In order to investigate the transmission mechanism of the infectious individual with Ebola virus, we establish an SEIT (susceptible, exposed in the latent period, infectious, and treated/recovery) epidemic model. The basic reproduction number is defined. The mathematical analysis on the existence and stability of the disease-free equilibrium and endemic equilibrium is given. As the applications of the model, we use the recognized infectious and death cases in Guinea to estimate parameters of the model by the least square method. With suitable parameter values, we obtain the estimated value of the basic reproduction number and analyze the sensitivity and uncertainty property by partial rank correlation coefficients.
1. Introduction
Ebola virus disease (EVD) is first identified in the Democratic Republic of Congo (formerly Zäire) in 1976. It is a lethal viral hemorrhagic fever and can cause a high case fatality rate lying between 50 and 90% [1, 2]. Since 1976, the outbreak of Ebola epidemic has happened more than 20 times, most of which appeared in Africa. Sudan (1976, 1979, and 2004), Democratic Republic of Congo (1976, 1977, 1995, 2007, 2008, and 2012), Gabon (1994, 1996, 2001, and 2002), Republic of Côte d’Ivoire (1994), Uganda (2000, 2007, 2011, and 2012), and Republic of the Congo (2001, 2002, 2003, and 2005) have reported EVD epidemics [3, 4].
Mathematical modeling has emerged as an important tool for gaining understanding of the dynamics of the spread of EVD. Reference [5] established a stochastic discrete-time susceptible-exposed-infectious-recovered (SEIR) model to estimate parameters from daily incidence and mortality time series for an outbreak of Ebola in the Democratic Republic of Congo in 1995. Different from [5], reference [1] used both onset and death date to constrain the optimization of SEIR model parameters by Bayesian inference. To develop a better understanding of Ebola transmission dynamics, [6] introduced a compartmental model to quantify transmission in different settings (illness in the community, hospitalization, and traditional burial). Further, [7] used the above model to analyze the temporal dynamics of Ebola.
The current Ebola outbreak began in December 2013 in Guinea [8], initially in the Prefecture of Gueckedou, and shortly spread to other West African countries such as Liberia, Sierra Leone, Nigeria, and Senegal [6, 9–11]. The outbreak is the largest to date: as of 25th January 2015, 22,092 cases have been reported by World Health Organisation (WHO), as well as 8,810 deaths [12], which contain 2,917 infected cases and 1,910 deaths of Guinea. Reference [13] analyzed transmission dynamics of EVD in Nigeria and showed the time window for successful containment of EVD outbreaks caused by infected air travelers. Reference [14] designed a model that is formulated by splitting the total population into two main subgroups, namely a subgroup of individuals in the community and another for those in health-care settings.
To assess the effect these various intervention strategies could have on controlling the spread of Ebola virus, we develop a mathematical model for transmission, fitted probabilistically to epidemiological data of reported cases in Guinea. Experimental vaccines and treatments for Ebola are under development, but they have not yet been fully tested for safety or effectiveness. Thus, we establish an SEIT model and apply it to analyze the spread of EVD in Guinea, where S, E, I, and T denote the number of the susceptible, exposed in the latent period, infectious, and treated/recovered population, respectively. For the SEIT model, [15–17] constructed a discrete version with latent age structure to investigate tuberculosis transmission. Along the paper, we assume that infected individuals in SEIT model can develop EVD by contacting the exposed in the latent period or infectious individuals.
This paper is organized as follows. Section 2 introduces an SEIT model by a four-dimensional differential equations system. In Section 3, the basic reproduction number R0 is defined. Based on R0<1 and R0>1, the local stability of the disease-free equilibrium and endemic equilibrium is obtained according to Hurwitz criterion. In Section 4, by the least square method we propose the estimation of parameters by onset and death data in Guinea, which can be used to calculate R0. Moreover, we provide the sensitivity and uncertainty analysis of R0. Section 5 presents our concluding remark.
2. Model Formulation
The transmission of EVD is by direct contact with body fluids, secretions, tissues, or semen from the infectious individuals [1], which starts with acute fever, diarrhea that can be bloody, and vomiting followed by headache, nausea, and abdominal pain [2]. Its incubation period ranges from 2 to 21 days (5–12 days in most cases) [1]. Since the diseases had caused the loss of thousands of lives and brought great pain to families, there have been lots of mathematical models for gaining understanding of the dynamics of the spread of EVD [1, 2, 6, 7, 18, 19]. In this work, we establish an SEIT model to extend the SEIR and SEIRS types and apply the model to describe the dynamics of EVD during 2014 outbreak in Guinea.
