + 119898120583 (119886 + 119903

1))

+ (1199032119908120579 + 119903

1(119908 + 120583) 120579

+ 119887119886120579 + 119898120583 (119886 + 1199031)) 120575


1198621= Π (120575 ((119891119901 + 120598119902) (119887120583119905 + (119898 + 119887) 120583) + 119898119908)

+ 120590 ((]119902120598 + 119891ℎ119901) 120579 + 119898 (120583119891119901 + ] 119902)))

1198622= 120583 (119870

1120579 + 119898120583 (119886 + 119903

1)) 120575 + 120583 (119870

2120579 + 119898120583 (119886 + 119903

1)) 120590

+ ℎ (1198701120579 + 119898120583 (119886 + 119903

1))

(A2)


119861 = 1205732119891119861(120573)

119862 = 120573119891119862(120573)

(A3)

where119891119861(120573) = minus119861

1120573 + 1198612

119891119862(120573) = minus119862

1120573 + 119862

2

119860 = 1205733120575120590 (120583 + 120583

119879) (120583 + 120583

119879+ 119888 + (1 minus 119902) 119903

1+ 1199021199032) gt 0

119863 = ℎ ([119886119887 + 1199081199032+ 1199031(119908 + 120583)] (120583 + 120583119905)

+ 120583119898 (1199031+ 119886)) (1 minus 119877

0)

(A4)





Acknowledgments


References
















































of






Disease Markers



OncologyJournal of





PPAR Research



Journal of

ObesityJournal of


















((Π120583) 0 0 0 0) is

119869 =

[[[[[[[[

[


1205830 0


1205830 0

0 119902]119891119901120573Π


0 (1 minus 119902) ](1 minus 119891) 119901120573Π


0 0 1199031 + 119888 1199032 + 119888 minus2119908 minus 120583

]]]]]]]]

]

(9)


(120582 + 120583) (1205824+ 11988631205823+ 11988621205822+ 1198861120582 + 1198860) = 0 (10)


119875 (120582) = 1205824+ 11988631205823+ 11988621205822+ 1198861120582 + 1198860= 0 (11)







0of the




119894(120573) = minus119860

119894120573+

119861119894with 119894 = 0 1 2 3 and 119860

119894 119861119894gt 0 We can define 120573

119894= 119861119894119860119894


119894(120573) gt 0

For example

1198863(120573) = minus

119891119901120573 Π

120583+ ] + 3120583 + 120583

119879+ 119888 + 119903

2+ 2119908

1205733=

(] + 3120583 + 120583119879+ 119888 + 119903

2+ 2119908) 120583

119891119901Π

(12)



1198862(1205733) lt 0 997904rArr 120573

2lt 1205733

1198861(1205732) lt 0 997904rArr 120573

1lt 1205732

1198860(1205731) lt 0 997904rArr 120573

0lt 1205731

(13)


0 all the




0of


1198860(120573) = minus119860

0120573 + 1198610= 1198610(1 minus

1198600120573

1198610

) = 1198610(1 minus 119877

0) (14)


1198770lt 1 then 119886

0(120573) gt 0 so 120573 lt 120573

0




4



real part


1198601198683

119868+ 1198611198682

119868+ 119862119868119868+ 119863 = 0 (15)


119860 = 1205733120575120590 (120583 + 120583

119879) (120583 + 120583

119879+ 119888 + (1 minus 119902) 119903

1+ 1199021199032) gt 0 (16)

119863 = ℎ ([119886119887 + 1199081199032+ 1199031(119908 + 120583)] (120583 + 120583119905)

+ 120583119898 (1199031+ 119886)) (1 minus 119877

0)

(17)


general form

119861 = 1205732119891119861(120573)

119862 = 120573119891119862(120573)

(18)

where119891119861(120573) = minus119861

1120573 + 1198612

119891119862(120573) = minus119862

1120573 + 119862

2

(19)


and 119862119894=12



119861(120573) and 119891


functions 119891119861(120573) and 119891



119875 (119909) = 1198601199093+ 1198611199092+ 119862119909 + 119863 (20)






2+ 2119861119909+119862 and then we


(1) If Δ = 1198612minus 3119860119862 le 0 119875

1015840(119909) ge 0 for all 119909



given by

11990921


3119860

(21)


and 1198751015840(119909) ge 0 for all 119909 ge 119909

2and 119909 le 119909

1 So we need to


2in the real line


have 1199091lt 0 119909

2gt 0 and 119875

1015840(119909) gt 0 for all 119909 ge 119909

2ge 0



2are



2


1) ge 0


119861(120573119861) = 0 and 120573

119862the


1198770be the value






(Table 3)


1198770lt 120573119862lt 120573119861 that



For 1205731198770



For 120573119862



Finally for 120573 gt 120573119861





lt 120573119862lt 120573119861


119861(120573119862) gt 0 and 119863(120573

119861) gt 0


lt 120573119862lt 120573119861


lt 120573119862

lt 120573119861



1198770lt 120573119862

lt 120573119861is

fulfilled


120573 lt 1205731198770


1205731198770

lt 120573 lt 120573119862 119860 gt 0 119861 gt 0 119862 gt 0119863 lt 0


120573119862lt 120573 lt 120573


120573 gt 120573119861 119860 gt 0 119861 lt 0 119862 lt 0119863 lt 0


800

600

400

200

0

minus200

minus400

119909

1198770 gt 1

1205731198770120573119862 120573119861

00002 00004 00006 00008 0001120573


1198770lt 120573119862

lt 120573119861 1205731198770


0gt 1


1198770 that is when 119877

0= 1


lt 120573119862



1205731198770 that is119877



1198770





119861lt 120573119862

lt 1205731198770

isfulfilled Here Δ



120573 lt 120573119861 119860 gt 0 119861 gt 0 119862 gt 0119863 gt 0


120573119861lt 120573 lt 120573

119862 119860 gt 0 119861 lt 0 119862 gt 0119863 gt 0


120573119862lt 120573 lt 120573

1198770119860 gt 0 119861 lt 0 119862 lt 0119863 gt 0


1205731198770





0



0


0= 1 and


119875 (119909) = 1198601199093+ 1198611199092+ 119862119909

= 119909 (1198601199092+ 1198611199092+ 119862) = 0

(22)



1and 119909

2



means 119891119861(1205731198770) lt 0 and




1198770lt 120573119862


1205731198770






times10minus3

119875(119909)

01

005

minus005

minus01

minus200 0 200 400 600119909


119861lt 1205731198770





119861lt 120573119862

lt 1205731198770

isfulfilled



For 120573119861

lt 120573 lt 120573119862and 120573

119862lt 120573 lt 120573





1lt 0







case we have

Δ1=

1

4

1198632

1198602+

1

27

1198623

1198603gt 0 (23)




Δ1(120573lowast) = 0 (24)


600

500

400

300

200

100

0

minus100

minus200

minus300

119909

1198770 gt 1

1205731198770120573119862120573119861 120573lowast

000005 00001 000015 00002120573


lt 120573119862

lt 1205731198770




(seeFigure 4)













1205731198770










119868050 (assumed)


119873020 (assumed)






parameters 1205731198770 120573119861 and 120573



1198770 120573119861 120573119862) then the system


1198770 120573119861 120573119862) then








1198770lt 120573119862

lt 120573119861 120575 = 09 120578 = 001) Let us


lt 120573119862lt 120573119861is











were

119878 (0) = 4980 119864 (0) = 0 119868119868 (0) = 20

119868119873 (0) = 0 119877 (0) = 0

(25)


1205731198770

= 00001450317354

120573119861= 001087387065

120573119862= 00002715343808

(26)


lt 120573119862

lt 120573119861


1198770


119878infin

= 1616 119877infin

= 4080 119868119873infin

= 103

119868119868infin

= 195 119864infin

= 1150

(27)





lt 120573119862lt 120573119861 120575 = 00 120578 = 09) For our



0= 3585422172 The used


119878 (0) = 4980 119864 (0) = 0 119868119868 (0) = 20

119868119873 (0) = 0 119877 (0) = 0

(28)

119905

7000

6000

5000

4000

3000

2000

1000

00 50 100 150 200

119878

119877

119868119873119864

119868119868 + 119868119873119868119868119873

119868119868 + 119868119873 + 119864 + 119877



4000

4000 5000

3000

3000

2000

2000

1000

10004000

6000

20000

119868 119868+119868 119873+119864

119878 119877 4000 500

3000000000000000000000000000000

3000

200202000020200202002020200000000200202020020020200200202020202002002000202002000020202020200020202000220000200202000020202222000222020200220000202022022220022002202222220022222 02 02222 0000000000000000000000000000000000000000000000000000000000000000

2000

100110010010010010010010010100100100100010010010010010010010010000010100100100100100100100100110000001001000000110010110010010010010001000001110100010000010000000001010000011000001110000000001001001000000000000001000010000111000101 000000000000000000000000000000000000000000000000000000000000000000000000000000000000000

10004000

6000

20000

119868119868119868119868119868119868119868119868119868119868119868119868119868119868119868119868119868119868119868119868119868119868119868119868119868119868119868119868119868119868119868119868119868119868119868119868119868119868119868119868 119868119868119868119868119868119868119868119868119868119868119868119868119868119868++++++++++++++++++++++++++119868 119873119868+119864

119878 119877


0=

3585422172 120575 = 09 120578 = 001 and 120573 = 000052


1205731198770

= 00001450317354

120573119861= 001226355348

120573119862= 00003132229272

(29)


lt 120573119861lt 120573119862 and as in



119878infin

= 1938 119877infin

= 974 119868119873infin

= 60

119868119868infin

= 156 119864infin

= 4530

(30)


119905

7000

6000

5000

4000

3000

2000

1000

00 50 100 150 200

119878

119877

119868119873119864

119868119868 + 119868119873119868119868119873

119868119868 + 119868119873 + 119864 + 119877




1198770lt 120573119862







119861lt 120573119862

lt 1205731198770 120575 = 30 120578 = 25) There


0=


1205731198770

= 00001450317354

120573119861= 00001066568066

120573119862= 00001225687204

(31)


1205731198770


119894 119868119899 119864)

1198751= (9009 0 0 0 0)

1198752= (8507 182 9 5 2166)

1198753= (3221 1406 285 103 1566)

(32)

800

700

600

500

400

300

200

100

07006005004003002001000

119868 119868

119905



= 0



11987531198751198751198751198751198751198751198751198751198751198751198751198751198751198751198751198751198751198751198751198751198751198751198751198751198751198751198751198751198751198751198751198751198751198751198751198751198751198751198751198751198751198751198751198751198751198751198751198751198751198751198751198751198751198751198751198751198751198751198751198751198751198751198751198751198751198751198751198751198751198751198751198751198753119875119875

16001200

800400

900080007000

600050004000300020000

1000

2000

3000

119868 119868+119868 119873+119864

1198781198771198752

1198751




3



119868infin= 0 or to 119868

119868infin=



3according




119868+ 119868119873+ 119864


lowast=


lowast the system


119861lt 1205731198770

lt 120573119862 120575 = 30 120578 = 25) Consider


119861lt 1205731198770



1205731198770

= 00001679568390

120573119862= 00001729256777

120573119861= 00001489092005

(33)










00001688612368 with 1205731198770



condition 1205731lt 120573 lt 120573

1198770




(119878 119877 119868119894 119868119899 119864)

1198750= (5148 0 0 0 0)

1198751= (3372 1041 122 60 482)

1198752= (2828 1283 190 88 651)

(34)



2is


119868infin= 0 or to 119868




2






lt 1205731198770


1205731198770

1205731

1205732

120573119862120573119861

000015 000016 000017 000018120573

600

500

400

300

200

100

0

119909


119861lt 1205731198770



2

500

400

300

200

100

0

119868 119868

0 500 1000 1500 2000119905



= 0

or 119868119868infin



(119878 119877 119868119894 119868119899 119864)

1198750= (5148 0 0 0 0)

1198751= (5042 76 5 3 20)

1198752= (3971 734 69 36 298)

1198753= (2491 1413 246 109 750)

(35)


160014001200800600400200

500

2000

0

1000

1500

1000

2000

3000

4000

5000

119878119877

11987521198751

1198750 160014001200800600444400420000002000200

5005005000

000

1000

1500

1000

20020020000000

300003003003003000000000000

400404004004004000040040040000000 119878119877

1198752119875119875

1198751119875119875

119868 119868+119868 119873+119877





2







gt








lowast= 120583 = 003885 120583

119905= 001520 119901 = 08

] = 00266 119891 = 08 119902 = 085 119908 = 0005 119888 = 04 1199031= 05





160014001200

800600400200

500

2000

0

1000

1500

1000

2000

3000

4000

5000

119868 119868+119868 119873+

119878119877

119877

1198752

1198751

1198753

1198750 160014001200

8006004004040200200200200200200

500

000

1000

1500

1000

2002002002002 000000

3000030033000000000000

4004004004004000040000400 00000

119868 119868119868+119868 119873119868+

119878119877

119877

1198752119875119875

1198751119875119875

1198753119875119875

0




0and







0is near one













5

5

4

4

3

3

2

2

1

10

0

120578

119861 gt 0 119862 gt 0

120575

119861 lt 0 119862 lt 0

119861 gt 0 119862 lt 0

(a) 1198770 = 1

5

5

4

4

3

3

2

2

1

10

0

120578

120575

(b) 1198770 = 1689504264

5

5

4

4

3

3

2

2

1

10

0

120578

120575

(c) 1198770 = 2379008527

5

5

4

4

3

3

2

2

1

10

0

120578

120575

(d) 1198770 = 7895042636



1198770= 00001450317354 (b) 120573 = 00002450317354 (c) 120573 = 00003450317354 and (d)

120573 = 0001145031735


1198770 120573119861 120573119862 and 120573 We divided



0has to do solely






5

5

4

4

3

3

2

2

1

10

0

120578

120575

Δ gt 0

119891(1199091) gt 0

119861 gt 0 119862 lt 0

119861 lt 0 119862 lt 0

119861 lt 0 119862 gt 0

119861 lt 0119862 gt 0

119861 gt 0 119862 gt 0

(a) 1198770 = 1

5

5

4

4

3

3

2

2

1

10

0

120578

120575

(b) 1198770 = 1004390340

5

5

4

4

3

3

2

2

1

10

0

120578

120575

(c) 1198770 = 1087806817

5

5

4

4

3

3

2

2

1

10

0

120578

120575

(d) 1198770 = 2317102260



1198770= 00002277727471 (b) 120573 = 00002287727471 (c)

