CC BY
0Hindawi Publishing Corporation Journal of Applied Mathematics Volume 2014, Article ID 324767, 16 pages http://dx.doi.org/10.1155/2014/324767
Research Article
Mathematical Analysis of a Cholera Model with Vaccination
Jing'an Cui,1 Zhanmin Wu,1 and Xueyong Zhou2
1 School of Science, Beijing University of Civil Engineering and Architecture, Beijing 100044, China
2 College of Mathematics and Information Science, Xinyang Normal University, Xinyang, Henan 464000, China
Correspondence should be addressed to Jing'an Cui; cuijingan@bucea.edu.cn
Received 14 March 2013; Revised 18 May 2013; Accepted 26 September 2013; Published 13 February 2014 Academic Editor: Tin-Tai Chow
Copyright © 2014 Jing'an Cui et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
We consider a SVR-B cholera model with imperfect vaccination. By analyzing the corresponding characteristic equations, the local stability of a disease-free equilibrium and an endemic equilibrium is established. We calculate the certain threshold known as the control reproduction number RV. If RV < 1, we obtain sufficient conditions for the global asymptotic stability of the disease-free equilibrium; the diseases will be eliminated from the community. By comparison of arguments, it is proved that if RV > 1, the disease persists and the unique endemic equilibrium is globally asymptotically stable, which is obtained by the second compound matrix techniques and autonomous convergence theorems. We perform sensitivity analysis of RV on the parameters in order to determine their relative importance to disease transmission and show that an imperfect vaccine is always beneficial in reducing disease spread within the community.
1. Introduction
Cholera is an acute intestinal infection caused by the ingestion of food or water contaminated with the bacterium Vibrio cholera. Among the 200 serogroups of Vibrio cholera, it is only Vibrio cholera o1 and o139 that are known to be the cause of the cholera disease [1]. The etiological agent, Vibrio cholera o1 (and more recently vibrio cholera o139), passes through and survives the gastric acid barrier of the stomach and then penetrates the mucus lining that coats the intestinal epithelial [2]. Once they colonise the intestinal gut, then produce enterotoxin (which stimulates water and electrolyte secretion by the endothelial cells of the small intestine) that leads to copious, painless, and watery diarrhoea that can quickly lead to severe dehydration and death if treatment is not promptly given. Vomiting also occurs in most patients. In human volunteer studies, the infection was determined to be 102-103 [3]. Cholera can either be transmitted through interaction between humans (i.e., fecal-oral) or through interaction between humans and their environment (i.e., ingestion of contaminated water and food from the environment). To come on urgent, transmission fast, sweeping range widely are the characteristics of cholera which is one of the international quarantine infectious diseases as stipulated by
the International Health Regulations (IHR), as well as one of a class of infectious diseases as stipulated by law for infectious diseases prevention and control of China.
Globally, cholera incidence has increased steadily since 2005 with cholera outbreaks affecting several continents (see Figure 1). Cholera continues to pose a serious public health problem among developing world populations which have no access to adequate water and sanitation resources. In 2011, 32% of cases were reported from Africa whereas between 2001 and 2009, 93% to 98% of total cases worldwide were reported from that continent [4]. In 2011, 61.2% of cases were reported from Americas where a large outbreak that started in Haiti at the end of October 2010 also affected the Dominican Republic. The outbreak was still ongoing at the end of 2011 with 522335 cases including 7001 deaths that were reported by 25 December in Haiti [4]. So, cholera remains a global threat and is one of the key indicators of social development. The history and reality have warned: we are facing the serious threat of cholera, the importance for the study of cholera's pathogenesis, regular transmission and prevention and control strategy have become increasingly prominent, which has also become a major problem that needs to be solved. Up till now, a number of mathematical models have been used to study the transmission dynamics
600,000
(N<N<N<N<N<N<N<N<N
Africa Asia
Americas Oceania
Figure 1: In 1989-2011, cholera cases were reported to World Health Organization by year and by continent [4].
of cholera. Capasso and Serio [5] introduced an incidence rate in the form of kSI/(\ + al) (with human-to-human transmission model only) in 1973 [6]. Codefo [7] proposed an incidence form of aSB/(K + B) (with environment-to-human transmission model only) in 2001 which, in the first time, explicitly incorporated the pathogen concentration into cholera modeling. Mukandavire et al. [8] included both transmission pathways in the form of phSI + (peSB/(K + B)). In 2012, Liao and Wang [9] generalised Codeco's model [7], incorporating the theory of Volterra-Lyapunov stable matrices into the classic method of Lyapunov functions; they studied the global dynamics of the mathematical model.