The total host population is partitioned into susceptibles, exposed (in the latent period), infectious, and treated/recovered individuals, respectively, denoted by S(t), E(t), I(t), and T(t) at time t. After one unit time, a susceptible individual can be infected through contacting with the exposed or infectious individuals and enter the latent class or is still in the susceptible class or dies. A latent individual may become infectious and enter the infectious class or still stay in the latent class or die. An infectious individual may be treated and enter the treated/recovery class or stay in the infectious class or die. For a treated individual, he (or she) may recover by effective treatment. The recovered individual from Ebola depends on good supportive care and the patients immune response. People who recover from Ebola infection develop antibodies that last for at least 10 years, possibly longer. However, it is not known whether people who recover are immune for life or whether they can become infected with a different species of Ebola [20]. Thus, we consider that the recovered individual may enter the susceptible class. Otherwise, the treated individual may stay in the treated/recovered class, or die. Figure 1 shows the relationship between the four variables of our SEIT model.
Transfer diagram of SEIT epidemic model.
Using Figure 1, we formulate the following SEIT model, which is a four-dimensional differential equations model:(1)S˙=Λ-β10IS-β20ES-μS+γ0T,E˙=β10IS+β20ES-ε0E-μ+μ10E,I˙=ε0E-v0I-μ+μ20I,T˙=v0I-γ0T-μ+μ30T,where Λ is the recruitment rate; the positive parameter μ is the rate of natural death; μ10, μ20, and μ30 are nonnegative constants and denote rates of disease-caused death. Parameters β10 and β20 are the rate of the efficient contact in the infected, latent, and exposed period; ε0 and v0, respectively, denote the transfer rates between the exposed and the infectious, the infectious and treated infectious; and γ0 denotes the rate of the effectively treated individuals. In model (1), when γ0=0, the SEIT model is an SEIR type. If γ0≠0, the SEIT model is an SEIRS type. Thus, the SEIT model is a general version of SEIR or SEIRS type.
For convenience, denote β1=β10/μ, β2=β20/μ, μ1=μ10/μ, μ2=μ20/μ, μ3=μ30/μ, ε=ε0/μ, v=v0/μ, and γ=γ0/μ. Further, define(2)ω1=1+ɛ+μ1,ω2=1+v+μ2,ω3=1+γ+μ3.Let τ=μt; then model (1) is equivalent to the following form:(3)dSdτ=Λμ-β1IS-β2ES-S+γT,dEdτ=β1IS+β2ES-ω1E,dIdτ=ɛE-ω2I,dTdτ=vI-ω3T.
3. Equilibria and Local Stability
It is clear that model (3) always has a disease-free equilibrium P0(S0,0,0,0) with S0=Λ/μ. Let P∗=(S∗,E∗,I∗,T∗) be the endemic equilibrium; then we have(4)Λμ-β1I∗S∗-β2E∗S∗-S∗+γT∗=0,β1I∗S∗+β2E∗S∗-ω1E∗=0,ɛE∗-ω2I∗=0,vI∗-ω3T∗=0.Define the basic reproduction number as(5)R0=Λɛβ1+ω2β2μω1ω2.Based on (2), we know that ω1,ω2,ω3>0 and ω1ω2ω3-ɛγv>0. When R0>1, solving (4) we obtain the endemic equilibrium P∗:(6)S∗=ω1ω2ɛβ1+ω2β2,E∗=ω2ω3Λɛβ1+ω2β2-μω1ω2μω1ω2ω3-ɛγvɛβ1+ω2β2,I∗=ɛω3Λɛβ1+ω2β2-μω1ω2μω1ω2ω3-ɛγvɛβ1+ω2β2,T∗=ɛvΛɛβ1+ω2β2-μω1ω2μω1ω2ω3-ɛγvɛβ1+ω2β2.
Now, we discuss the local stability of equilibria. Firstly, on the stability of disease-free equilibrium P0 we have the following result.
Theorem 1.
If R0<1, then disease-free equilibrium P0=(Λ/μ,0,0,0) is locally asymptotically stable.
Proof.
The Jacobian of model (3) is(7)J=-β1I-β2E-1-β2S-β1Sγβ1I+β2Eβ2S-ω1β1S00ɛ-ω2000v-ω3.Thus, the Jacobian at point P0 is (8)JP0=-1-β2Λμ-β1Λμγ0β2Λμ-ω1β1Λμ00ɛ-ω2000v-ω3and its characteristic equation is given by det(λI-J(P0))=0, where I is the unit matrix, since (9)detλI-JP0=λ+1β2Λμβ1Λμ-γ0λ-β2Λμ+ω1-β1Λμ00-ɛλ+ω2000-vλ+ω3=λ+1·λ+ω3λ-β2Λμ+ω1λ+ω2-ɛβ1Λμ.Clearly, there exist two roots λ1=-1 and λ2=-ω3, and other roots satisfy (10)f1λ≜λ-β2Λμ+ω1λ+ω2-ɛβ1Λμ=λ2+a1λ+a2,where (11)a1=ω1+ω2-β2Λμ,a2=ω1ω2-Λμβ2ω2+ɛβ1.