120573 = 00002477727471 and (d) 120573 = 00005277727471


0gt 1






(1205731198770

lt 120573119862






0 the


0 Some



5

5

4

4

3

3

2

2

1

10

0

120578

120575

Δ gt 0

119891(1199091) gt 0

119861 gt 0 119862 lt 0

119861 lt 0 119862 lt 0

119861 lt 0 119862 gt 0

119861 lt 0119862 gt 0

119861 gt 0 119862 gt 0

(a) 1198770 = 09560965911

5

5

4

4

3

3

2

2

1

10

0

120578

120575

(b) 1198770 = 08902414781

5

5

4

4

3

3

2

2

1

10

0

120578

120575

(c) 1198770 = 07804829561

5

5

4

4

3

3

2

2

1

10

0

120578

120575

(d) 1198770 = 05609659131



00001777727471 and (d) 120573 = 00001277727471






0




0such that disease-











lt 120573119862lt 120573119861

120573 lt 1205731198770


lt 120573 lt 120573119862




120573119862lt 1205731198770

lt 120573119861

120573 lt 120573119862



2) le 0 or Δ

1lt 0 none if 119875(119909

2) ge 0 or Δ

1gt 0

1205731198770

lt 120573 lt 120573119861


1gt 0


120573119862lt 120573119861lt 1205731198770

120573 lt 120573119862



2) le 0 or Δ

1lt 0 none if 119875(119909

2) ge 0 or Δ

1gt 0

120573119861lt 120573 lt 120573


1lt 0) or none (Δ

1gt 0)

1205731198770


lt 120573119861lt 120573119862

120573 lt 1205731198770


lt 120573 lt 120573119861




1lt 0)


120573119861lt 1205731198770

lt 120573119862

120573 lt 120573119861



1lt 0) or none (Δ

1gt 0)

1205731198770

lt 120573 lt 120573119862


1lt 0)


120573119861lt 120573119862lt 1205731198770

120573 lt 120573119861



1lt 0) or none (Δ

1gt 0)

120573119862lt 120573 lt 120573


1lt 0) or none (Δ

1gt 0)

1205731198770



Appendices



119886 = 120583 + 120583119879+ 119888

119887 = 2119908 + 120583

ℎ = ] + 120583

119892 = 1199031+ 119888

119898 = 1199032+ 119888

120579 = 120583 + 120583119905

120598 = 1 minus 119901

1198711= 120579 + 119892 (1 minus 119902) + 119898119902

1198712= ]1199021199032+ ℎ119886 + 119903

1] (1 minus 119902) + 119903

1120583

1198701= 1199032119908 + 119887119886 + 119903

1(119908 + 120583)

1198702= ]1199021199032+ ℎ119886 + 119903

1] (1 minus 119902) + 119903

1120583

(A1)

We have

1198611= (120579 (119891119901 + (1 minus 119901) 119902) + 119898119902)Π120575120590

1198612= 120590 (120583119871

1120579 120575 + 119871

2120579 + 119898120583 (119886 + 119903

1))

+ (1199032119908120579 + 119903

1(119908 + 120583) 120579

+ 119887119886120579 + 119898120583 (119886 + 1199031)) 120575


1198621= Π (120575 ((119891119901 + 120598119902) (119887120583119905 + (119898 + 119887) 120583) + 119898119908)

+ 120590 ((]119902120598 + 119891ℎ119901) 120579 + 119898 (120583119891119901 + ] 119902)))

1198622= 120583 (119870

1120579 + 119898120583 (119886 + 119903

1)) 120575 + 120583 (119870

2120579 + 119898120583 (119886 + 119903

1)) 120590

+ ℎ (1198701120579 + 119898120583 (119886 + 119903

1))

(A2)


119861 = 1205732119891119861(120573)

119862 = 120573119891119862(120573)

(A3)

where119891119861(120573) = minus119861

1120573 + 1198612

119891119862(120573) = minus119862

1120573 + 119862

2

119860 = 1205733120575120590 (120583 + 120583

119879) (120583 + 120583

119879+ 119888 + (1 minus 119902) 119903

1+ 1199021199032) gt 0

119863 = ℎ ([119886119887 + 1199081199032+ 1199031(119908 + 120583)] (120583 + 120583119905)

+ 120583119898 (1199031+ 119886)) (1 minus 119877

0)

(A4)





Acknowledgments


References
















































of






Disease Markers



OncologyJournal of





PPAR Research



Journal of

ObesityJournal of

















and 1198751015840(119909) ge 0 for all 119909 ge 119909

2and 119909 le 119909

1 So we need to


2in the real line


have 1199091lt 0 119909

2gt 0 and 119875

1015840(119909) gt 0 for all 119909 ge 119909

2ge 0



2are



2


1) ge 0


119861(120573119861) = 0 and 120573

119862the


1198770be the value






(Table 3)


1198770lt 120573119862lt 120573119861 that



For 1205731198770



For 120573119862



Finally for 120573 gt 120573119861





lt 120573119862lt 120573119861


119861(120573119862) gt 0 and 119863(120573

119861) gt 0


lt 120573119862lt 120573119861


lt 120573119862

lt 120573119861



1198770lt 120573119862

lt 120573119861is

fulfilled


120573 lt 1205731198770


1205731198770

lt 120573 lt 120573119862 119860 gt 0 119861 gt 0 119862 gt 0119863 lt 0


120573119862lt 120573 lt 120573


120573 gt 120573119861 119860 gt 0 119861 lt 0 119862 lt 0119863 lt 0


800

600

400

200

0

minus200

minus400

119909

1198770 gt 1

1205731198770120573119862 120573119861

00002 00004 00006 00008 0001120573


1198770lt 120573119862

lt 120573119861 1205731198770


0gt 1


1198770 that is when 119877

0= 1


lt 120573119862



1205731198770 that is119877



1198770





119861lt 120573119862

lt 1205731198770

isfulfilled Here Δ



120573 lt 120573119861 119860 gt 0 119861 gt 0 119862 gt 0119863 gt 0


120573119861lt 120573 lt 120573

119862 119860 gt 0 119861 lt 0 119862 gt 0119863 gt 0


120573119862lt 120573 lt 120573

1198770119860 gt 0 119861 lt 0 119862 lt 0119863 gt 0


1205731198770





0



0


0= 1 and


119875 (119909) = 1198601199093+ 1198611199092+ 119862119909

= 119909 (1198601199092+ 1198611199092+ 119862) = 0

(22)



1and 119909

2



means 119891119861(1205731198770) lt 0 and




1198770lt 120573119862


1205731198770






times10minus3

119875(119909)

01

005

minus005

minus01

minus200 0 200 400 600119909


119861lt 1205731198770





119861lt 120573119862

lt 1205731198770

isfulfilled



For 120573119861

lt 120573 lt 120573119862and 120573

119862lt 120573 lt 120573





1lt 0







case we have

Δ1=

1

4

1198632

1198602+

1

27

1198623

1198603gt 0 (23)




Δ1(120573lowast) = 0 (24)


600

500

400

300

200

100

0

minus100

minus200

minus300

119909

1198770 gt 1

1205731198770120573119862120573119861 120573lowast

000005 00001 000015 00002120573


lt 120573119862

lt 1205731198770




(seeFigure 4)













1205731198770










119868050 (assumed)


119873020 (assumed)






parameters 1205731198770 120573119861 and 120573



1198770 120573119861 120573119862) then the system


1198770 120573119861 120573119862) then








1198770lt 120573119862

lt 120573119861 120575 = 09 120578 = 001) Let us


lt 120573119862lt 120573119861is











were

119878 (0) = 4980 119864 (0) = 0 119868119868 (0) = 20

119868119873 (0) = 0 119877 (0) = 0

(25)


1205731198770

= 00001450317354

120573119861= 001087387065

120573119862= 00002715343808

(26)


lt 120573119862

lt 120573119861


1198770


119878infin

= 1616 119877infin

= 4080 119868119873infin

= 103

119868119868infin

= 195 119864infin

= 1150

(27)





lt 120573119862lt 120573119861 120575 = 00 120578 = 09) For our



0= 3585422172 The used


119878 (0) = 4980 119864 (0) = 0 119868119868 (0) = 20

119868119873 (0) = 0 119877 (0) = 0

(28)

119905

7000

6000

5000

4000

3000

2000

1000

00 50 100 150 200

119878

119877

119868119873119864

119868119868 + 119868119873119868119868119873

119868119868 + 119868119873 + 119864 + 119877



4000

4000 5000

3000

3000

2000

2000

1000

10004000

6000

20000

119868 119868+119868 119873+119864

119878 119877 4000 500

3000000000000000000000000000000

3000

200202000020200202002020200000000200202020020020200200202020202002002000202002000020202020200020202000220000200202000020202222000222020200220000202022022220022002202222220022222 02 02222 0000000000000000000000000000000000000000000000000000000000000000

2000

100110010010010010010010010100100100100010010010010010010010010000010100100100100100100100100110000001001000000110010110010010010010001000001110100010000010000000001010000011000001110000000001001001000000000000001000010000111000101 000000000000000000000000000000000000000000000000000000000000000000000000000000000000000

10004000

6000

20000

119868119868119868119868119868119868119868119868119868119868119868119868119868119868119868119868119868119868119868119868119868119868119868119868119868119868119868119868119868119868119868119868119868119868119868119868119868119868119868119868 119868119868119868119868119868119868119868119868119868119868119868119868119868119868++++++++++++++++++++++++++119868 119873119868+119864

119878 119877


0=

3585422172 120575 = 09 120578 = 001 and 120573 = 000052


1205731198770

= 00001450317354

120573119861= 001226355348

120573119862= 00003132229272

(29)


lt 120573119861lt 120573119862 and as in



119878infin

= 1938 119877infin

= 974 119868119873infin

= 60

119868119868infin

= 156 119864infin

= 4530

(30)


119905

7000

6000

5000

4000

3000

2000

1000

00 50 100 150 200

119878

119877

119868119873119864

119868119868 + 119868119873119868119868119873

119868119868 + 119868119873 + 119864 + 119877




1198770lt 120573119862







119861lt 120573119862

lt 1205731198770 120575 = 30 120578 = 25) There


0=


1205731198770

= 00001450317354

120573119861= 00001066568066

120573119862= 00001225687204

(31)


1205731198770


119894 119868119899 119864)

1198751= (9009 0 0 0 0)

1198752= (8507 182 9 5 2166)

1198753= (3221 1406 285 103 1566)

(32)

800

700

600

500

400

300

200

100

07006005004003002001000

119868 119868

119905



= 0



11987531198751198751198751198751198751198751198751198751198751198751198751198751198751198751198751198751198751198751198751198751198751198751198751198751198751198751198751198751198751198751198751198751198751198751198751198751198751198751198751198751198751198751198751198751198751198751198751198751198751198751198751198751198751198751198751198751198751198751198751198751198751198751198751198751198751198751198751198751198751198751198751198751198753119875119875

16001200

800400

900080007000

600050004000300020000

1000

2000

3000

119868 119868+119868 119873+119864

1198781198771198752

1198751




3



119868infin= 0 or to 119868

119868infin=



3according




119868+ 119868119873+ 119864


lowast=


lowast the system


119861lt 1205731198770

lt 120573119862 120575 = 30 120578 = 25) Consider


119861lt 1205731198770



1205731198770

= 00001679568390

120573119862= 00001729256777

120573119861= 00001489092005

(33)










00001688612368 with 1205731198770



condition 1205731lt 120573 lt 120573

1198770




(119878 119877 119868119894 119868119899 119864)

1198750= (5148 0 0 0 0)

1198751= (3372 1041 122 60 482)

1198752= (2828 1283 190 88 651)

(34)



2is


119868infin= 0 or to 119868




2






lt 1205731198770


1205731198770

1205731

1205732

120573119862120573119861

000015 000016 000017 000018120573

600

500

400

300

200

100

0

119909


119861lt 1205731198770



2

500

400

300

200

100

0

119868 119868

0 500 1000 1500 2000119905



= 0

or 119868119868infin



(119878 119877 119868119894 119868119899 119864)

1198750= (5148 0 0 0 0)

1198751= (5042 76 5 3 20)

1198752= (3971 734 69 36 298)

1198753= (2491 1413 246 109 750)

(35)


160014001200800600400200

500

2000

0

1000

1500

1000

2000

3000

4000

5000

119878119877

11987521198751

1198750 160014001200800600444400420000002000200

5005005000

000

1000

1500

1000

20020020000000

300003003003003000000000000

400404004004004000040040040000000 119878119877

1198752119875119875

1198751119875119875

119868 119868+119868 119873+119877





2







gt








lowast= 120583 = 003885 120583

119905= 001520 119901 = 08

] = 00266 119891 = 08 119902 = 085 119908 = 0005 119888 = 04 1199031= 05





160014001200

800600400200

500

2000

0

1000

1500

1000

2000

3000

4000

5000

119868 119868+119868 119873+

119878119877

119877

1198752

1198751

1198753

1198750 160014001200

8006004004040200200200200200200

500

000

1000

1500

1000

2002002002002 000000

3000030033000000000000

4004004004004000040000400 00000

119868 119868119868+119868 119873119868+

119878119877

119877

1198752119875119875

1198751119875119875

1198753119875119875

0




0and







0is near one













5

5

4

4

3

3

2

2

1

10

0

120578

119861 gt 0 119862 gt 0

120575

119861 lt 0 119862 lt 0

119861 gt 0 119862 lt 0

(a) 1198770 = 1

5

5

4

4

3

3

2

2

1

10

0

120578

120575

(b) 1198770 = 1689504264

5

5

4

4

3

3

2

2

1

10

0

120578

120575

(c) 1198770 = 2379008527

5

5

4

4

3

3

2

2

1

10

0

120578

120575

(d) 1198770 = 7895042636



1198770= 00001450317354 (b) 120573 = 00002450317354 (c) 120573 = 00003450317354 and (d)