Vaccination is a major factor in the resurgence and epidemic outbreaks of some infectious diseases. Since the pioneering work of Edward Jenner on smallpox [10], vaccination has been a commonly used method for diseases control [11-13] and works by reducing the number of susceptible individuals in a population. In modern times, vaccination has a large impact on the incidence and persistence of children infections, such as measles and whooping cough [14]. Hethcote [11] investigated a pertussis infectious model and showed that the vaccination can make the infection undergo fluctuation. Although vaccination offers a very powerful tool for disease control, generally, vaccines are not 100% affective and sometimes they only provide limited immunity due to the natural waning of immunity in the host or antigenic variation in the pathogen [15].
Here, we develop a cholera model with an additional equation for the vaccinated individuals in the population. Since Koch found Vibrio cholera in 1883, the research for cholera vaccine had been going on for over one hundred years. People have developed a variety of vaccines. However, these vaccines were parenteral, which have short effective protection and big side effects. In 1973, the World Health Organization canceled the vaccine inoculation which attracted a major concern to oral vaccines. At present, there are three kinds of oral vaccines (i.e., WC/BS vaccine, WC/rBS vaccine, and CVD103-HgR vaccine) have been proved to be
safe, effective, and immunogenic, which were approved to apply in some countries [16].
In this paper, according to the natural history of cholera, we improve the model of [9] in the following two aspects. Firstly, if the cholera persists for a long time, it will cause death [17], especially in the area where water and sanitation resources are not adequate [4]; a parameter d is added to describe the rate of disease-related death. Secondly, we propose a proportion of the vaccination in susceptible individuals as shown in the following differential equations:
dS (t) dt
ßiS(t)B(t) ß2S(t)I(t) 1 + aiB (t) 1+a2I(t)
dV(t) dt
- (t) - faS (t) + GV (t), = $S(t)-GV(t)-faV(t)..
dl(t) ßi S(t)B(t) ß2S(t)I(t) . .
—;— = -+ --— - (d + a + fa)I(t),
dt 1 + aiB (t) 1 + a2I (t) riJ
dt dB (t) dt
= al(t)-fa R(t), = ni(t)-^B(t).
The flow diagram of the model is depicted in Figure 2. Since the first three and last equations in (1) are independent of the variable R, it suffices to consider the following reduced model:
dS (t) dt
ßiS(t)B(t) ß2S(t)I(t) 1 + aiB (t) 1 + a2I(t)
-4S(t)
-piS(t) + GV(t) dV(t)
= $S(t)-GV(t)-faV(t)..
dl (t) dt
ßi S(t)B(t) ß2S(t)I(t) . .
+ --rrr- - (d + a + fa) I (t),
1 + aiB(t)
dB (t) dt
1 + a2I (t) = nI(t)-faB(t).
Here, S, I, V, and R refer to the susceptible individuals, infected individuals, vaccinated individuals, and recovered individuals, respectively. The pathogen population at time t is given by B(t). The parameter fa denotes the natural human birth and death rate, a denotes the rate of recovery from the disease, q represents the rate of human contribution to the growth of the pathogen, and fa2 represents the death rate of the pathogen in the environment. The coefficients p1 and p2 represent the contact rates for the human-environment and human-human interactions, respectively. Constants a1 and a2 adjust the appropriate form of the incidence which determines the rate of new infection. If a2 = 0, the corresponding incidence is reduced to the standard bilinear
H /N. S ev(t) ß\S(t)B(t) + ß2S(t)I(t) l + a1B(í) l + a2I(í) |- + )I(t) aI(t) ftR(í)
> L ftS(t) > |_KJ > People
nm ->[ Pathogen ft B(t) x El->
Figure 2: Progression of infection from susceptible (S) and vaccinated (V) individuals through the infected (I) and recovered (R) compartments for the combined human-environment epidemiological model with an environmental component.
form based on the mass action law, which is most common in epidemiological models. If a2 > 0, then the corresponding incidence represents a consequence of saturation effects: when the infected number is high, the incidence rate will respond more slowly than linearly to the increase in I. Similar meanings stand for al. The rate at which the susceptible population is vaccinated is 0,and therateatwhich thevaccine wears off is d. All parameters are assumed nonnegative.