If R0=Λ(ɛβ1+ω2β2)/μω1ω2<1, then ω1>Λɛβ1/μω2+Λβ2/μ and ω1ω2>Λɛβ1/μ+(Λβ2/μ)ω2. Clearly, a1,a2>0 and a1a2>0. According to Hurwitz criterion, all roots of f1(λ) have negative real parts. Hence, disease-free equilibrium P0 is local asymptotical stability.
Next, on the stability of endemic equilibrium P∗, we have the result as follows.
Theorem 2.
If R0>1, then endemic equilibrium P∗ is locally asymptotically stable.
Proof.
By (7), the matrix of the linearization of model (3) at equilibrium P∗ is (12)JP∗=-β1I∗-β2E∗-1-β2S∗-β1S∗γβ1I∗+β2E∗β2S∗-ω1β1S∗00ɛ-ω2000v-ω3.The corresponding characteristic equation is (13)f2λ≜detλI-JP∗=λ4+b1λ3+b2λ2+b3λ+b4,where (14)b1=β1I∗+β2E∗-β2S∗+ω1+ω2+ω3+1,b2=ω1+ω2+ω3β1I∗+β2E∗+1-1+ω2+ω3β2+ɛβ1S∗+ω1ω2+ω3+ω2ω3,b3=ω2ω3+ω1ω2+ω3β1I∗+β2E∗-ω2ω3+ω2+ω3β2+ɛ1+ω3β1S∗+ω1ω2ω3+ω2+ω3+ω2ω3,b4=ω1ω2ω3-ɛγvβ1I∗+β2E∗.Denote A=(Λ(ɛω3β1+ω2ω3β2)-μω1ω2ω3)/μ(ω1ω2ω3-ɛγv). Based on (6), we have (15)β1I∗+β2E∗=ɛβ1+ω2β2ɛI∗=A.Thus,(16)b1=A+ɛω1β1ɛβ1+ω2β2+ω2+ω3+1,b2=ω1+ω2+ω3A+ɛω1β1+ɛω1ω3β1ɛβ1+ω2β2+ω2+ω3+ω2ω3,b3=ω2ω3+ω1ω2+ω3A+ω2ω3+ɛω1ω3β1ɛβ1+ω2β2,b4=ω1ω2ω3-ɛγvA.The calculation of bi, i=1,2,3,4, is placed in Appendix A.
In order to obtain the local asymptotical stability of P∗, we need to verify the following conditions according to Hurwitz criterion: (i) bi>0 for i=1,2,3,4; (ii) b1b2-b3>0; and (iii) b3(b1b2-b3)-b12b4>0:
Since β1,β2,ω1,ω2,ω3>0 and ω1ω2ω3-ɛγv>0, it is easy to get b1,b2,b3,b4>0.
For b1b2-b3, we have (17)b1b2-b3=ω1+ω2+ω3A2+ɛω1β1+ɛω1ω3β1ɛβ1+ω2β2A+ω1+ω2+ω3A+ɛω1β1ɛβ1+ω2β2ω1+ω2+ω3A+ω2+ω3ω2+ω3+1A+ɛω1β1ɛβ1+ω2β2ω2+ω3+ω2ω3+ɛω1β1ɛω1β1+ɛω1ω3β1ɛβ1+ω2β22+ɛω1ω2β1+ɛω1ω2ω3β1ɛβ1+ω2β2+ɛω1ω32β1ɛβ1+ω2β2+ɛω1β1+ɛω1ω3β1ɛβ1+ω2β2+ω2+ω3ω2+ω3+1+ω2ω3ω2+ω3.
Clearly, b1b2-b3>0.
Based on (16) and (ii), we have b3(b1b2-b3)-b12b4>0. Because the expression of b3(b1b2-b3)-b12b4 is too long, we place it in Appendix B. According to Hurwitz criterion, all roots of f2(λ) have negative real parts. Hence, endemic equilibrium P∗ is local asymptotical stability.
4. Application of the Model
In this section, model (1) will be applied to analyze the characteristics of EVD, which is equivalent to system (3). In order to estimate its parameters and calculate the basic reproduced number R0, we use the onset and death data of EVD to fit the observed variables by least square method.
4.1. Data Sets
The 2014 Ebola outbreak began in Guinea. The patient zero is a 2-year-old toddler named Emile Ouamouno, who lived in the village of Meliandou, sitting close to Guinea’s borders with Sierra Leone and Liberia. The boy was infected in December 2013 and it is not clear exactly how he got infected [8]. In December, Emile had a fever, black stool, and started vomiting. Four days later, on December 6, he was dead. Within a month, so were his young sister, his mother, and his grandmother.