120573 = 0001145031735


1198770 120573119861 120573119862 and 120573 We divided



0has to do solely






5

5

4

4

3

3

2

2

1

10

0

120578

120575

Δ gt 0

119891(1199091) gt 0

119861 gt 0 119862 lt 0

119861 lt 0 119862 lt 0

119861 lt 0 119862 gt 0

119861 lt 0119862 gt 0

119861 gt 0 119862 gt 0

(a) 1198770 = 1

5

5

4

4

3

3

2

2

1

10

0

120578

120575

(b) 1198770 = 1004390340

5

5

4

4

3

3

2

2

1

10

0

120578

120575

(c) 1198770 = 1087806817

5

5

4

4

3

3

2

2

1

10

0

120578

120575

(d) 1198770 = 2317102260



1198770= 00002277727471 (b) 120573 = 00002287727471 (c)

120573 = 00002477727471 and (d) 120573 = 00005277727471


0gt 1






(1205731198770

lt 120573119862






0 the


0 Some



5

5

4

4

3

3

2

2

1

10

0

120578

120575

Δ gt 0

119891(1199091) gt 0

119861 gt 0 119862 lt 0

119861 lt 0 119862 lt 0

119861 lt 0 119862 gt 0

119861 lt 0119862 gt 0

119861 gt 0 119862 gt 0

(a) 1198770 = 09560965911

5

5

4

4

3

3

2

2

1

10

0

120578

120575

(b) 1198770 = 08902414781

5

5

4

4

3

3

2

2

1

10

0

120578

120575

(c) 1198770 = 07804829561

5

5

4

4

3

3

2

2

1

10

0

120578

120575

(d) 1198770 = 05609659131



00001777727471 and (d) 120573 = 00001277727471






0




0such that disease-











lt 120573119862lt 120573119861

120573 lt 1205731198770


lt 120573 lt 120573119862




120573119862lt 1205731198770

lt 120573119861

120573 lt 120573119862



2) le 0 or Δ

1lt 0 none if 119875(119909

2) ge 0 or Δ

1gt 0

1205731198770

lt 120573 lt 120573119861


1gt 0


120573119862lt 120573119861lt 1205731198770

120573 lt 120573119862



2) le 0 or Δ

1lt 0 none if 119875(119909

2) ge 0 or Δ

1gt 0

120573119861lt 120573 lt 120573


1lt 0) or none (Δ

1gt 0)

1205731198770


lt 120573119861lt 120573119862

120573 lt 1205731198770


lt 120573 lt 120573119861




1lt 0)


120573119861lt 1205731198770

lt 120573119862

120573 lt 120573119861



1lt 0) or none (Δ

1gt 0)

1205731198770

lt 120573 lt 120573119862


1lt 0)


120573119861lt 120573119862lt 1205731198770

120573 lt 120573119861



1lt 0) or none (Δ

1gt 0)

120573119862lt 120573 lt 120573


1lt 0) or none (Δ

1gt 0)

1205731198770



Appendices



119886 = 120583 + 120583119879+ 119888

119887 = 2119908 + 120583

ℎ = ] + 120583

119892 = 1199031+ 119888

119898 = 1199032+ 119888

120579 = 120583 + 120583119905

120598 = 1 minus 119901

1198711= 120579 + 119892 (1 minus 119902) + 119898119902

1198712= ]1199021199032+ ℎ119886 + 119903

1] (1 minus 119902) + 119903

1120583

1198701= 1199032119908 + 119887119886 + 119903

1(119908 + 120583)

1198702= ]1199021199032+ ℎ119886 + 119903

1] (1 minus 119902) + 119903

1120583

(A1)

We have

1198611= (120579 (119891119901 + (1 minus 119901) 119902) + 119898119902)Π120575120590

1198612= 120590 (120583119871

1120579 120575 + 119871

2120579 + 119898120583 (119886 + 119903

1))

+ (1199032119908120579 + 119903

1(119908 + 120583) 120579

+ 119887119886120579 + 119898120583 (119886 + 1199031)) 120575


1198621= Π (120575 ((119891119901 + 120598119902) (119887120583119905 + (119898 + 119887) 120583) + 119898119908)

+ 120590 ((]119902120598 + 119891ℎ119901) 120579 + 119898 (120583119891119901 + ] 119902)))

1198622= 120583 (119870

1120579 + 119898120583 (119886 + 119903

1)) 120575 + 120583 (119870

2120579 + 119898120583 (119886 + 119903

1)) 120590

+ ℎ (1198701120579 + 119898120583 (119886 + 119903

1))

(A2)


119861 = 1205732119891119861(120573)

119862 = 120573119891119862(120573)

(A3)

where119891119861(120573) = minus119861

1120573 + 1198612

119891119862(120573) = minus119862

1120573 + 119862

2

119860 = 1205733120575120590 (120583 + 120583

119879) (120583 + 120583

119879+ 119888 + (1 minus 119902) 119903

1+ 1199021199032) gt 0

119863 = ℎ ([119886119887 + 1199081199032+ 1199031(119908 + 120583)] (120583 + 120583119905)

+ 120583119898 (1199031+ 119886)) (1 minus 119877

0)

(A4)





Acknowledgments


References
















































of






Disease Markers



OncologyJournal of





PPAR Research



Journal of

ObesityJournal of


















119861lt 120573119862

lt 1205731198770

isfulfilled Here Δ



120573 lt 120573119861 119860 gt 0 119861 gt 0 119862 gt 0119863 gt 0


120573119861lt 120573 lt 120573

119862 119860 gt 0 119861 lt 0 119862 gt 0119863 gt 0


120573119862lt 120573 lt 120573

1198770119860 gt 0 119861 lt 0 119862 lt 0119863 gt 0


1205731198770





0



0


0= 1 and


119875 (119909) = 1198601199093+ 1198611199092+ 119862119909

= 119909 (1198601199092+ 1198611199092+ 119862) = 0

(22)



1and 119909

2



means 119891119861(1205731198770) lt 0 and




1198770lt 120573119862


1205731198770






times10minus3

119875(119909)

01

005

minus005

minus01

minus200 0 200 400 600119909


119861lt 1205731198770





119861lt 120573119862

lt 1205731198770

isfulfilled



For 120573119861

lt 120573 lt 120573119862and 120573

119862lt 120573 lt 120573





1lt 0







case we have

Δ1=

1

4

1198632

1198602+

1

27

1198623

1198603gt 0 (23)




Δ1(120573lowast) = 0 (24)


600

500

400

300

200

100

0

minus100

minus200

minus300

119909

1198770 gt 1

1205731198770120573119862120573119861 120573lowast

000005 00001 000015 00002120573


lt 120573119862

lt 1205731198770




(seeFigure 4)













1205731198770










119868050 (assumed)


119873020 (assumed)






parameters 1205731198770 120573119861 and 120573



1198770 120573119861 120573119862) then the system


1198770 120573119861 120573119862) then








1198770lt 120573119862

lt 120573119861 120575 = 09 120578 = 001) Let us


lt 120573119862lt 120573119861is











were

119878 (0) = 4980 119864 (0) = 0 119868119868 (0) = 20

119868119873 (0) = 0 119877 (0) = 0

(25)


1205731198770

= 00001450317354

120573119861= 001087387065

120573119862= 00002715343808

(26)


lt 120573119862

lt 120573119861


1198770


119878infin

= 1616 119877infin

= 4080 119868119873infin

= 103

119868119868infin

= 195 119864infin

= 1150

(27)





lt 120573119862lt 120573119861 120575 = 00 120578 = 09) For our



0= 3585422172 The used


119878 (0) = 4980 119864 (0) = 0 119868119868 (0) = 20

119868119873 (0) = 0 119877 (0) = 0

(28)

119905

7000

6000

5000

4000

3000

2000

1000

00 50 100 150 200

119878

119877

119868119873119864

119868119868 + 119868119873119868119868119873

119868119868 + 119868119873 + 119864 + 119877



4000

4000 5000

3000

3000

2000

2000

1000

10004000

6000

20000

119868 119868+119868 119873+119864

119878 119877 4000 500

3000000000000000000000000000000

3000

200202000020200202002020200000000200202020020020200200202020202002002000202002000020202020200020202000220000200202000020202222000222020200220000202022022220022002202222220022222 02 02222 0000000000000000000000000000000000000000000000000000000000000000

2000

100110010010010010010010010100100100100010010010010010010010010000010100100100100100100100100110000001001000000110010110010010010010001000001110100010000010000000001010000011000001110000000001001001000000000000001000010000111000101 000000000000000000000000000000000000000000000000000000000000000000000000000000000000000

10004000

6000

20000

119868119868119868119868119868119868119868119868119868119868119868119868119868119868119868119868119868119868119868119868119868119868119868119868119868119868119868119868119868119868119868119868119868119868119868119868119868119868119868119868 119868119868119868119868119868119868119868119868119868119868119868119868119868119868++++++++++++++++++++++++++119868 119873119868+119864

119878 119877


0=

3585422172 120575 = 09 120578 = 001 and 120573 = 000052


1205731198770

= 00001450317354

120573119861= 001226355348

120573119862= 00003132229272

(29)


lt 120573119861lt 120573119862 and as in



119878infin

= 1938 119877infin

= 974 119868119873infin

= 60

119868119868infin

= 156 119864infin

= 4530

(30)


119905

7000

6000

5000

4000

3000

2000

1000

00 50 100 150 200

119878

119877

119868119873119864

119868119868 + 119868119873119868119868119873

119868119868 + 119868119873 + 119864 + 119877




1198770lt 120573119862







119861lt 120573119862

lt 1205731198770 120575 = 30 120578 = 25) There


0=


1205731198770

= 00001450317354

120573119861= 00001066568066

120573119862= 00001225687204

(31)


1205731198770


119894 119868119899 119864)

1198751= (9009 0 0 0 0)

1198752= (8507 182 9 5 2166)

1198753= (3221 1406 285 103 1566)

(32)

800

700

600

500

400

300

200

100

07006005004003002001000

119868 119868

119905



= 0



11987531198751198751198751198751198751198751198751198751198751198751198751198751198751198751198751198751198751198751198751198751198751198751198751198751198751198751198751198751198751198751198751198751198751198751198751198751198751198751198751198751198751198751198751198751198751198751198751198751198751198751198751198751198751198751198751198751198751198751198751198751198751198751198751198751198751198751198751198751198751198751198751198751198753119875119875

16001200

800400

900080007000

600050004000300020000

1000

2000

3000

119868 119868+119868 119873+119864

1198781198771198752

1198751




3



119868infin= 0 or to 119868

119868infin=



3according




119868+ 119868119873+ 119864


lowast=


lowast the system


119861lt 1205731198770

lt 120573119862 120575 = 30 120578 = 25) Consider


119861lt 1205731198770



1205731198770

= 00001679568390

120573119862= 00001729256777

120573119861= 00001489092005

(33)










00001688612368 with 1205731198770



condition 1205731lt 120573 lt 120573

1198770




(119878 119877 119868119894 119868119899 119864)

1198750= (5148 0 0 0 0)

1198751= (3372 1041 122 60 482)

1198752= (2828 1283 190 88 651)

(34)



2is


119868infin= 0 or to 119868




2






lt 1205731198770


1205731198770

1205731

1205732

120573119862120573119861

000015 000016 000017 000018120573

600

500

400

300

200

100

0

119909


119861lt 1205731198770



2

500

400

300

200

100

0

119868 119868

0 500 1000 1500 2000119905



= 0

or 119868119868infin



(119878 119877 119868119894 119868119899 119864)

1198750= (5148 0 0 0 0)

1198751= (5042 76 5 3 20)

1198752= (3971 734 69 36 298)

1198753= (2491 1413 246 109 750)

(35)


160014001200800600400200

500

2000

0

1000

1500

1000

2000

3000

4000

5000

119878119877

11987521198751

1198750 160014001200800600444400420000002000200

5005005000

000

1000

1500

1000

20020020000000

300003003003003000000000000

400404004004004000040040040000000 119878119877

1198752119875119875

1198751119875119875

119868 119868+119868 119873+119877





2







gt








lowast= 120583 = 003885 120583

119905= 001520 119901 = 08

] = 00266 119891 = 08 119902 = 085 119908 = 0005 119888 = 04 1199031= 05





160014001200

800600400200

500

2000

0

1000

1500

1000

2000

3000

4000

5000

119868 119868+119868 119873+

119878119877

119877

1198752

1198751

1198753

1198750 160014001200

8006004004040200200200200200200

500

000

1000

1500

1000

2002002002002 000000

3000030033000000000000

4004004004004000040000400 00000

119868 119868119868+119868 119873119868+

119878119877

119877

1198752119875119875

1198751119875119875

1198753119875119875

0




0and







0is near one













5

5

4

4

3

3

2

2

1

10

0

120578

119861 gt 0 119862 gt 0

120575

119861 lt 0 119862 lt 0

119861 gt 0 119862 lt 0

(a) 1198770 = 1

5

5

4

4

3

3

2

2

1

10

0

120578

120575

(b) 1198770 = 1689504264

5

5

4

4

3

3

2

2

1

10

0

120578

120575

(c) 1198770 = 2379008527

5

5

4

4

3

3

2

2

1

10

0

120578

120575

(d) 1198770 = 7895042636



1198770= 00001450317354 (b) 120573 = 00002450317354 (c) 120573 = 00003450317354 and (d)

120573 = 0001145031735


1198770 120573119861 120573119862 and 120573 We divided



0has to do solely






5

5

4

4

3

3

2

2

1

10

0

120578

120575

Δ gt 0

119891(1199091) gt 0

119861 gt 0 119862 lt 0

119861 lt 0 119862 lt 0

119861 lt 0 119862 gt 0

119861 lt 0119862 gt 0

119861 gt 0 119862 gt 0

(a) 1198770 = 1

5

5

4

4

3

3

2

2

1

10

0

120578

120575

(b) 1198770 = 1004390340

5

5

4

4

3

3

2

2

1

10

0

120578

120575

(c) 1198770 = 1087806817

5

5

4

4

3

3

2

2

1

10

0

120578

120575

(d) 1198770 = 2317102260



1198770= 00002277727471 (b) 120573 = 00002287727471 (c)