The organization of this paper is as follows: the positivity and boundedness of solutions are obtained in Section 2. In Section 3, we obtain the existence of the endemic equilibrium. We get the local and global stability of the disease-free equilibrium in Section 4. In Section 5, we present the persistence of the system. In Section 6, we show the local and global stability of the endemic equilibrium. We analyze the sensitivity of RV on the parameters, and we present the numerical simulation in Section 7. The paper ends with a conclusion in Section 8.
2. Positivity and Boundedness of Solutions
In the following, we show that the solutions of system (2) are positive with the nonnegative initial conditions.
Theorem 1. The solutions (S(t),V(t),I(t),B(t)) of model (2) are nonnegative for all t > 0 with the non-negative initial conditions.
Proof. System (2) can be put into the matrix form
X' = M (X),
-J^-WL-ts-HS + evX
n 1 + a1B 1 + a2I n \
We have
Therefore,
$S-dV- faV ßiSB ß2SI
1 + a1B 1 + a2I
nI - fa2B
- (d + a + I
dS (t)
= +9V >0,
dt dI(t)
= $S>0,
I=o 1 + a1B
= qI > 0.
x,(t)=o,xt,c+ >0' i =12,3,4.
Due to Lemma 2 in [18], any solution of system (2) is such that
X(t) e R+ for all t > 0. This completes the proof of Theorem 1.
Theorem 2. All solutions (S(t),V(t),I(t),B(t)) of model (2) are bounded.
where X = (S, V, I, B)T e R4 and M(X) is given by
M(X) =
(Mx (X)\ M2 (X) M3 (X) \M4 (X)J
Proof. System (2) is split into two parts, the human population (i.e., S(t), V(t), and I(t)) and pathogen population (i.e., B(t)). It follows from the first three equations of system (2) that
d(S + V + I) dt
= (l-S-I-V)-dI-aI (1-S-I-V),
then it follows that lim supt_^ (S + V+I) < 1. From the first
equation, we can get
dS(t) dt
< fa - faS -<pS + 9V
< fa + 9 - (fa + 9 + <f>)S.
Thus dS(t)/dt < 0, as S(t) > (fa + 9)/(fa + 9 + ip). It is easyto obtain
dV (t) dt
<$(l-V)-(fa +9)V = $- (fa + 9+ $)V.
Thus dV(t)/dt < 0, as V(t) > $/(fa +0 + 0). From the last equation, we can obtain
dB (t) dt
Hence, dB(t)/dt < 0, when B > q/fa. Therefore, all solutions (S(t),V(t),I(t),B(t)) of model (2) arebounded. □
From above discussion, we can see that the feasible region of human population for system (2) is
QH = {(S,V,7) \S + V + I<1,0<S<
fa + 9 fa + 9 + <p'
fa + 9 + ip
, I>0\,
and the feasible region of pathogen population for system (2) is
nB = {B \ 0 < B <
Define Q = QH x QB. Let int Q denote the interior of Q. It is easy to verify that the region Q is a positively invariant region (i.e., the solutions with initial conditions in Q remain in Q) with respect to system (2). Hence, we will consider the global stability of (2) in region Q.
3. Equilibria
In this section, we investigate the existence of equilibria of system (2). Solving the right hand side of model system (2) by equating it to zero, we obtain the following biologically relevant equilibria.
It is easy to see that model (2) always has a disease-free equilibrium (the absence of infection, i.e., I = B = 0) E0(S0, V0, 0,0), where S0 = (fa1 + 9)/(fa1 + 9 + <p) and V0 = <f/(fal + 9 + $).
In the following, we will discuss the existence and uniqueness of the endemic equilibrium. The components of the endemic equilibrium E*(S*,V*, I*,B*) satisfy
6,S*B* 62S*I* t n t * fai --T-¿7 - 2 Tt - №* + 9V* - faiS* = 0, (13a)
1 + a1B* 1 + a2I*
- 9V* - faV* = 0, ß1S*B* ß2S*I
-(d + a + fa)l* = 0, (13c)
1+a1B* 1+a2I*
nl* - fa2B* = 0. (13d)
Substituting (13a), (13b), and (13d) into (13c), we obtain a single equation for I*:
ß2 \ (fa1 + - (d + a + fa) I*
fa2 + 1 + a2I
- (d + a + fa) I* = 0.