In March 22, 2014, the Ministry of health of Guinea has reported the acute infectious disease named EVD, which began with fever, severe diarrhea, vomiting, and high case fatality rate (59%). WHO publicly announced outbreak of EVD on its web site on March 23; 49 cases and 29 deaths were officially reported in March 22 [21]. By January 25, 2015, 2,917 reported cases of Ebola in Guinea were identified, of which 1,910 individuals had died. From the first case in December 1, 2013, to January 18, 2015, in Guinea, the Ebola epidemics lasted 400 days, but the recorded data starting time and evolution of the epidemic is March 22, 2014. Therefore, the onset and death data were collected from March 22, 2014, to January 25, 2015 [22]; see Figures 2 and 3.
Number of recognized infectious cases.
Number of recognized death cases.
4.2. Least Square Method for Parameters Estimation
In this subsection, we use the least square method to estimate the parameters of model (1). The spread of EVD started in December 2013, in which the whole population of Guinea was 11,745,000 in this year [23]. Based on the data sets (see Figures 2 and 3), we know that I(0)=49, T(0)=49-29=20, and S(0)=11,745,000-49=11,744,951. Moreover, since Ebola incubation period ranges from 5 to 12 days in most cases [1], then we take 5 days as the incubation period. Therefore, E(0)=37 (that is the reported infected cases at the fifth day minus the reported infected cases at the first day).
In model (1), there are ten parameters that correspond to the recruitment rate Λ, the natural and disease-caused death rates μ, μ10, μ20, and μ30, the efficient contact rates in the infected and the latent period β10 and β20, and the transfer rates ε0, v0, and γ0. Among these parameters, parts of them need to be fixed and others will be estimated as follows:
The recruitment rate Λ can be given by birth rate, which is the total number of births per 1,000 of a population in a year. Based on [24–26], the 2014 birth rate of Guinea is 0.03602. Thus, Λ=(0.03602/365)×11,744,951=1,159.
The natural death rate μ is typically expressed in units of deaths per 1,000 individuals per year. Thus, we use the natural death rate μ=0.0097/365=2.657×10-5 of Guinea in 2013 [27]. For two disease-caused death rates μ10 in the latent period and μ30 in treated/recovery period, their values are smaller than that of μ20 in the infectious period. Thus, we assume that μ10=μ30=0. The rate μ20 will be estimated. In order to provide the value range of μ20, we use the method in which the accumulative death number is divided by the accumulative cases. Then, it is obtained that 0.5≤μ20≤0.8 (see Figure 4).
In this paper, we consider that Ebola virus during the latent and exposed period is contagious. In general, the efficient contact rates β10 and β20 are different. In order to reduce the risk of human-to-human transmission, WHO raises the awareness of the risk factors of Ebola infection and the protective measures [28, 29]. Thus, we consider that 0<β10≤β20<1 and all of them will be estimated.
Since Ebola incubation period ranges from 2 to 21 days (5–12 days in most cases) [1], the transfer rate ε0 is set following the constraint 2<1/ε0<21; that is, 0.04761<ε0<0.5. For the transfer rates v0, we will estimate it and let it satisfy 0<v0<1.
Based on [2, 5], we know that the effective treated rate 3.5<1/γ0<10.7; that is, 0.093<γ0<0.2857.
Ebola disease-caused death rate of recognized infectious cases.
Based on the conditions (P1)–(P5), parameters of model (1) to be estimated are β10, β20, μ20, ε0, v0, and γ0. We list their estimated values in Table 1.
The estimated values of model parameters by least square method (unit: year^{−1}).
Parameters
Definition
Initial values
Estimates
β10
Contact rate in infectious period
5.0138×10-11
6.474×10-11
β20
Contact rate in latent and exposed period
9.3133×10-8
5.685×10-9
μ20
Disease-caused death rate in infectious period
0.5061
0.6647
ε0
Transfer rate from latent and exposed to the infectious
0.4
0.0596
v0
Transfer rate form the infectious to treatment state
0.5140
0.8613
γ0
Effective treated rate
0.2
0.2999
4.3. Numerical Simulations
Using the parameter values in Table 1, numerical simulations of model (1), respectively, give the comparison curves between the real values and fitting values of the accumulative infectious cases and accumulative death cases from March 22, 2014, to January 25, 2015 [22]; see Figure 5.
Real values and fitting values of the accumulative infectious cases and death cases.
Through Figure 5, we observed that the EVD infection still does not get the effective control. This point can be confirmed by threshold value R0. Based on the estimated values of parameters in Table 1, we have R0=Λ(ɛβ1+ω2β2)/μω1ω2=4.16, where βi=βi0/μ, i=1,2. Since R0>1, by Theorem 2, endemic equilibrium P∗ of model (1) is locally asymptotically stable. This indicates that EVD infection still infects humans and will be endemic in Guinea without the effective control measures.