120573 = 00002477727471 and (d) 120573 = 00005277727471


0gt 1






(1205731198770

lt 120573119862






0 the


0 Some



5

5

4

4

3

3

2

2

1

10

0

120578

120575

Δ gt 0

119891(1199091) gt 0

119861 gt 0 119862 lt 0

119861 lt 0 119862 lt 0

119861 lt 0 119862 gt 0

119861 lt 0119862 gt 0

119861 gt 0 119862 gt 0

(a) 1198770 = 09560965911

5

5

4

4

3

3

2

2

1

10

0

120578

120575

(b) 1198770 = 08902414781

5

5

4

4

3

3

2

2

1

10

0

120578

120575

(c) 1198770 = 07804829561

5

5

4

4

3

3

2

2

1

10

0

120578

120575

(d) 1198770 = 05609659131



00001777727471 and (d) 120573 = 00001277727471






0




0such that disease-











lt 120573119862lt 120573119861

120573 lt 1205731198770


lt 120573 lt 120573119862




120573119862lt 1205731198770

lt 120573119861

120573 lt 120573119862



2) le 0 or Δ

1lt 0 none if 119875(119909

2) ge 0 or Δ

1gt 0

1205731198770

lt 120573 lt 120573119861


1gt 0


120573119862lt 120573119861lt 1205731198770

120573 lt 120573119862



2) le 0 or Δ

1lt 0 none if 119875(119909

2) ge 0 or Δ

1gt 0

120573119861lt 120573 lt 120573


1lt 0) or none (Δ

1gt 0)

1205731198770


lt 120573119861lt 120573119862

120573 lt 1205731198770


lt 120573 lt 120573119861




1lt 0)


120573119861lt 1205731198770

lt 120573119862

120573 lt 120573119861



1lt 0) or none (Δ

1gt 0)

1205731198770

lt 120573 lt 120573119862


1lt 0)


120573119861lt 120573119862lt 1205731198770

120573 lt 120573119861



1lt 0) or none (Δ

1gt 0)

120573119862lt 120573 lt 120573


1lt 0) or none (Δ

1gt 0)

1205731198770



Appendices



119886 = 120583 + 120583119879+ 119888

119887 = 2119908 + 120583

ℎ = ] + 120583

119892 = 1199031+ 119888

119898 = 1199032+ 119888

120579 = 120583 + 120583119905

120598 = 1 minus 119901

1198711= 120579 + 119892 (1 minus 119902) + 119898119902

1198712= ]1199021199032+ ℎ119886 + 119903

1] (1 minus 119902) + 119903

1120583

1198701= 1199032119908 + 119887119886 + 119903

1(119908 + 120583)

1198702= ]1199021199032+ ℎ119886 + 119903

1] (1 minus 119902) + 119903

1120583

(A1)

We have

1198611= (120579 (119891119901 + (1 minus 119901) 119902) + 119898119902)Π120575120590

1198612= 120590 (120583119871

1120579 120575 + 119871

2120579 + 119898120583 (119886 + 119903

1))

+ (1199032119908120579 + 119903

1(119908 + 120583) 120579

+ 119887119886120579 + 119898120583 (119886 + 1199031)) 120575


1198621= Π (120575 ((119891119901 + 120598119902) (119887120583119905 + (119898 + 119887) 120583) + 119898119908)

+ 120590 ((]119902120598 + 119891ℎ119901) 120579 + 119898 (120583119891119901 + ] 119902)))

1198622= 120583 (119870

1120579 + 119898120583 (119886 + 119903

1)) 120575 + 120583 (119870

2120579 + 119898120583 (119886 + 119903

1)) 120590

+ ℎ (1198701120579 + 119898120583 (119886 + 119903

1))

(A2)


119861 = 1205732119891119861(120573)

119862 = 120573119891119862(120573)

(A3)

where119891119861(120573) = minus119861

1120573 + 1198612

119891119862(120573) = minus119862

1120573 + 119862

2

119860 = 1205733120575120590 (120583 + 120583

119879) (120583 + 120583

119879+ 119888 + (1 minus 119902) 119903

1+ 1199021199032) gt 0

119863 = ℎ ([119886119887 + 1199081199032+ 1199031(119908 + 120583)] (120583 + 120583119905)

+ 120583119898 (1199031+ 119886)) (1 minus 119877

0)

(A4)





Acknowledgments


References
















































of






Disease Markers



OncologyJournal of





PPAR Research



Journal of

ObesityJournal of

















600

500

400

300

200

100

0

minus100

minus200

minus300

119909

1198770 gt 1

1205731198770120573119862120573119861 120573lowast

000005 00001 000015 00002120573


lt 120573119862

lt 1205731198770




(seeFigure 4)













1205731198770










119868050 (assumed)


119873020 (assumed)






parameters 1205731198770 120573119861 and 120573



1198770 120573119861 120573119862) then the system


1198770 120573119861 120573119862) then








1198770lt 120573119862

lt 120573119861 120575 = 09 120578 = 001) Let us


lt 120573119862lt 120573119861is











were

119878 (0) = 4980 119864 (0) = 0 119868119868 (0) = 20

119868119873 (0) = 0 119877 (0) = 0

(25)


1205731198770

= 00001450317354

120573119861= 001087387065

120573119862= 00002715343808

(26)


lt 120573119862

lt 120573119861


1198770


119878infin

= 1616 119877infin

= 4080 119868119873infin

= 103

119868119868infin

= 195 119864infin

= 1150

(27)





lt 120573119862lt 120573119861 120575 = 00 120578 = 09) For our



0= 3585422172 The used


119878 (0) = 4980 119864 (0) = 0 119868119868 (0) = 20

119868119873 (0) = 0 119877 (0) = 0

(28)

119905

7000

6000

5000

4000

3000

2000

1000

00 50 100 150 200

119878

119877

119868119873119864

119868119868 + 119868119873119868119868119873

119868119868 + 119868119873 + 119864 + 119877



4000

4000 5000

3000

3000

2000

2000

1000

10004000

6000

20000

119868 119868+119868 119873+119864

119878 119877 4000 500

3000000000000000000000000000000

3000

200202000020200202002020200000000200202020020020200200202020202002002000202002000020202020200020202000220000200202000020202222000222020200220000202022022220022002202222220022222 02 02222 0000000000000000000000000000000000000000000000000000000000000000

2000

100110010010010010010010010100100100100010010010010010010010010000010100100100100100100100100110000001001000000110010110010010010010001000001110100010000010000000001010000011000001110000000001001001000000000000001000010000111000101 000000000000000000000000000000000000000000000000000000000000000000000000000000000000000

10004000

6000

20000

119868119868119868119868119868119868119868119868119868119868119868119868119868119868119868119868119868119868119868119868119868119868119868119868119868119868119868119868119868119868119868119868119868119868119868119868119868119868119868119868 119868119868119868119868119868119868119868119868119868119868119868119868119868119868++++++++++++++++++++++++++119868 119873119868+119864

119878 119877


0=

3585422172 120575 = 09 120578 = 001 and 120573 = 000052


1205731198770

= 00001450317354

120573119861= 001226355348

120573119862= 00003132229272

(29)


lt 120573119861lt 120573119862 and as in



119878infin

= 1938 119877infin

= 974 119868119873infin

= 60

119868119868infin

= 156 119864infin

= 4530

(30)


119905

7000

6000

5000

4000

3000

2000

1000

00 50 100 150 200

119878

119877

119868119873119864

119868119868 + 119868119873119868119868119873

119868119868 + 119868119873 + 119864 + 119877




1198770lt 120573119862







119861lt 120573119862

lt 1205731198770 120575 = 30 120578 = 25) There


0=


1205731198770

= 00001450317354

120573119861= 00001066568066

120573119862= 00001225687204

(31)


1205731198770


119894 119868119899 119864)

1198751= (9009 0 0 0 0)

1198752= (8507 182 9 5 2166)

1198753= (3221 1406 285 103 1566)

(32)

800

700

600

500

400

300

200

100

07006005004003002001000

119868 119868

119905



= 0



11987531198751198751198751198751198751198751198751198751198751198751198751198751198751198751198751198751198751198751198751198751198751198751198751198751198751198751198751198751198751198751198751198751198751198751198751198751198751198751198751198751198751198751198751198751198751198751198751198751198751198751198751198751198751198751198751198751198751198751198751198751198751198751198751198751198751198751198751198751198751198751198751198751198753119875119875

16001200

800400

900080007000

600050004000300020000

1000

2000

3000

119868 119868+119868 119873+119864

1198781198771198752

1198751




3



119868infin= 0 or to 119868

119868infin=



3according




119868+ 119868119873+ 119864


lowast=


lowast the system


119861lt 1205731198770

lt 120573119862 120575 = 30 120578 = 25) Consider


119861lt 1205731198770



1205731198770

= 00001679568390

120573119862= 00001729256777

120573119861= 00001489092005

(33)










00001688612368 with 1205731198770



condition 1205731lt 120573 lt 120573

1198770




(119878 119877 119868119894 119868119899 119864)

1198750= (5148 0 0 0 0)

1198751= (3372 1041 122 60 482)

1198752= (2828 1283 190 88 651)

(34)



2is


119868infin= 0 or to 119868




2






lt 1205731198770


1205731198770

1205731

1205732

120573119862120573119861

000015 000016 000017 000018120573

600

500

400

300

200

100

0

119909


119861lt 1205731198770



2

500

400

300

200

100

0

119868 119868

0 500 1000 1500 2000119905



= 0

or 119868119868infin



(119878 119877 119868119894 119868119899 119864)

1198750= (5148 0 0 0 0)

1198751= (5042 76 5 3 20)

1198752= (3971 734 69 36 298)

1198753= (2491 1413 246 109 750)

(35)


160014001200800600400200

500

2000

0

1000

1500

1000

2000

3000

4000

5000

119878119877

11987521198751

1198750 160014001200800600444400420000002000200

5005005000

000

1000

1500

1000

20020020000000

300003003003003000000000000

400404004004004000040040040000000 119878119877

1198752119875119875

1198751119875119875

119868 119868+119868 119873+119877





2







gt








lowast= 120583 = 003885 120583

119905= 001520 119901 = 08

] = 00266 119891 = 08 119902 = 085 119908 = 0005 119888 = 04 1199031= 05





160014001200

800600400200

500

2000

0

1000

1500

1000

2000

3000

4000

5000

119868 119868+119868 119873+

119878119877

119877

1198752

1198751

1198753

1198750 160014001200

8006004004040200200200200200200

500

000

1000

1500

1000

2002002002002 000000

3000030033000000000000

4004004004004000040000400 00000

119868 119868119868+119868 119873119868+

119878119877

119877

1198752119875119875

1198751119875119875

1198753119875119875

0




0and







0is near one













5

5

4

4

3

3

2

2

1

10

0

120578

119861 gt 0 119862 gt 0

120575

119861 lt 0 119862 lt 0

119861 gt 0 119862 lt 0

(a) 1198770 = 1

5

5

4

4

3

3

2

2

1

10

0

120578

120575

(b) 1198770 = 1689504264

5

5

4

4

3

3

2

2

1

10

0

120578

120575

(c) 1198770 = 2379008527

5

5

4

4

3

3

2

2

1

10

0

120578

120575

(d) 1198770 = 7895042636



1198770= 00001450317354 (b) 120573 = 00002450317354 (c) 120573 = 00003450317354 and (d)

120573 = 0001145031735


1198770 120573119861 120573119862 and 120573 We divided



0has to do solely






5

5

4

4

3

3

2

2

1

10

0

120578

120575

Δ gt 0

119891(1199091) gt 0

119861 gt 0 119862 lt 0

119861 lt 0 119862 lt 0

119861 lt 0 119862 gt 0

119861 lt 0119862 gt 0

119861 gt 0 119862 gt 0

(a) 1198770 = 1

5

5

4

4

3

3

2

2

1

10

0

120578

120575

(b) 1198770 = 1004390340

5

5

4

4

3

3

2

2

1

10

0

120578

120575

(c) 1198770 = 1087806817

5

5

4

4

3

3

2

2

1

10

0

120578

120575

(d) 1198770 = 2317102260



1198770= 00002277727471 (b) 120573 = 00002287727471 (c)

120573 = 00002477727471 and (d) 120573 = 00005277727471


0gt 1






(1205731198770

lt 120573119862






0 the


0 Some



5

5

4

4

3

3

2

2

1

10

0

120578

120575

Δ gt 0

119891(1199091) gt 0

119861 gt 0 119862 lt 0

119861 lt 0 119862 lt 0

119861 lt 0 119862 gt 0

119861 lt 0119862 gt 0

119861 gt 0 119862 gt 0

(a) 1198770 = 09560965911

5

5

4

4

3

3

2

2

1

10

0

120578

120575

(b) 1198770 = 08902414781

5

5

4

4

3

3

2

2

1

10

0

120578

120575

(c) 1198770 = 07804829561

5

5

4

4

3

3

2

2

1

10

0

120578

120575

(d) 1198770 = 05609659131



00001777727471 and (d) 120573 = 00001277727471






0




0such that disease-











lt 120573119862lt 120573119861

120573 lt 1205731198770


lt 120573 lt 120573119862




120573119862lt 1205731198770

lt 120573119861

120573 lt 120573119862



2) le 0 or Δ

1lt 0 none if 119875(119909

2) ge 0 or Δ

1gt 0

1205731198770

lt 120573 lt 120573119861


1gt 0


120573119862lt 120573119861lt 1205731198770

120573 lt 120573119862



2) le 0 or Δ

1lt 0 none if 119875(119909

2) ge 0 or Δ

1gt 0

120573119861lt 120573 lt 120573


1lt 0) or none (Δ

1gt 0)

1205731198770


lt 120573119861lt 120573119862

120573 lt 1205731198770


lt 120573 lt 120573119861




1lt 0)


120573119861lt 1205731198770

lt 120573119862

120573 lt 120573119861



1lt 0) or none (Δ

1gt 0)

1205731198770

lt 120573 lt 120573119862


1lt 0)


120573119861lt 120573119862lt 1205731198770

120573 lt 120573119861



1lt 0) or none (Δ

1gt 0)

120573119862lt 120573 lt 120573


1lt 0) or none (Δ

1gt 0)

1205731198770



Appendices



119886 = 120583 + 120583119879+ 119888

119887 = 2119908 + 120583

ℎ = ] + 120583

119892 = 1199031+ 119888

119898 = 1199032+ 119888

120579 = 120583 + 120583119905

120598 = 1 minus 119901

1198711= 120579 + 119892 (1 minus 119902) + 119898119902

1198712= ]1199021199032+ ℎ119886 + 119903

1] (1 minus 119902) + 119903

1120583

1198701= 1199032119908 + 119887119886 + 119903

1(119908 + 120583)

1198702= ]1199021199032+ ℎ119886 + 119903

1] (1 minus 119902) + 119903

1120583

(A1)