fa (fa +9 + (fi
After dropping the solution I* = 0, we obtain 01(l*) = 02 (ll'
(fa +9) [fa -(d + a + fa) I]
01 W = 02 (I) =
fa (fa +9 + tf d + a + fa
(ß^Kfa +«1ll)) + (&/(l+a2l)ï
Note that g1 (I) represents a straight line with a negative slope and a vertical intercept g1(0) = (fa1 + 9)/(fa1 + 9 + <f). Meanwhile, we have
02 (i) =
+ a + fa
(fa2 + Wfy + (fa2/ (1 + «2I)))
ß1a2l2
(fa2 + <*211I) (1+K21)
We see that g2(I) is increasing for I > 0, and g2(0) = ((fa + 9)/(fa + 9 + <p))(l/RV), where
(fa1 + 8) (ß2fa2 + ß1l) fa2 (fa1 + 9 + (d + a + fa)
is the control reproduction number of infection. When RV > 1, g2(0) < g1(0). Hence, there is one and only one intersection between the curves of g1(I) and g2(I); that is, there is a unique solution I* to the equation g1(I*) = g2(I*). Consequently, S*, V*, and B* are uniquely determined by I*.
Theorem 3. System (2) has a unique endemic equilibrium when RV > 1 and no positive endemic equilibrium when RV < 1.
4. Stability of Disease-Free Equilibrium
Now, we will discuss the local and global stability of the disease-free equilibrium.
Theorem 4. The disease-free equilibrium E0 is locally asymptotically stable for RV < 1 and unstable for RV > 1.
Proof. The Jacobian matrix of system (2) at X = E0 is
(-fai -<t> 0 -p2So -fa1So\
4> -e-fai o o
0 0 №0 -(d + a + fa1) faSo
\ 0 0 n -K J
The characteristic polynomial of the matrix J(E0) is given by det (XI - J (E0)) = a0X4 + a1X3 + a2X2 + a3X + a4, (20)
a0 = 1,
a1 = 2 fai + fa + d + (f> + ^
02 = (9 + 2 fa +4>)$i +$2 + fa2 + 2fafa + (Q + <f) (fai +fa), a3 = (2fai +$ + d)$2 + (faiQ + fa2 + fai<t>) Si
+ faifa2 (fai + 9 + <t>),
a4 = fafa (d + <f> + fa) (d + a + fa) (1 - RV), aia2 -a3 = ^i ((2fai + 41)2 + 4fai (d + hi)
+ 2fa ($ + e) + e2 + 29$)
+ $2h +$2 (d + 2fa + 4>) + 4fafa (d + $ + fai) + (2fa +9)fa + 3fa (<f> + 0)
i + h) + 2d$fa + 29(pfa + 2 fa3. =d + a + fai - faS0, $2 = fa2 (d + a + fai)- (faS0fa2 + faS0n).
If RV < 1, then
fa2 (d + a + fai) > fa2S0fa2 + faiS0n, i.e $2 > 0. (22) Further,
d + a + fai > fa2S0, i.e ^ > 0. (23)
So ^ > 0, a2 > 0, a3 > 0, a4 > 0, aia2 - a3 > 0, and aia2a3 > a^ + a^a4 (see Appendix A). Thus, using the Routh-Hurwitz criterion, all eigenvalues of J(E0) have negative real
part; E0 is local asymptotically stable for system (2). If RV > 1, then a4 < 0 and we show that J(E0) has at least one eigenvalue with nonnegative real parts. Consequently, E0 is not asymptotically stable. □
Theorem 5. When RV < 1, the disease-free equilibrium is globally asymptotically stable.
We will prove the global stability of the disease-free equilibrium using Lemma 6.
Lemma 6 (see [19]). If a model system can be written in the form
= P (X, Z),
— =G(X, Z), G(Z,0) = 0, dt
where X e Rm denotes (its components) the number of uninfected individuals and Z e Rn denotes (its components) the number of infected individuals including latent and so forth, U0 = (X*, 0) denotes the disease-free equilibrium of the system.
Assume that
(H1) for dX/dt = P(X,0), X* is globally asymptotically stable;
(H2) G(X, Z) = AZ - G(X, Z), G(X, Z) > 0 for (X, Z) e Q, where the Jacobian matrix A = (dG/dZ)(X* ,0) is an Metzler matrix (the off-diagonal elements of A are nonnegative) and Q is the region where the model makes biological sense. Then the fixed point U0 = (X*, 0) is a globally asymptotically stable equilibrium of cholera model system (2) provided that RV < 1.