4.4. Sensitivity and Uncertainty Analysis of <inline-formula><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M163"><mml:mrow><mml:msub><mml:mrow><mml:mi>R</mml:mi></mml:mrow><mml:mrow><mml:mn mathvariant="normal">0</mml:mn></mml:mrow></mml:msub></mml:mrow></mml:math></inline-formula>
Because R0 is an important threshed value for the spread of EVD, we perform uncertainty and sensitivity analysis of R0 in model (1) using partial rank correlation coefficients (PRCCs) [30]. Based on (1), (2), and (5), we get the expression of R0 as follows: (18)R0=Λε0β10+β20μ+v0+μ20μμ+v0+μ20μ+ε0+μ10.
Among all parameters of model (1), we only analyze the influence of seven parameters in determining the magnitude of R0 except the fixed numbers Λ, μ, μ10, and μ30, which have been discussed in Section 4.2. The ordering of these PRCCs directly corresponds to the level of statistical influence, the impact that uncertainty in the estimate of a parameter has on the variability of R0. A positive PRCC value indicates that an increase in that parameter leads to an increase in R0, while a negative value shows that increasing that parameter decreases R0. For the Ebola SEIT model, seven parameters were significantly different from 0 (p value < 0.05). Among these parameters, β10, β20, μ20, and v0 have a positive influence on R0, while ε0 and γ0 have a negative influence on R0; see Figure 6.
PRCCs for the effect of six parameters on R0.
5. Conclusions
In this paper, we establish the SEIT model to analyze the dynamical properties of EVD transmission in Guinea. One of the reasons is that there is still no effective treatments for EVD and the current response is only support treatment [31]. In particular, if parameters γ0=0 or γ0≠0, the SEIT model will become the SEIR or SEIRS types.
The SEIT model is formed by four-dimensional differential equations. In order to understand the transmission of Ebola virus, we discuss the local stability of the disease-free equilibrium and the endemic equilibrium by basic reproduction number R0<1 and R0>1. Since there exist ten parameters in the expression of R0, it is important to estimate the value of each parameter. By the least square method and the recorded data in Guinea we obtain the estimation of seven parameters except for four fixed constants in Table 1. In Table 1, we observe that the effective treated rate is only 0.2999, and the disease-caused death rate in infectious period is 0.6647. Thus, it is urgent to provide the effective vaccines to cure the infectious people and protect the susceptible.
With suitable parameter values, we obtain estimation value of R0. The result shows that the Ebola virus still infects people in Guinea and does not disappear in short time. Without the effective control measures, the EVD may be endemic in Guinea. By the PRCCs method, we analyze the sensitivity and uncertainty of R0. The result shows that the rates of the efficient contact, especially in latent and exposed period, lead to the significant increase in R0 (see Figure 6).
On the other hand, it would be interesting to study more properties of the present model. In particular, a study involving both stability properties of pulse vaccination strategy and global stability is worth pursuing. We leave these for future consideration.
AppendicesA. Calculation of <inline-formula><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M194"><mml:msub><mml:mrow><mml:mi>b</mml:mi></mml:mrow><mml:mrow><mml:mn>1</mml:mn></mml:mrow></mml:msub><mml:mo>,</mml:mo><mml:mi /><mml:msub><mml:mrow><mml:mi>b</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msub><mml:mo>,</mml:mo><mml:mi /><mml:msub><mml:mrow><mml:mi>b</mml:mi></mml:mrow><mml:mrow><mml:mn>3</mml:mn></mml:mrow></mml:msub><mml:mo>,</mml:mo><mml:mi /><mml:msub><mml:mrow><mml:mi>b</mml:mi></mml:mrow><mml:mrow><mml:mn>4</mml:mn></mml:mrow></mml:msub></mml:math></inline-formula>
By (6), (16), and the definition of A, we obtain (A.1)b1=β1I∗+β2E∗-β2S∗+ω1+ω2+ω3+1=A-β2ω1ω2ɛβ1+ω2β2+ω1+ω2+ω3+1=A+ω1-ω1ω2β2ɛβ1+ω2β2+ω2+ω3+1=A+ɛω1β1ɛβ1+ω2β2+ω2+ω3+1.
Calculating b2, we have (A.2)b2=ω1+ω2+ω3A+1-1+ω2+ω3β2+ɛβ1S∗+ω1ω2+ω3+ω2ω3=ω1+ω2+ω3A+1-1+ω2+ω3β2+ɛβ1ω1ω2ɛβ1+ω2β2+ω1ω2+ω3+ω2ω3=ω1+ω2+ω3A+1+ɛω1ω3β1-ω1ω2β1ɛβ1+ω2β2+ω2ω3=ω1+ω2+ω3A+ω1ɛβ1+ω2β2+ɛω1ω3β1-ω1ω2β1ɛβ1+ω2β2+ω2+ω3+ω2ω3=ω1+ω2+ω3A+ɛω1β1+ɛω1ω3β1ɛβ1+ω2β2+ω2+ω3+ω2ω3.