We have

1198611= (120579 (119891119901 + (1 minus 119901) 119902) + 119898119902)Π120575120590

1198612= 120590 (120583119871

1120579 120575 + 119871

2120579 + 119898120583 (119886 + 119903

1))

+ (1199032119908120579 + 119903

1(119908 + 120583) 120579

+ 119887119886120579 + 119898120583 (119886 + 1199031)) 120575


1198621= Π (120575 ((119891119901 + 120598119902) (119887120583119905 + (119898 + 119887) 120583) + 119898119908)

+ 120590 ((]119902120598 + 119891ℎ119901) 120579 + 119898 (120583119891119901 + ] 119902)))

1198622= 120583 (119870

1120579 + 119898120583 (119886 + 119903

1)) 120575 + 120583 (119870

2120579 + 119898120583 (119886 + 119903

1)) 120590

+ ℎ (1198701120579 + 119898120583 (119886 + 119903

1))

(A2)


119861 = 1205732119891119861(120573)

119862 = 120573119891119862(120573)

(A3)

where119891119861(120573) = minus119861

1120573 + 1198612

119891119862(120573) = minus119862

1120573 + 119862

2

119860 = 1205733120575120590 (120583 + 120583

119879) (120583 + 120583

119879+ 119888 + (1 minus 119902) 119903

1+ 1199021199032) gt 0

119863 = ℎ ([119886119887 + 1199081199032+ 1199031(119908 + 120583)] (120583 + 120583119905)

+ 120583119898 (1199031+ 119886)) (1 minus 119877

0)

(A4)





Acknowledgments


References
















































of






Disease Markers



OncologyJournal of





PPAR Research



Journal of

ObesityJournal of


























were

119878 (0) = 4980 119864 (0) = 0 119868119868 (0) = 20

119868119873 (0) = 0 119877 (0) = 0

(25)


1205731198770

= 00001450317354

120573119861= 001087387065

120573119862= 00002715343808

(26)


lt 120573119862

lt 120573119861


1198770


119878infin

= 1616 119877infin

= 4080 119868119873infin

= 103

119868119868infin

= 195 119864infin

= 1150

(27)





lt 120573119862lt 120573119861 120575 = 00 120578 = 09) For our



0= 3585422172 The used


119878 (0) = 4980 119864 (0) = 0 119868119868 (0) = 20

119868119873 (0) = 0 119877 (0) = 0

(28)

119905

7000

6000

5000

4000

3000

2000

1000

00 50 100 150 200

119878

119877

119868119873119864

119868119868 + 119868119873119868119868119873

119868119868 + 119868119873 + 119864 + 119877



4000

4000 5000

3000

3000

2000

2000

1000

10004000

6000

20000

119868 119868+119868 119873+119864

119878 119877 4000 500

3000000000000000000000000000000

3000

200202000020200202002020200000000200202020020020200200202020202002002000202002000020202020200020202000220000200202000020202222000222020200220000202022022220022002202222220022222 02 02222 0000000000000000000000000000000000000000000000000000000000000000

2000

100110010010010010010010010100100100100010010010010010010010010000010100100100100100100100100110000001001000000110010110010010010010001000001110100010000010000000001010000011000001110000000001001001000000000000001000010000111000101 000000000000000000000000000000000000000000000000000000000000000000000000000000000000000

10004000

6000

20000

119868119868119868119868119868119868119868119868119868119868119868119868119868119868119868119868119868119868119868119868119868119868119868119868119868119868119868119868119868119868119868119868119868119868119868119868119868119868119868119868 119868119868119868119868119868119868119868119868119868119868119868119868119868119868++++++++++++++++++++++++++119868 119873119868+119864

119878 119877


0=

3585422172 120575 = 09 120578 = 001 and 120573 = 000052


1205731198770

= 00001450317354

120573119861= 001226355348

120573119862= 00003132229272

(29)


lt 120573119861lt 120573119862 and as in



119878infin

= 1938 119877infin

= 974 119868119873infin

= 60

119868119868infin

= 156 119864infin

= 4530

(30)


119905

7000

6000

5000

4000

3000

2000

1000

00 50 100 150 200

119878

119877

119868119873119864

119868119868 + 119868119873119868119868119873

119868119868 + 119868119873 + 119864 + 119877




1198770lt 120573119862







119861lt 120573119862

lt 1205731198770 120575 = 30 120578 = 25) There


0=


1205731198770

= 00001450317354

120573119861= 00001066568066

120573119862= 00001225687204

(31)


1205731198770


119894 119868119899 119864)

1198751= (9009 0 0 0 0)

1198752= (8507 182 9 5 2166)

1198753= (3221 1406 285 103 1566)

(32)

800

700

600

500

400

300

200

100

07006005004003002001000

119868 119868

119905



= 0



11987531198751198751198751198751198751198751198751198751198751198751198751198751198751198751198751198751198751198751198751198751198751198751198751198751198751198751198751198751198751198751198751198751198751198751198751198751198751198751198751198751198751198751198751198751198751198751198751198751198751198751198751198751198751198751198751198751198751198751198751198751198751198751198751198751198751198751198751198751198751198751198751198751198753119875119875

16001200

800400

900080007000

600050004000300020000

1000

2000

3000

119868 119868+119868 119873+119864

1198781198771198752

1198751




3



119868infin= 0 or to 119868

119868infin=



3according




119868+ 119868119873+ 119864


lowast=


lowast the system


119861lt 1205731198770

lt 120573119862 120575 = 30 120578 = 25) Consider


119861lt 1205731198770



1205731198770

= 00001679568390

120573119862= 00001729256777

120573119861= 00001489092005

(33)










00001688612368 with 1205731198770



condition 1205731lt 120573 lt 120573

1198770




(119878 119877 119868119894 119868119899 119864)

1198750= (5148 0 0 0 0)

1198751= (3372 1041 122 60 482)

1198752= (2828 1283 190 88 651)

(34)



2is


119868infin= 0 or to 119868




2






lt 1205731198770


1205731198770

1205731

1205732

120573119862120573119861

000015 000016 000017 000018120573

600

500

400

300

200

100

0

119909


119861lt 1205731198770



2

500

400

300

200

100

0

119868 119868

0 500 1000 1500 2000119905



= 0

or 119868119868infin



(119878 119877 119868119894 119868119899 119864)

1198750= (5148 0 0 0 0)

1198751= (5042 76 5 3 20)

1198752= (3971 734 69 36 298)

1198753= (2491 1413 246 109 750)

(35)


160014001200800600400200

500

2000

0

1000

1500

1000

2000

3000

4000

5000

119878119877

11987521198751

1198750 160014001200800600444400420000002000200

5005005000

000

1000

1500

1000

20020020000000

300003003003003000000000000

400404004004004000040040040000000 119878119877

1198752119875119875

1198751119875119875

119868 119868+119868 119873+119877





2







gt








lowast= 120583 = 003885 120583

119905= 001520 119901 = 08

] = 00266 119891 = 08 119902 = 085 119908 = 0005 119888 = 04 1199031= 05





160014001200

800600400200

500

2000

0

1000

1500

1000

2000

3000

4000

5000

119868 119868+119868 119873+

119878119877

119877

1198752

1198751

1198753

1198750 160014001200

8006004004040200200200200200200

500

000

1000

1500

1000

2002002002002 000000

3000030033000000000000

4004004004004000040000400 00000

119868 119868119868+119868 119873119868+

119878119877

119877

1198752119875119875

1198751119875119875

1198753119875119875

0




0and







0is near one













5

5

4

4

3

3

2

2

1

10

0

120578

119861 gt 0 119862 gt 0

120575

119861 lt 0 119862 lt 0

119861 gt 0 119862 lt 0

(a) 1198770 = 1

5

5

4

4

3

3

2

2

1

10

0

120578

120575

(b) 1198770 = 1689504264

5

5

4

4

3

3

2

2

1

10

0

120578

120575

(c) 1198770 = 2379008527

5

5

4

4

3

3

2

2

1

10

0

120578

120575

(d) 1198770 = 7895042636



1198770= 00001450317354 (b) 120573 = 00002450317354 (c) 120573 = 00003450317354 and (d)

120573 = 0001145031735


1198770 120573119861 120573119862 and 120573 We divided



0has to do solely






5

5

4

4

3

3

2

2

1

10

0

120578

120575

Δ gt 0

119891(1199091) gt 0

119861 gt 0 119862 lt 0

119861 lt 0 119862 lt 0

119861 lt 0 119862 gt 0

119861 lt 0119862 gt 0

119861 gt 0 119862 gt 0

(a) 1198770 = 1

5

5

4

4

3

3

2

2

1

10

0

120578

120575

(b) 1198770 = 1004390340

5

5

4

4

3

3

2

2

1

10

0

120578

120575

(c) 1198770 = 1087806817

5

5

4

4

3

3

2

2

1

10

0

120578

120575

(d) 1198770 = 2317102260



1198770= 00002277727471 (b) 120573 = 00002287727471 (c)

120573 = 00002477727471 and (d) 120573 = 00005277727471


0gt 1






(1205731198770

lt 120573119862






0 the


0 Some



5

5

4

4

3

3

2

2

1

10

0

120578

120575

Δ gt 0

119891(1199091) gt 0

119861 gt 0 119862 lt 0

119861 lt 0 119862 lt 0

119861 lt 0 119862 gt 0

119861 lt 0119862 gt 0

119861 gt 0 119862 gt 0

(a) 1198770 = 09560965911

5

5

4

4

3

3

2

2

1

10

0

120578

120575

(b) 1198770 = 08902414781

5

5

4

4

3

3

2

2

1

10

0

120578

120575

(c) 1198770 = 07804829561

5

5

4

4

3

3

2

2

1

10

0

120578

120575

(d) 1198770 = 05609659131



00001777727471 and (d) 120573 = 00001277727471






0




0such that disease-











lt 120573119862lt 120573119861

120573 lt 1205731198770


lt 120573 lt 120573119862




120573119862lt 1205731198770

lt 120573119861

120573 lt 120573119862



2) le 0 or Δ

1lt 0 none if 119875(119909

2) ge 0 or Δ

1gt 0

1205731198770

lt 120573 lt 120573119861


1gt 0


120573119862lt 120573119861lt 1205731198770

120573 lt 120573119862



2) le 0 or Δ

1lt 0 none if 119875(119909

2) ge 0 or Δ

1gt 0

120573119861lt 120573 lt 120573


1lt 0) or none (Δ

1gt 0)

1205731198770


lt 120573119861lt 120573119862

120573 lt 1205731198770


lt 120573 lt 120573119861




1lt 0)


120573119861lt 1205731198770

lt 120573119862

120573 lt 120573119861



1lt 0) or none (Δ

1gt 0)

1205731198770

lt 120573 lt 120573119862


1lt 0)


120573119861lt 120573119862lt 1205731198770

120573 lt 120573119861



1lt 0) or none (Δ

1gt 0)

120573119862lt 120573 lt 120573


1lt 0) or none (Δ

1gt 0)

1205731198770



Appendices



119886 = 120583 + 120583119879+ 119888

119887 = 2119908 + 120583

ℎ = ] + 120583

119892 = 1199031+ 119888

119898 = 1199032+ 119888

120579 = 120583 + 120583119905

120598 = 1 minus 119901

1198711= 120579 + 119892 (1 minus 119902) + 119898119902

1198712= ]1199021199032+ ℎ119886 + 119903

1] (1 minus 119902) + 119903

1120583

1198701= 1199032119908 + 119887119886 + 119903

1(119908 + 120583)

1198702= ]1199021199032+ ℎ119886 + 119903

1] (1 minus 119902) + 119903

1120583

(A1)

We have

1198611= (120579 (119891119901 + (1 minus 119901) 119902) + 119898119902)Π120575120590

1198612= 120590 (120583119871

1120579 120575 + 119871

2120579 + 119898120583 (119886 + 119903

1))

+ (1199032119908120579 + 119903

1(119908 + 120583) 120579

+ 119887119886120579 + 119898120583 (119886 + 1199031)) 120575


1198621= Π (120575 ((119891119901 + 120598119902) (119887120583119905 + (119898 + 119887) 120583) + 119898119908)

+ 120590 ((]119902120598 + 119891ℎ119901) 120579 + 119898 (120583119891119901 + ] 119902)))

1198622= 120583 (119870

1120579 + 119898120583 (119886 + 119903

1)) 120575 + 120583 (119870

2120579 + 119898120583 (119886 + 119903

1)) 120590

+ ℎ (1198701120579 + 119898120583 (119886 + 119903

1))

(A2)


119861 = 1205732119891119861(120573)

119862 = 120573119891119862(120573)

(A3)

where119891119861(120573) = minus119861

1120573 + 1198612

119891119862(120573) = minus119862

1120573 + 119862

2

119860 = 1205733120575120590 (120583 + 120583

119879) (120583 + 120583

119879+ 119888 + (1 minus 119902) 119903

1+ 1199021199032) gt 0

119863 = ℎ ([119886119887 + 1199081199032+ 1199031(119908 + 120583)] (120583 + 120583119905)

+ 120583119898 (1199031+ 119886)) (1 minus 119877

0)

(A4)





Acknowledgments


References
















































of






Disease Markers



OncologyJournal of





PPAR Research



Journal of

ObesityJournal of

















119905

7000

6000

5000

4000

3000

2000

1000

00 50 100 150 200

119878

119877

119868119873119864

119868119868 + 119868119873119868119868119873

119868119868 + 119868119873 + 119864 + 119877




1198770lt 120573119862







119861lt 120573119862

lt 1205731198770 120575 = 30 120578 = 25) There


0=


1205731198770

= 00001450317354

120573119861= 00001066568066

120573119862= 00001225687204

(31)


1205731198770


119894 119868119899 119864)

1198751= (9009 0 0 0 0)

1198752= (8507 182 9 5 2166)

1198753= (3221 1406 285 103 1566)

(32)

800

700

600

500

400

300

200

100

07006005004003002001000

119868 119868

119905



= 0



11987531198751198751198751198751198751198751198751198751198751198751198751198751198751198751198751198751198751198751198751198751198751198751198751198751198751198751198751198751198751198751198751198751198751198751198751198751198751198751198751198751198751198751198751198751198751198751198751198751198751198751198751198751198751198751198751198751198751198751198751198751198751198751198751198751198751198751198751198751198751198751198751198751198753119875119875

16001200

800400

900080007000

600050004000300020000

1000

2000

3000

119868 119868+119868 119873+119864

1198781198771198752

1198751




3



119868infin= 0 or to 119868

119868infin=



3according




119868+ 119868119873+ 119864


lowast=


lowast the system


119861lt 1205731198770

lt 120573119862 120575 = 30 120578 = 25) Consider


119861lt 1205731198770



1205731198770

= 00001679568390

120573119862= 00001729256777

120573119861= 00001489092005

(33)










00001688612368 with 1205731198770



condition 1205731lt 120573 lt 120573

1198770




(119878 119877 119868119894 119868119899 119864)

1198750= (5148 0 0 0 0)

1198751= (3372 1041 122 60 482)

1198752= (2828 1283 190 88 651)

(34)



2is


119868infin= 0 or to 119868




2






lt 1205731198770


1205731198770

1205731

1205732

120573119862120573119861

000015 000016 000017 000018120573

600

500

400

300

200

100

0

119909


119861lt 1205731198770



2

500

400

300

200

100

0

119868 119868

0 500 1000 1500 2000119905



= 0

or 119868119868infin



(119878 119877 119868119894 119868119899 119864)

1198750= (5148 0 0 0 0)

1198751= (5042 76 5 3 20)

1198752= (3971 734 69 36 298)

1198753= (2491 1413 246 109 750)

(35)


160014001200800600400200

500

2000

0

1000

1500

1000

2000

3000

4000

5000

119878119877

11987521198751

1198750 160014001200800600444400420000002000200

5005005000

000

1000

1500

1000

20020020000000

300003003003003000000000000

400404004004004000040040040000000 119878119877

1198752119875119875

1198751119875119875

119868 119868+119868 119873+119877





2







gt








lowast= 120583 = 003885 120583

119905= 001520 119901 = 08

] = 00266 119891 = 08 119902 = 085 119908 = 0005 119888 = 04 1199031= 05





160014001200

800600400200

500

2000

0

1000

1500

1000

2000

3000

4000

5000

119868 119868+119868 119873+

119878119877

119877

1198752

1198751

1198753

1198750 160014001200

8006004004040200200200200200200

500

000

1000

1500

1000

2002002002002 000000

3000030033000000000000

4004004004004000040000400 00000

119868 119868119868+119868 119873119868+

119878119877

119877

1198752119875119875

1198751119875119875

1198753119875119875

0




0and







0is near one













5

5

4

4

3

3

2

2

1

10

0

120578

119861 gt 0 119862 gt 0

120575

119861 lt 0 119862 lt 0

119861 gt 0 119862 lt 0

(a) 1198770 = 1

5

5

4

4

3

3

2

2

1

10

0

120578

120575

(b) 1198770 = 1689504264

5

5

4

4

3

3

2

2

1

10

0

120578

120575

(c) 1198770 = 2379008527

5

5

4

4

3

3

2

2

1

10

0

120578

120575

(d) 1198770 = 7895042636



1198770= 00001450317354 (b) 120573 = 00002450317354 (c) 120573 = 00003450317354 and (d)

120573 = 0001145031735


1198770 120573119861 120573119862 and 120573 We divided



0has to do solely






5

5

4

4

3

3

2

2

1

10

0

120578

120575

Δ gt 0

119891(1199091) gt 0

119861 gt 0 119862 lt 0

119861 lt 0 119862 lt 0

119861 lt 0 119862 gt 0

119861 lt 0119862 gt 0

119861 gt 0 119862 gt 0

(a) 1198770 = 1

5

5

4

4

3

3

2

2

1

10

0

120578

120575

(b) 1198770 = 1004390340

5

5

4

4

3

3

2

2

1

10

0

120578

120575

(c) 1198770 = 1087806817

5

5

4

4

3

3

2

2

1

10

0

120578

120575

(d) 1198770 = 2317102260



1198770= 00002277727471 (b) 120573 = 00002287727471 (c)

120573 = 00002477727471 and (d) 120573 = 00005277727471


0gt 1






(1205731198770

lt 120573119862






0 the


0 Some



5

5

4

4

3

3

2

2

1

10

0

120578

120575

Δ gt 0

119891(1199091) gt 0

119861 gt 0 119862 lt 0

119861 lt 0 119862 lt 0

119861 lt 0 119862 gt 0

119861 lt 0119862 gt 0

119861 gt 0 119862 gt 0

(a) 1198770 = 09560965911

5

5

4

4

3

3

2

2

1

10

0

120578

120575

(b) 1198770 = 08902414781

5

5

4

4

3

3

2

2

1

10

0

120578

120575

(c) 1198770 = 07804829561

5

5

4

4

3

3

2

2

1

10

0

120578

120575

(d) 1198770 = 05609659131



00001777727471 and (d) 120573 = 00001277727471






0




0such that disease-











lt 120573119862lt 120573119861

120573 lt 1205731198770


lt 120573 lt 120573119862




120573119862lt 1205731198770

lt 120573119861

120573 lt 120573119862



2) le 0 or Δ

1lt 0 none if 119875(119909

2) ge 0 or Δ

1gt 0

1205731198770

lt 120573 lt 120573119861


1gt 0


120573119862lt 120573119861lt 1205731198770

120573 lt 120573119862



2) le 0 or Δ

1lt 0 none if 119875(119909

2) ge 0 or Δ

1gt 0

120573119861lt 120573 lt 120573


1lt 0) or none (Δ

1gt 0)

1205731198770


lt 120573119861lt 120573119862

120573 lt 1205731198770


lt 120573 lt 120573119861




1lt 0)


120573119861lt 1205731198770

lt 120573119862

120573 lt 120573119861



1lt 0) or none (Δ

1gt 0)

1205731198770

lt 120573 lt 120573119862


1lt 0)


120573119861lt 120573119862lt 1205731198770

120573 lt 120573119861



1lt 0) or none (Δ

1gt 0)

120573119862lt 120573 lt 120573


1lt 0) or none (Δ

1gt 0)

1205731198770



Appendices



119886 = 120583 + 120583119879+ 119888

119887 = 2119908 + 120583

ℎ = ] + 120583

119892 = 1199031+ 119888

119898 = 1199032+ 119888

120579 = 120583 + 120583119905

120598 = 1 minus 119901

1198711= 120579 + 119892 (1 minus 119902) + 119898119902

1198712= ]1199021199032+ ℎ119886 + 119903

1] (1 minus 119902) + 119903

1120583

1198701= 1199032119908 + 119887119886 + 119903

1(119908 + 120583)

1198702= ]1199021199032+ ℎ119886 + 119903

1] (1 minus 119902) + 119903

1120583

(A1)

We have

1198611= (120579 (119891119901 + (1 minus 119901) 119902) + 119898119902)Π120575120590

1198612= 120590 (120583119871

1120579 120575 + 119871

2120579 + 119898120583 (119886 + 119903

1))

+ (1199032119908120579 + 119903

1(119908 + 120583) 120579

+ 119887119886120579 + 119898120583 (119886 + 1199031)) 120575


1198621= Π (120575 ((119891119901 + 120598119902) (119887120583119905 + (119898 + 119887) 120583) + 119898119908)

+ 120590 ((]119902120598 + 119891ℎ119901) 120579 + 119898 (120583119891119901 + ] 119902)))

1198622= 120583 (119870

1120579 + 119898120583 (119886 + 119903

1)) 120575 + 120583 (119870

2120579 + 119898120583 (119886 + 119903

1)) 120590

+ ℎ (1198701120579 + 119898120583 (119886 + 119903

1))

(A2)


119861 = 1205732119891119861(120573)

119862 = 120573119891119862(120573)

(A3)

where119891119861(120573) = minus119861

1120573 + 1198612

119891119862(120573) = minus119862

1120573 + 119862

2

119860 = 1205733120575120590 (120583 + 120583

119879) (120583 + 120583

119879+ 119888 + (1 minus 119902) 119903

1+ 1199021199032) gt 0

119863 = ℎ ([119886119887 + 1199081199032+ 1199031(119908 + 120583)] (120583 + 120583119905)

+ 120583119898 (1199031+ 119886)) (1 minus 119877

0)

(A4)





Acknowledgments


References
















































of






Disease Markers



OncologyJournal of





PPAR Research



Journal of

ObesityJournal of


















119868+ 119868119873+ 119864


lowast=


lowast the system


119861lt 1205731198770

lt 120573119862 120575 = 30 120578 = 25) Consider


119861lt 1205731198770



1205731198770

= 00001679568390

120573119862= 00001729256777

120573119861= 00001489092005

(33)










00001688612368 with 1205731198770



condition 1205731lt 120573 lt 120573

1198770




(119878 119877 119868119894 119868119899 119864)

1198750= (5148 0 0 0 0)

1198751= (3372 1041 122 60 482)

1198752= (2828 1283 190 88 651)

(34)



2is


119868infin= 0 or to 119868




2






lt 1205731198770


1205731198770

1205731

1205732

120573119862120573119861

000015 000016 000017 000018120573

600

500

400

300

200

100

0

119909


119861lt 1205731198770



2

500

400

300

200

100

0

119868 119868

0 500 1000 1500 2000119905



= 0

or 119868119868infin



(119878 119877 119868119894 119868119899 119864)

1198750= (5148 0 0 0 0)

1198751= (5042 76 5 3 20)

1198752= (3971 734 69 36 298)

1198753= (2491 1413 246 109 750)

(35)


160014001200800600400200

500

2000

0

1000

1500

1000

2000

3000

4000

5000

119878119877

11987521198751

1198750 160014001200800600444400420000002000200

5005005000

000

1000

1500

1000

20020020000000

300003003003003000000000000

400404004004004000040040040000000 119878119877

1198752119875119875

1198751119875119875

119868 119868+119868 119873+119877





2







gt








lowast= 120583 = 003885 120583

119905= 001520 119901 = 08

] = 00266 119891 = 08 119902 = 085 119908 = 0005 119888 = 04 1199031= 05





160014001200

800600400200

500

2000

0

1000

1500

1000

2000

3000

4000

5000

119868 119868+119868 119873+

119878119877

119877

1198752

1198751

1198753

1198750 160014001200

8006004004040200200200200200200

500

000

1000

1500

1000

2002002002002 000000

3000030033000000000000

4004004004004000040000400 00000

119868 119868119868+119868 119873119868+

119878119877

119877

1198752119875119875

1198751119875119875

1198753119875119875

0




0and







0is near one













5

5

4

4

3

3

2

2

1

10

0

120578

119861 gt 0 119862 gt 0

120575

119861 lt 0 119862 lt 0

119861 gt 0 119862 lt 0

(a) 1198770 = 1

5

5

4

4

3

3

2

2

1

10

0

120578

120575

(b) 1198770 = 1689504264

5

5

4

4

3

3

2

2

1

10

0

120578

120575

(c) 1198770 = 2379008527

5

5

4

4

3

3

2

2

1

10

0

120578

120575

(d) 1198770 = 7895042636



1198770= 00001450317354 (b) 120573 = 00002450317354 (c) 120573 = 00003450317354 and (d)

120573 = 0001145031735


1198770 120573119861 120573119862 and 120573 We divided



0has to do solely






5

5

4

4

3

3

2

2

1

10

0

120578

120575

Δ gt 0

119891(1199091) gt 0

119861 gt 0 119862 lt 0

119861 lt 0 119862 lt 0

119861 lt 0 119862 gt 0

119861 lt 0119862 gt 0

119861 gt 0 119862 gt 0

(a) 1198770 = 1

5

5

4

4

3

3

2

2

1

10

0

120578

120575

(b) 1198770 = 1004390340

5

5

4

4

3

3

2

2

1

10

0

120578

120575

(c) 1198770 = 1087806817

5

5

4

4

3

3

2

2

1

10

0

120578

120575

(d) 1198770 = 2317102260



1198770= 00002277727471 (b) 120573 = 00002287727471 (c)

120573 = 00002477727471 and (d) 120573 = 00005277727471


0gt 1






(1205731198770

lt 120573119862






0 the


0 Some



5

5

4

4

3

3

2

2

1

10

0

120578

120575

Δ gt 0

119891(1199091) gt 0

119861 gt 0 119862 lt 0

119861 lt 0 119862 lt 0

119861 lt 0 119862 gt 0

119861 lt 0119862 gt 0

119861 gt 0 119862 gt 0

(a) 1198770 = 09560965911

5

5

4

4

3

3

2

2

1

10

0

120578

120575

(b) 1198770 = 08902414781

5

5

4

4

3

3

2

2

1

10

0

120578

120575

(c) 1198770 = 07804829561

5

5

4

4

3

3

2

2

1

10

0

120578

120575

(d) 1198770 = 05609659131



00001777727471 and (d) 120573 = 00001277727471






0




0such that disease-











lt 120573119862lt 120573119861

120573 lt 1205731198770


lt 120573 lt 120573119862




120573119862lt 1205731198770

lt 120573119861

120573 lt 120573119862



2) le 0 or Δ

1lt 0 none if 119875(119909

2) ge 0 or Δ

1gt 0

1205731198770

lt 120573 lt 120573119861


1gt 0


120573119862lt 120573119861lt 1205731198770

120573 lt 120573119862



2) le 0 or Δ

1lt 0 none if 119875(119909

2) ge 0 or Δ

1gt 0

120573119861lt 120573 lt 120573


1lt 0) or none (Δ

1gt 0)

1205731198770


lt 120573119861lt 120573119862

120573 lt 1205731198770


lt 120573 lt 120573119861




1lt 0)


120573119861lt 1205731198770

lt 120573119862

120573 lt 120573119861



1lt 0) or none (Δ

1gt 0)

1205731198770

lt 120573 lt 120573119862


1lt 0)


120573119861lt 120573119862lt 1205731198770

120573 lt 120573119861



1lt 0) or none (Δ

1gt 0)

120573119862lt 120573 lt 120573


1lt 0) or none (Δ

1gt 0)

1205731198770



Appendices



119886 = 120583 + 120583119879+ 119888

119887 = 2119908 + 120583

ℎ = ] + 120583

119892 = 1199031+ 119888

119898 = 1199032+ 119888

120579 = 120583 + 120583119905

120598 = 1 minus 119901

1198711= 120579 + 119892 (1 minus 119902) + 119898119902

1198712= ]1199021199032+ ℎ119886 + 119903

1] (1 minus 119902) + 119903

1120583

1198701= 1199032119908 + 119887119886 + 119903

1(119908 + 120583)