For b3, we have (A.3)b3=ω2ω3+ω1ω2+ω3β1I∗+β2E∗-ω2ω3+ω2+ω3β2+ɛ1+ω3β1S∗+ω1ω2ω3+ω2+ω3+ω2ω3=ω2ω3+ω1ω2+ω3A+1-ω2ω3β2+ɛω3β1+ω2+ω3β2+ɛβ1S∗+ω1ω2ω3=ω2ω3+ω1ω2+ω3A+1-ω2+ω3β2+ɛβ1ω1ω2ɛβ1+ω2β2=ω2ω3+ω1ω2+ω3A+ω2ω3+ɛω1ω3β1ɛβ1+ω2β2.
Lastly, it is easy to get (A.4)b4=ω1ω2ω3-ɛγvβ1I∗+β2E∗=ω1ω2ω3-ɛγvA.
B. Calculation of <inline-formula><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M202"><mml:msub><mml:mrow><mml:mi>b</mml:mi></mml:mrow><mml:mrow><mml:mn>3</mml:mn></mml:mrow></mml:msub><mml:mo mathvariant="bold">(</mml:mo><mml:msub><mml:mrow><mml:mi>b</mml:mi></mml:mrow><mml:mrow><mml:mn>1</mml:mn></mml:mrow></mml:msub><mml:msub><mml:mrow><mml:mi>b</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msub><mml:mo mathvariant="bold"> </mml:mo><mml:mo mathvariant="bold">-</mml:mo><mml:mo mathvariant="bold"> </mml:mo><mml:msub><mml:mrow><mml:mi>b</mml:mi></mml:mrow><mml:mrow><mml:mn>3</mml:mn></mml:mrow></mml:msub><mml:mo mathvariant="bold">)</mml:mo><mml:mo mathvariant="bold"> </mml:mo><mml:mo mathvariant="bold">-</mml:mo><mml:mo mathvariant="bold"> </mml:mo><mml:msubsup><mml:mrow><mml:mi>b</mml:mi></mml:mrow><mml:mrow><mml:mn>1</mml:mn></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msubsup><mml:msub><mml:mrow><mml:mi>b</mml:mi></mml:mrow><mml:mrow><mml:mn>4</mml:mn></mml:mrow></mml:msub></mml:math></inline-formula>
By calculating, we have (B.1)b3b1b2-b3=ω1+ω2+ω3ω2ω3+ω1ω2+ω3A3+ɛω1β1+ɛω1ω3β1ɛβ1+ω2β2ω2ω3+ω1ω2+ω3A2+ω2ω3+ω1ω2+ω3ω1+ω2+ω3A2+ɛω1β1ɛβ1+ω2β2ω2ω3+ω1ω2+ω3ω1+ω2+ω3A2+ω2+ω3·ω2ω3+ω1ω2+ω3ω2+ω3+1A2+ɛω1ω3β1ɛβ1+ω2β2ω1+ω2+ω3A2+ω1+ω2+ω3ω2ω3A2+ɛω1β1ɛβ1+ω2β2ω2+ω3+ω2ω3ω2ω3+ω1ω2+ω3A+ɛω1β1ɛω1β1+ɛω1ω3β1ɛβ1+ω2β22ω2ω3+ω1ω2+ω3A+ɛω1ω2β1+ɛω1ω2ω3β1ɛβ1+ω2β2ω2ω3+ω1ω2+ω3A+ɛω1ω32β1ɛβ1+ω2β2ω2ω3+ω1ω2+ω3A+ɛω1β1+ɛω1ω3β1ɛβ1+ω2β2ω2ω3+ω1ω2+ω3A+ω2+ω3ω2+ω3+1ω2ω3+ω1ω2+ω3A+ω2ω3ω2+ω3ω2ω3+ω1ω2+ω3A+ɛω1ω3β1ɛβ1+ω2β2ω1+ω2+ω3A+ɛω1β1+ɛω1ω3β1ɛβ1+ω2β2ω2ω3A+ω1+ω2+ω3·ω2ω3A+ɛω1β1ɛβ1+ω2β2ω1+ω2+ω3ω2ω3A+ω2+ω3ω2+ω3+1ω2ω3A+ɛω1ω3β1ɛβ1+ω2β2ω2+ω3ω2+ω3+1A+ɛω1ω3β1ɛω1β1+ɛω1ω3β1ɛβ1+ω2β22A+ε2ω12ω3β12ɛβ1+ω2β2ω1+ω2+ω3A+C1C2,where (B.2)C1=ω2ω2+ɛω1ω3β1ɛβ1+ω2β2,C2=ɛω1β1ɛβ1+ω2β2ω2+ω3+ω2ω3+ɛω1β1ɛω1β1+ɛω1ω3β1ɛβ1+ω2β22+ɛω1ω2β1+ɛω1ω2ω3β1ɛβ1+ω2β2+ɛω1ω32β1ɛβ1+ω2β2+ɛω1β1+ɛω1ω3β1ɛβ1+ω2β2+ω2+ω3ω2+ω3+1+ω2ω3ω2+ω3.