1198702= ]1199021199032+ ℎ119886 + 119903

1] (1 minus 119902) + 119903

1120583

(A1)

We have

1198611= (120579 (119891119901 + (1 minus 119901) 119902) + 119898119902)Π120575120590

1198612= 120590 (120583119871

1120579 120575 + 119871

2120579 + 119898120583 (119886 + 119903

1))

+ (1199032119908120579 + 119903

1(119908 + 120583) 120579

+ 119887119886120579 + 119898120583 (119886 + 1199031)) 120575


1198621= Π (120575 ((119891119901 + 120598119902) (119887120583119905 + (119898 + 119887) 120583) + 119898119908)

+ 120590 ((]119902120598 + 119891ℎ119901) 120579 + 119898 (120583119891119901 + ] 119902)))

1198622= 120583 (119870

1120579 + 119898120583 (119886 + 119903

1)) 120575 + 120583 (119870

2120579 + 119898120583 (119886 + 119903

1)) 120590

+ ℎ (1198701120579 + 119898120583 (119886 + 119903

1))

(A2)


119861 = 1205732119891119861(120573)

119862 = 120573119891119862(120573)

(A3)

where119891119861(120573) = minus119861

1120573 + 1198612

119891119862(120573) = minus119862

1120573 + 119862

2

119860 = 1205733120575120590 (120583 + 120583

119879) (120583 + 120583

119879+ 119888 + (1 minus 119902) 119903

1+ 1199021199032) gt 0

119863 = ℎ ([119886119887 + 1199081199032+ 1199031(119908 + 120583)] (120583 + 120583119905)

+ 120583119898 (1199031+ 119886)) (1 minus 119877

0)

(A4)





Acknowledgments


References
















































of






Disease Markers



OncologyJournal of





PPAR Research



Journal of

ObesityJournal of

















160014001200800600400200

500

2000

0

1000

1500

1000

2000

3000

4000

5000

119878119877

11987521198751

1198750 160014001200800600444400420000002000200

5005005000

000

1000

1500

1000

20020020000000

300003003003003000000000000

400404004004004000040040040000000 119878119877

1198752119875119875

1198751119875119875

119868 119868+119868 119873+119877





2







gt








lowast= 120583 = 003885 120583

119905= 001520 119901 = 08

] = 00266 119891 = 08 119902 = 085 119908 = 0005 119888 = 04 1199031= 05





160014001200

800600400200

500

2000

0

1000

1500

1000

2000

3000

4000

5000

119868 119868+119868 119873+

119878119877

119877

1198752

1198751

1198753

1198750 160014001200

8006004004040200200200200200200

500

000

1000

1500

1000

2002002002002 000000

3000030033000000000000

4004004004004000040000400 00000

119868 119868119868+119868 119873119868+

119878119877

119877

1198752119875119875

1198751119875119875

1198753119875119875

0




0and







0is near one













5

5

4

4

3

3

2

2

1

10

0

120578

119861 gt 0 119862 gt 0

120575

119861 lt 0 119862 lt 0

119861 gt 0 119862 lt 0

(a) 1198770 = 1

5

5

4

4

3

3

2

2

1

10

0

120578

120575

(b) 1198770 = 1689504264

5

5

4

4

3

3

2

2

1

10

0

120578

120575

(c) 1198770 = 2379008527

5

5

4

4

3

3

2

2

1

10

0

120578

120575

(d) 1198770 = 7895042636



1198770= 00001450317354 (b) 120573 = 00002450317354 (c) 120573 = 00003450317354 and (d)

120573 = 0001145031735


1198770 120573119861 120573119862 and 120573 We divided



0has to do solely






5

5

4

4

3

3

2

2

1

10

0

120578

120575

Δ gt 0

119891(1199091) gt 0

119861 gt 0 119862 lt 0

119861 lt 0 119862 lt 0

119861 lt 0 119862 gt 0

119861 lt 0119862 gt 0

119861 gt 0 119862 gt 0

(a) 1198770 = 1

5

5

4

4

3

3

2

2

1

10

0

120578

120575

(b) 1198770 = 1004390340

5

5

4

4

3

3

2

2

1

10

0

120578

120575

(c) 1198770 = 1087806817

5

5

4

4

3

3

2

2

1

10

0

120578

120575

(d) 1198770 = 2317102260



1198770= 00002277727471 (b) 120573 = 00002287727471 (c)

120573 = 00002477727471 and (d) 120573 = 00005277727471


0gt 1






(1205731198770

lt 120573119862






0 the


0 Some



5

5

4

4

3

3

2

2

1

10

0

120578

120575

Δ gt 0

119891(1199091) gt 0

119861 gt 0 119862 lt 0

119861 lt 0 119862 lt 0

119861 lt 0 119862 gt 0

119861 lt 0119862 gt 0

119861 gt 0 119862 gt 0

(a) 1198770 = 09560965911

5

5

4

4

3

3

2

2

1

10

0

120578

120575

(b) 1198770 = 08902414781

5

5

4

4

3

3

2

2

1

10

0

120578

120575

(c) 1198770 = 07804829561

5

5

4

4

3

3

2

2

1

10

0

120578

120575

(d) 1198770 = 05609659131



00001777727471 and (d) 120573 = 00001277727471






0




0such that disease-











lt 120573119862lt 120573119861

120573 lt 1205731198770


lt 120573 lt 120573119862




120573119862lt 1205731198770

lt 120573119861

120573 lt 120573119862



2) le 0 or Δ

1lt 0 none if 119875(119909

2) ge 0 or Δ

1gt 0

1205731198770

lt 120573 lt 120573119861


1gt 0


120573119862lt 120573119861lt 1205731198770

120573 lt 120573119862



2) le 0 or Δ

1lt 0 none if 119875(119909

2) ge 0 or Δ

1gt 0

120573119861lt 120573 lt 120573


1lt 0) or none (Δ

1gt 0)

1205731198770


lt 120573119861lt 120573119862

120573 lt 1205731198770


lt 120573 lt 120573119861




1lt 0)


120573119861lt 1205731198770

lt 120573119862

120573 lt 120573119861



1lt 0) or none (Δ

1gt 0)

1205731198770

lt 120573 lt 120573119862


1lt 0)


120573119861lt 120573119862lt 1205731198770

120573 lt 120573119861



1lt 0) or none (Δ

1gt 0)

120573119862lt 120573 lt 120573


1lt 0) or none (Δ

1gt 0)

1205731198770



Appendices



119886 = 120583 + 120583119879+ 119888

119887 = 2119908 + 120583

ℎ = ] + 120583

119892 = 1199031+ 119888

119898 = 1199032+ 119888

120579 = 120583 + 120583119905

120598 = 1 minus 119901

1198711= 120579 + 119892 (1 minus 119902) + 119898119902

1198712= ]1199021199032+ ℎ119886 + 119903

1] (1 minus 119902) + 119903

1120583

1198701= 1199032119908 + 119887119886 + 119903

1(119908 + 120583)

1198702= ]1199021199032+ ℎ119886 + 119903

1] (1 minus 119902) + 119903

1120583

(A1)

We have

1198611= (120579 (119891119901 + (1 minus 119901) 119902) + 119898119902)Π120575120590

1198612= 120590 (120583119871

1120579 120575 + 119871

2120579 + 119898120583 (119886 + 119903

1))

+ (1199032119908120579 + 119903

1(119908 + 120583) 120579

+ 119887119886120579 + 119898120583 (119886 + 1199031)) 120575


1198621= Π (120575 ((119891119901 + 120598119902) (119887120583119905 + (119898 + 119887) 120583) + 119898119908)

+ 120590 ((]119902120598 + 119891ℎ119901) 120579 + 119898 (120583119891119901 + ] 119902)))

1198622= 120583 (119870

1120579 + 119898120583 (119886 + 119903

1)) 120575 + 120583 (119870

2120579 + 119898120583 (119886 + 119903

1)) 120590

+ ℎ (1198701120579 + 119898120583 (119886 + 119903

1))

(A2)


119861 = 1205732119891119861(120573)

119862 = 120573119891119862(120573)

(A3)

where119891119861(120573) = minus119861

1120573 + 1198612

119891119862(120573) = minus119862

1120573 + 119862

2

119860 = 1205733120575120590 (120583 + 120583

119879) (120583 + 120583

119879+ 119888 + (1 minus 119902) 119903

1+ 1199021199032) gt 0

119863 = ℎ ([119886119887 + 1199081199032+ 1199031(119908 + 120583)] (120583 + 120583119905)

+ 120583119898 (1199031+ 119886)) (1 minus 119877

0)

(A4)





Acknowledgments


References
















































of






Disease Markers



OncologyJournal of





PPAR Research



Journal of

ObesityJournal of

















5

5

4

4

3

3

2

2

1

10

0

120578

119861 gt 0 119862 gt 0

120575

119861 lt 0 119862 lt 0

119861 gt 0 119862 lt 0

(a) 1198770 = 1

5

5

4

4

3

3

2

2

1

10

0

120578

120575

(b) 1198770 = 1689504264

5

5

4

4

3

3

2

2

1

10

0

120578

120575

(c) 1198770 = 2379008527

5

5

4

4

3

3

2

2

1

10

0

120578

120575

(d) 1198770 = 7895042636



1198770= 00001450317354 (b) 120573 = 00002450317354 (c) 120573 = 00003450317354 and (d)

120573 = 0001145031735


1198770 120573119861 120573119862 and 120573 We divided



0has to do solely






5

5

4

4

3

3

2

2

1

10

0

120578

120575

Δ gt 0

119891(1199091) gt 0

119861 gt 0 119862 lt 0

119861 lt 0 119862 lt 0

119861 lt 0 119862 gt 0

119861 lt 0119862 gt 0

119861 gt 0 119862 gt 0

(a) 1198770 = 1

5

5

4

4

3

3

2

2

1

10

0

120578

120575

(b) 1198770 = 1004390340

5

5

4

4

3

3

2

2

1

10

0

120578

120575

(c) 1198770 = 1087806817

5

5

4

4

3

3

2

2

1

10

0

120578

120575

(d) 1198770 = 2317102260



1198770= 00002277727471 (b) 120573 = 00002287727471 (c)

120573 = 00002477727471 and (d) 120573 = 00005277727471


0gt 1






(1205731198770

lt 120573119862






0 the


0 Some



5

5

4

4

3

3

2

2

1

10

0

120578

120575

Δ gt 0

119891(1199091) gt 0

119861 gt 0 119862 lt 0

119861 lt 0 119862 lt 0

119861 lt 0 119862 gt 0

119861 lt 0119862 gt 0

119861 gt 0 119862 gt 0

(a) 1198770 = 09560965911

5

5

4

4

3

3

2

2

1

10

0

120578

120575

(b) 1198770 = 08902414781

5

5

4

4

3

3

2

2

1

10

0

120578

120575

(c) 1198770 = 07804829561

5

5

4

4

3

3

2

2

1

10

0

120578

120575

(d) 1198770 = 05609659131



00001777727471 and (d) 120573 = 00001277727471






0




0such that disease-











lt 120573119862lt 120573119861

120573 lt 1205731198770


lt 120573 lt 120573119862




120573119862lt 1205731198770

lt 120573119861

120573 lt 120573119862



2) le 0 or Δ

1lt 0 none if 119875(119909

2) ge 0 or Δ

1gt 0

1205731198770

lt 120573 lt 120573119861


1gt 0


120573119862lt 120573119861lt 1205731198770

120573 lt 120573119862



2) le 0 or Δ

1lt 0 none if 119875(119909

2) ge 0 or Δ

1gt 0

120573119861lt 120573 lt 120573


1lt 0) or none (Δ

1gt 0)

1205731198770


lt 120573119861lt 120573119862

120573 lt 1205731198770


lt 120573 lt 120573119861




1lt 0)


120573119861lt 1205731198770

lt 120573119862

120573 lt 120573119861



1lt 0) or none (Δ

1gt 0)

1205731198770

lt 120573 lt 120573119862


1lt 0)


120573119861lt 120573119862lt 1205731198770

120573 lt 120573119861



1lt 0) or none (Δ

1gt 0)

120573119862lt 120573 lt 120573


1lt 0) or none (Δ

1gt 0)

1205731198770



Appendices



119886 = 120583 + 120583119879+ 119888

119887 = 2119908 + 120583

ℎ = ] + 120583

119892 = 1199031+ 119888

119898 = 1199032+ 119888

120579 = 120583 + 120583119905

120598 = 1 minus 119901

1198711= 120579 + 119892 (1 minus 119902) + 119898119902

1198712= ]1199021199032+ ℎ119886 + 119903

1] (1 minus 119902) + 119903

1120583

1198701= 1199032119908 + 119887119886 + 119903

1(119908 + 120583)

1198702= ]1199021199032+ ℎ119886 + 119903

1] (1 minus 119902) + 119903

1120583

(A1)

We have

1198611= (120579 (119891119901 + (1 minus 119901) 119902) + 119898119902)Π120575120590

1198612= 120590 (120583119871

1120579 120575 + 119871

2120579 + 119898120583 (119886 + 119903

1))

+ (1199032119908120579 + 119903

1(119908 + 120583) 120579

+ 119887119886120579 + 119898120583 (119886 + 1199031)) 120575


1198621= Π (120575 ((119891119901 + 120598119902) (119887120583119905 + (119898 + 119887) 120583) + 119898119908)

+ 120590 ((]119902120598 + 119891ℎ119901) 120579 + 119898 (120583119891119901 + ] 119902)))

1198622= 120583 (119870

1120579 + 119898120583 (119886 + 119903

1)) 120575 + 120583 (119870

2120579 + 119898120583 (119886 + 119903

1)) 120590

+ ℎ (1198701120579 + 119898120583 (119886 + 119903

1))

(A2)


119861 = 1205732119891119861(120573)

119862 = 120573119891119862(120573)

(A3)

where119891119861(120573) = minus119861

1120573 + 1198612

119891119862(120573) = minus119862

1120573 + 119862

2

119860 = 1205733120575120590 (120583 + 120583

119879) (120583 + 120583

119879+ 119888 + (1 minus 119902) 119903

1+ 1199021199032) gt 0

119863 = ℎ ([119886119887 + 1199081199032+ 1199031(119908 + 120583)] (120583 + 120583119905)