On the other hand, (B.3)b12b4=ω1ω2ω3A3+2ɛω1β1ɛβ1+ω2β2ω1ω2ω3A2+2ω2+ω3+1ω1ω2ω3A2+ω2+ω3+12ω1ω2ω3A+ɛω1β1ɛβ1+ω2β22ω1ω2ω3A+2ɛω1β1ɛβ1+ω2β2ω2+ω3+1ω1ω2ω3A-ɛγvA3-2ɛω1β1ɛβ1+ω2β2ɛγvA2-2ω2+ω3+1ɛγvA2-ω2+ω3+12ɛγvA-ɛω1β1ɛβ1+ω2β22ɛγvA-2ɛω1β1ɛβ1+ω2β2ω2+ω3+1ɛγvA.Therefore, we further have (B.4)b3b1b2-b3-b12b4=ω1+ω2+ω3ω2ω3+ω1ω2+ω3A3-ω1ω2ω3A3+ɛω1β1ɛβ1+ω2β2ω2ω3+ω1ω2+ω3ω1+ω2+ω3A2-2ɛω1β1ɛβ1+ω2β2ω1ω2ω3A2+ω2+ω3·ω2ω3+ω1ω2+ω3ω2+ω3+1A2-2ω2+ω3+1ω1ω2ω3A2+ɛω1β1+ɛω1ω3β1ɛβ1+ω2β2ω2ω3+ω1ω2+ω3A2+ω2ω3+ω1ω2+ω3ω1+ω2+ω3A2+ɛω1ω3β1ɛβ1+ω2β2ω1+ω2+ω3A2+ω1+ω2+ω3·ω2ω3A2+ω2+ω3ω2+ω3+1ω2ω3+ω1ω2+ω3A+ω2ω3ω2+ω3ω2ω3+ω1ω2+ω3A-ω2+ω3+12ω1ω2ω3A+ɛω1β1ɛω1β1+ɛω1ω3β1ɛβ1+ω2β22ω2ω3+ω1ω2+ω3A-ɛω1β1ɛβ1+ω2β22ω1ω2ω3A+ɛω1β1ɛβ1+ω2β2ω2+ω3+ω2ω3ω2ω3+ω1ω2+ω3A+ɛω1ω2β1+ɛω1ω2ω3β1ɛβ1+ω2β2ω2ω3+ω1ω2+ω3·A-2ɛω1β1ɛβ1+ω2β2ω2+ω3+1ω1ω2ω3A+ɛω1ω32β1ɛβ1+ω2β2ω2ω3+ω1ω2+ω3A+ɛω1β1+ɛω1ω3β1ɛβ1+ω2β2ω2ω3+ω1ω2+ω3A+ɛω1ω3β1ɛβ1+ω2β2ω1+ω2+ω3A+ɛω1β1+ɛω1ω3β1ɛβ1+ω2β2ω2ω3A+ω1+ω2+ω3·ω2ω3A+ɛω1β1ɛβ1+ω2β2ω1+ω2+ω3ω2ω3A+ω2+ω3ω2+ω3+1ω2ω3A+ɛω1ω3β1ɛβ1+ω2β2ω2+ω3ω2+ω3+1A+ɛω1ω3β1ɛω1β1+ɛω1ω3β1ɛβ1+ω2β22A+ε2ω12ω3β12ɛβ1+ω2β2ω1+ω2+ω3A+ɛγvA3+2·ɛω1β1ɛβ1+ω2β2ɛγvA2+2ω2+ω3+1ɛγvA2+ω2+ω3+12ɛγvA+ɛω1β1ɛβ1+ω2β22ɛγvA+2ɛω1β1ɛβ1+ω2β2ω2+ω3+1ɛγvA+C1C2>ɛω1β1+ɛω1ω3β1ɛβ1+ω2β2ω2ω3+ω1ω2+ω3A2+ω2ω3+ω1ω2+ω3ω1+ω2+ω3A2+ɛω1ω3β1ɛβ1+ω2β2ω1+ω2+ω3A2+ω1+ω2+ω3·ω2ω3A2+ɛω1ω32β1ɛβ1+ω2β2ω2ω3+ω1ω2+ω3A+ɛω1β1+ɛω1ω3β1ɛβ1+ω2β2ω2ω3+ω1ω2+ω3A+ɛω1ω3β1ɛβ1+ω2β2ω1+ω2+ω3A+ɛω1β1+ɛω1ω3β1ɛβ1+ω2β2ω2ω3A+ω1+ω2+ω3·ω2ω3A+ɛω1β1ɛβ1+ω2β2ω1+ω2+ω3ω2ω3A+ω2+ω3ω2+ω3+1ω2ω3A+ɛω1ω3β1ɛβ1+ω2β2ω2+ω3ω2+ω3+1A+ɛω1ω3β1ɛω1β1+ɛω1ω3β1ɛβ1+ω2β22A+ε2ω12ω3β12ɛβ1+ω2β2ω1+ω2+ω3A+ɛγvA3+2·ɛω1β1ɛβ1+ω2β2ɛγvA2+2ω2+ω3+1ɛγvA2+ω2+ω3+12ɛγvA+ɛω1β1ɛβ1+ω2β22ɛγvA+2ɛω1β1ɛβ1+ω2β2ω2+ω3+1ɛγvA+C1C2>0.