+ 120583119898 (1199031+ 119886)) (1 minus 119877

0)

(A4)





Acknowledgments


References
















































of






Disease Markers



OncologyJournal of





PPAR Research



Journal of

ObesityJournal of

















5

5

4

4

3

3

2

2

1

10

0

120578

120575

Δ gt 0

119891(1199091) gt 0

119861 gt 0 119862 lt 0

119861 lt 0 119862 lt 0

119861 lt 0 119862 gt 0

119861 lt 0119862 gt 0

119861 gt 0 119862 gt 0

(a) 1198770 = 1

5

5

4

4

3

3

2

2

1

10

0

120578

120575

(b) 1198770 = 1004390340

5

5

4

4

3

3

2

2

1

10

0

120578

120575

(c) 1198770 = 1087806817

5

5

4

4

3

3

2

2

1

10

0

120578

120575

(d) 1198770 = 2317102260



1198770= 00002277727471 (b) 120573 = 00002287727471 (c)

120573 = 00002477727471 and (d) 120573 = 00005277727471


0gt 1






(1205731198770

lt 120573119862






0 the


0 Some



5

5

4

4

3

3

2

2

1

10

0

120578

120575

Δ gt 0

119891(1199091) gt 0

119861 gt 0 119862 lt 0

119861 lt 0 119862 lt 0

119861 lt 0 119862 gt 0

119861 lt 0119862 gt 0

119861 gt 0 119862 gt 0

(a) 1198770 = 09560965911

5

5

4

4

3

3

2

2

1

10

0

120578

120575

(b) 1198770 = 08902414781

5

5

4

4

3

3

2

2

1

10

0

120578

120575

(c) 1198770 = 07804829561

5

5

4

4

3

3

2

2

1

10

0

120578

120575

(d) 1198770 = 05609659131



00001777727471 and (d) 120573 = 00001277727471






0




0such that disease-











lt 120573119862lt 120573119861

120573 lt 1205731198770


lt 120573 lt 120573119862




120573119862lt 1205731198770

lt 120573119861

120573 lt 120573119862



2) le 0 or Δ

1lt 0 none if 119875(119909

2) ge 0 or Δ

1gt 0

1205731198770

lt 120573 lt 120573119861


1gt 0


120573119862lt 120573119861lt 1205731198770

120573 lt 120573119862



2) le 0 or Δ

1lt 0 none if 119875(119909

2) ge 0 or Δ

1gt 0

120573119861lt 120573 lt 120573


1lt 0) or none (Δ

1gt 0)

1205731198770


lt 120573119861lt 120573119862

120573 lt 1205731198770


lt 120573 lt 120573119861




1lt 0)


120573119861lt 1205731198770

lt 120573119862

120573 lt 120573119861



1lt 0) or none (Δ

1gt 0)

1205731198770

lt 120573 lt 120573119862


1lt 0)


120573119861lt 120573119862lt 1205731198770

120573 lt 120573119861



1lt 0) or none (Δ

1gt 0)

120573119862lt 120573 lt 120573


1lt 0) or none (Δ

1gt 0)

1205731198770



Appendices



119886 = 120583 + 120583119879+ 119888

119887 = 2119908 + 120583

ℎ = ] + 120583

119892 = 1199031+ 119888

119898 = 1199032+ 119888

120579 = 120583 + 120583119905

120598 = 1 minus 119901

1198711= 120579 + 119892 (1 minus 119902) + 119898119902

1198712= ]1199021199032+ ℎ119886 + 119903

1] (1 minus 119902) + 119903

1120583

1198701= 1199032119908 + 119887119886 + 119903

1(119908 + 120583)

1198702= ]1199021199032+ ℎ119886 + 119903

1] (1 minus 119902) + 119903

1120583

(A1)

We have

1198611= (120579 (119891119901 + (1 minus 119901) 119902) + 119898119902)Π120575120590

1198612= 120590 (120583119871

1120579 120575 + 119871

2120579 + 119898120583 (119886 + 119903

1))

+ (1199032119908120579 + 119903

1(119908 + 120583) 120579

+ 119887119886120579 + 119898120583 (119886 + 1199031)) 120575


1198621= Π (120575 ((119891119901 + 120598119902) (119887120583119905 + (119898 + 119887) 120583) + 119898119908)

+ 120590 ((]119902120598 + 119891ℎ119901) 120579 + 119898 (120583119891119901 + ] 119902)))

1198622= 120583 (119870

1120579 + 119898120583 (119886 + 119903

1)) 120575 + 120583 (119870

2120579 + 119898120583 (119886 + 119903

1)) 120590

+ ℎ (1198701120579 + 119898120583 (119886 + 119903

1))

(A2)


119861 = 1205732119891119861(120573)

119862 = 120573119891119862(120573)

(A3)

where119891119861(120573) = minus119861

1120573 + 1198612

119891119862(120573) = minus119862

1120573 + 119862

2

119860 = 1205733120575120590 (120583 + 120583

119879) (120583 + 120583

119879+ 119888 + (1 minus 119902) 119903

1+ 1199021199032) gt 0

119863 = ℎ ([119886119887 + 1199081199032+ 1199031(119908 + 120583)] (120583 + 120583119905)

+ 120583119898 (1199031+ 119886)) (1 minus 119877

0)

(A4)





Acknowledgments


References
















































of






Disease Markers



OncologyJournal of





PPAR Research



Journal of

ObesityJournal of

















5

5

4

4

3

3

2

2

1

10

0

120578

120575

Δ gt 0

119891(1199091) gt 0

119861 gt 0 119862 lt 0

119861 lt 0 119862 lt 0

119861 lt 0 119862 gt 0

119861 lt 0119862 gt 0

119861 gt 0 119862 gt 0

(a) 1198770 = 09560965911

5

5

4

4

3

3

2

2

1

10

0

120578

120575

(b) 1198770 = 08902414781

5

5

4

4

3

3

2

2

1

10

0

120578

120575

(c) 1198770 = 07804829561

5

5

4

4

3

3

2

2

1

10

0

120578

120575

(d) 1198770 = 05609659131



00001777727471 and (d) 120573 = 00001277727471






0




0such that disease-











lt 120573119862lt 120573119861

120573 lt 1205731198770


lt 120573 lt 120573119862




120573119862lt 1205731198770

lt 120573119861

120573 lt 120573119862



2) le 0 or Δ

1lt 0 none if 119875(119909

2) ge 0 or Δ

1gt 0

1205731198770

lt 120573 lt 120573119861


1gt 0


120573119862lt 120573119861lt 1205731198770

120573 lt 120573119862



2) le 0 or Δ

1lt 0 none if 119875(119909

2) ge 0 or Δ

1gt 0

120573119861lt 120573 lt 120573


1lt 0) or none (Δ

1gt 0)

1205731198770


lt 120573119861lt 120573119862

120573 lt 1205731198770


lt 120573 lt 120573119861




1lt 0)


120573119861lt 1205731198770

lt 120573119862

120573 lt 120573119861



1lt 0) or none (Δ

1gt 0)

1205731198770

lt 120573 lt 120573119862


1lt 0)


120573119861lt 120573119862lt 1205731198770

120573 lt 120573119861



1lt 0) or none (Δ

1gt 0)

120573119862lt 120573 lt 120573


1lt 0) or none (Δ

1gt 0)

1205731198770



Appendices



119886 = 120583 + 120583119879+ 119888

119887 = 2119908 + 120583

ℎ = ] + 120583

119892 = 1199031+ 119888

119898 = 1199032+ 119888

120579 = 120583 + 120583119905

120598 = 1 minus 119901

1198711= 120579 + 119892 (1 minus 119902) + 119898119902

1198712= ]1199021199032+ ℎ119886 + 119903

1] (1 minus 119902) + 119903

1120583

1198701= 1199032119908 + 119887119886 + 119903

1(119908 + 120583)

1198702= ]1199021199032+ ℎ119886 + 119903

1] (1 minus 119902) + 119903

1120583

(A1)

We have

1198611= (120579 (119891119901 + (1 minus 119901) 119902) + 119898119902)Π120575120590

1198612= 120590 (120583119871

1120579 120575 + 119871

2120579 + 119898120583 (119886 + 119903

1))

+ (1199032119908120579 + 119903

1(119908 + 120583) 120579

+ 119887119886120579 + 119898120583 (119886 + 1199031)) 120575


1198621= Π (120575 ((119891119901 + 120598119902) (119887120583119905 + (119898 + 119887) 120583) + 119898119908)

+ 120590 ((]119902120598 + 119891ℎ119901) 120579 + 119898 (120583119891119901 + ] 119902)))

1198622= 120583 (119870

1120579 + 119898120583 (119886 + 119903

1)) 120575 + 120583 (119870

2120579 + 119898120583 (119886 + 119903

1)) 120590

+ ℎ (1198701120579 + 119898120583 (119886 + 119903

1))

(A2)


119861 = 1205732119891119861(120573)

119862 = 120573119891119862(120573)

(A3)

where119891119861(120573) = minus119861

1120573 + 1198612

119891119862(120573) = minus119862

1120573 + 119862

2

119860 = 1205733120575120590 (120583 + 120583

119879) (120583 + 120583

119879+ 119888 + (1 minus 119902) 119903

1+ 1199021199032) gt 0

119863 = ℎ ([119886119887 + 1199081199032+ 1199031(119908 + 120583)] (120583 + 120583119905)

+ 120583119898 (1199031+ 119886)) (1 minus 119877

0)

(A4)





Acknowledgments


References
















































of






Disease Markers



OncologyJournal of





PPAR Research



Journal of

ObesityJournal of






















lt 120573119862lt 120573119861

120573 lt 1205731198770


lt 120573 lt 120573119862




120573119862lt 1205731198770

lt 120573119861

120573 lt 120573119862



2) le 0 or Δ

1lt 0 none if 119875(119909

2) ge 0 or Δ

1gt 0

1205731198770

lt 120573 lt 120573119861


1gt 0


120573119862lt 120573119861lt 1205731198770

120573 lt 120573119862



2) le 0 or Δ

1lt 0 none if 119875(119909

2) ge 0 or Δ

1gt 0

120573119861lt 120573 lt 120573


1lt 0) or none (Δ

1gt 0)

1205731198770


lt 120573119861lt 120573119862

120573 lt 1205731198770


lt 120573 lt 120573119861




1lt 0)


120573119861lt 1205731198770

lt 120573119862

120573 lt 120573119861



1lt 0) or none (Δ

1gt 0)

1205731198770

lt 120573 lt 120573119862


1lt 0)


120573119861lt 120573119862lt 1205731198770

120573 lt 120573119861



1lt 0) or none (Δ

1gt 0)

120573119862lt 120573 lt 120573


1lt 0) or none (Δ

1gt 0)

1205731198770



Appendices



119886 = 120583 + 120583119879+ 119888

119887 = 2119908 + 120583

ℎ = ] + 120583

119892 = 1199031+ 119888

119898 = 1199032+ 119888

120579 = 120583 + 120583119905

120598 = 1 minus 119901

1198711= 120579 + 119892 (1 minus 119902) + 119898119902

1198712= ]1199021199032+ ℎ119886 + 119903

1] (1 minus 119902) + 119903

1120583

1198701= 1199032119908 + 119887119886 + 119903

1(119908 + 120583)

1198702= ]1199021199032+ ℎ119886 + 119903

1] (1 minus 119902) + 119903

1120583

(A1)

We have

1198611= (120579 (119891119901 + (1 minus 119901) 119902) + 119898119902)Π120575120590

1198612= 120590 (120583119871

1120579 120575 + 119871

2120579 + 119898120583 (119886 + 119903

1))

+ (1199032119908120579 + 119903

1(119908 + 120583) 120579

+ 119887119886120579 + 119898120583 (119886 + 1199031)) 120575


1198621= Π (120575 ((119891119901 + 120598119902) (119887120583119905 + (119898 + 119887) 120583) + 119898119908)

+ 120590 ((]119902120598 + 119891ℎ119901) 120579 + 119898 (120583119891119901 + ] 119902)))

1198622= 120583 (119870

1120579 + 119898120583 (119886 + 119903

1)) 120575 + 120583 (119870

2120579 + 119898120583 (119886 + 119903

1)) 120590

+ ℎ (1198701120579 + 119898120583 (119886 + 119903

1))

(A2)


119861 = 1205732119891119861(120573)

119862 = 120573119891119862(120573)

(A3)

where119891119861(120573) = minus119861

1120573 + 1198612

119891119862(120573) = minus119862

1120573 + 119862

2

119860 = 1205733120575120590 (120583 + 120583

119879) (120583 + 120583

119879+ 119888 + (1 minus 119902) 119903

1+ 1199021199032) gt 0

119863 = ℎ ([119886119887 + 1199081199032+ 1199031(119908 + 120583)] (120583 + 120583119905)

+ 120583119898 (1199031+ 119886)) (1 minus 119877

0)

(A4)





Acknowledgments


References
















































of






Disease Markers



OncologyJournal of





PPAR Research



Journal of

ObesityJournal of

















1198621= Π (120575 ((119891119901 + 120598119902) (119887120583119905 + (119898 + 119887) 120583) + 119898119908)

+ 120590 ((]119902120598 + 119891ℎ119901) 120579 + 119898 (120583119891119901 + ] 119902)))

1198622= 120583 (119870

1120579 + 119898120583 (119886 + 119903

1)) 120575 + 120583 (119870

2120579 + 119898120583 (119886 + 119903

1)) 120590

+ ℎ (1198701120579 + 119898120583 (119886 + 119903

1))

(A2)


119861 = 1205732119891119861(120573)

119862 = 120573119891119862(120573)

(A3)

where119891119861(120573) = minus119861

1120573 + 1198612

119891119862(120573) = minus119862

1120573 + 119862

2

119860 = 1205733120575120590 (120583 + 120583

119879) (120583 + 120583

119879+ 119888 + (1 minus 119902) 119903

1+ 1199021199032) gt 0

119863 = ℎ ([119886119887 + 1199081199032+ 1199031(119908 + 120583)] (120583 + 120583119905)

+ 120583119898 (1199031+ 119886)) (1 minus 119877

0)

(A4)





Acknowledgments


References
















































of






Disease Markers



OncologyJournal of





PPAR Research



Journal of

ObesityJournal of










































of






Disease Markers



OncologyJournal of





PPAR Research



Journal of

ObesityJournal of
















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