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
Acknowledgments
The authors would like to thank the associate editor and referees for their valuable comments and structural suggestions for improving the paper. This work is supported by Scientific Research Program of the Higher Education Institution of Xinjiang (Grant no. XJUEDU2012S01), the Doctoral Program of Xinjiang University (Grant no. BS130107), and the Natural Science Foundation of China (Grant nos. 11271312, 11361058, and 41261087).
NdanguzaD.TchuencheJ. M.HaarioH.Statistical data analysis of the 1995 Ebola outbreak in the Democratic Republic of CongoChowellG.HengartnerN. W.Castillo-ChavezC.FenimoreP. W.HymanJ. M.The basic reproductive number of Ebola and the effects of public health measures: the cases of Congo and UgandaPierreF.Ebola virus diseaseZhangY. H.WangS.ChenY. Q.LiJ.Research advances in Ebola virus disease and disease update in the 2014 Ebola outbreakLekoneP. E.FinkenstädtB. F.Statistical inference in a stochastic epidemic SEIR model with control intervention: Ebola as a case studyCamachoA.KucharskiA.FunkS.BremanJ.PiotP.EdmundsW.Potential for large outbreaks of Ebola virus diseaseLewnardJ. A.Ndeffo MbahM. L.Alfaro-MurilloJ. A.AlticeF. L.BawoL.NyenswahT. G.GalvaniA. P.Dynamics and control of Ebola virus transmission in Montserrado, Liberia: a mathematical modelling analysisHollyY.EspritS.Ebola: who is patient zero? Disease traced back to 2-year old in GuineaDe ClercqE.Ebola virus (EBOV) infection: therapeutic strategiesGathererD.The 2014 Ebola virus disease outbreak in West AfricaBaizeS.PannetierD.OestereichL.RiegerT.KoivoguiL.MagassoubaN.SoropoguiB.SowM. S.Kei taS.de ClerckH.TiffanyA.DominguezG.Loua CaroV.CadarD.GabrielM.PahlmannM.TappeD.Schmidt-ChanasitJ.ImpoumaB.DialloA. K.FormentyP.van HerpM.GuntherS.BaizeS.Emergence of Zaire Ebola virus disease in Guinea: preliminary reportWorld Health OrganisationAlthausC. L.LowN.MusaE. O.ShuaibF.GsteigerS.Ebola virus disease outbreak in Nigeria: transmission dynamics and rapid controlAgustoF. B.Teboh-EwungkemM. I.GumelA. B.Mathematical assessment of the effect of traditional beliefs and customs on the transmission dynamics of the 2014 Ebola outbreaksCaoH.ZhouY.The discrete age-structured SEIT model with application to tuberculosis transmission in ChinaCaoH.XiaoY.ZhouY.The dynamics of a discrete {SEIT} model with age and infection age structuresZhouY.CaoH.SivaloganathanS.Discrete tuberculosis models and their applicationChenT.Ka-Kit LeungR.LiuR.ChenF.ZhangX.ZhaoJ.ChenS.Risk of imported Ebola virus disease in ChinaLekoneP. E.FinkenstädtB. F.Statistical inference in a stochastic epidemic SEIR momdel with control intervention: Ebola as a case studyCenters for Disease Control and PreventionEbola Virus Disease, 2015, http://www.cdc.gov/vhf/ebola/transmission/qas.htmlWorld Health OrganisationWorld Health OrganizationWorld Health OrganizationCountries: GuineaJanuary 2015, http://www.who.int/countries/gin/en/Central Intelligence AgencyThe world factbook, 2015, https://www.cia.gov/library/publications/the-world-factbook/rankorder/2054rank.htmlWikipediaThe Free Encyclopedia. Birth rateJanuary 2015, http://en.wikipedia.org/wiki/Birth ratesWikipedia, the Free Encyclopedia, List of sovereign states and dependent territories by birth rate, 2015, https://en.wikipedia.org/wiki/List_of_sovereign_states_and_dependent_territories_by_birth_rateWikipediaList of sovereign states and dependent territories by mortality rate2015, http://en.wikipedia.org/wiki/List of countries by death rateWorld Health OrganisationWorld Health OrganisationBlowerS. M.DowlatabadiH.Sensitivity and uncertainty analysis of comples models of disease transmission: an HIV model, as an exampleSmithaM.Ebola crisis: experimental vaccine ‘shipped to Liberia’January 2015, http://www.bbc.com/news/health-30943377