We begin by showing condition (H1) as
P(X,0) = (fai-fal-v-Si+/). (25)
For the equilibrium U0 = (X* ,0), the system reduces to dS(t)
dt dV(t) dt
= fa - (t) - faS (t) + 9V (t),
= $s(t)-9V(t)-faV(t).
The characteristic polynomial of the system is given by
(X + fa)(X + d + $ + fa) = 0. (27)
There are two negative characteristic foots: Xi = -fa, X2 = -Q-<p-fai. Hence, X* is always globally asymptotically stable.
Next, applying Lemma 6 to the cholera model system (2) gives
G (X, Z)
= AZ -G (X, Z) _ (ßi^o -(d + a + fa) ß1S0\ 11
^B{S0 -T+^B)++lS2I{S0 -T+ki)).
So A is a Metzler matrix. Meanwhile, we find G(X, Z) > 0. Hence, the disease-free equilibrium is globally asymptotically stable.
5. Persistence
Persistence is an important property of dynamical systems and of the systems in ecology, epidemics, and so forth, they are modeling. Biologically, persistence means the survival of all populations in future time. Mathematically, persistence of a system means that strictly positive solutions do not have any omega limit points on the boundary of the nonnegative cone [20]. In this section, we will present the persistence of system (2).
For various definitions of persistence [21, 22], we utilize the definitions of persistence developed by Freedman et al. [23]. System (2) can be defined to be uniformly persistent if
min {lim inf S (t), lim inf V (t), lim inf I (t),
[ t —TO t —TO t —TO
lim inf B (f)} > e
t — TO J
for some e > 0 for all initial points in int Q.
A uniform persistence result given in [23] requires the following hypothesis (H) to be satisfied.
We denote that E is a closed positively invariant subset of X on which a continuous flow F is defined and N is the maximal invariant set of dF on dE. Suppose N is a closed invariant set and there exists a cover |N"a.|a6A of N, where A is a nonempty index set; Na c dE, N c Ua£AN"a, and |N"a|(a e A) are pairwise disjoint closed invariant sets. Furthermore, we propose the following hypothesis.
Hypothesis (H):
(a) all Na are isolated invariant sets of the flow F;
(b) INJ
agA is acyclic; that is, any finite subset of {-^dagA
does not form a cycle [24];
(c) any compact subset of dE contains, at most, finitely many sets of {Nja€A.
Lemma 7 (see [24]). Let E be a closed positively invariant subset of X on which a continuous flow F is defined. Suppose there is a constant e > 0 such that F is point dissipative on
{X : x e X, d(x, dE) < e} n int E and the assumption (H) holds. Then the flow F is uniformly persistent if and only if
(Na) n{X: x eX, d(x, dE) < e} n int E = 0 (30)
for any a e A, where W+(Na) = {y e X : A+ (y) c NJ.
Now, we can obtain the following result. Theorem 8. System (2) is uniformly persistent in int Q ifRV >
(28) Proof. Suppose RV > 1. We show that system (2) satisfies all the conditions of Lemma 7. Choose X = R4 and E = Q.
The vector field of system (2) is transversal to the boundary of Q on its faces except the S-axis and V-axis, which are invariant with respect to system (2) and on the S-axis and V-axis the equations for S and V are dS(t)/dt = fa - $S(t) -faS(t) + 0V(t), anddV(t)/dt = <pS(t)-6V(t)-faV(t),which implies that S(t) ^ (fa + 9)/(fa + 9 + <p) and V(t) ^ <p/(fa + 9 + <f) as t ^ >x>. Therefore, E0 is the only w-limit point on the boundary of Q. As the maximal invariant set on the boundary dQ of Q is the singleton |£0} and E0 is isolated when RV < 1, thus the hypothesis (H) holds for system (2). The flow induced by f(x) is point dissipative dE by the positive invariance of E. Because of W+(N) = {y e X : A+(y) c N}, where A+(y) is the omega limit set of y, when RV < 1,we have that E is contained in the set W+(N) and for RV > 1, W+(N) = 0. Therefore, the uniform persistence of system (2) is equivalent to P0 being unstable, and the theorem is proved. □
Remark 9. Theorems 3 and 8 show that RV is a threshold parameter for the model; that is, when RV < 1, its epidemiological implication is that the infected fraction of the population vanishes, so the cholera dies out; when RV > 1, the disease is endemic and the infected fraction remains above a certain positive level for sufficiently large time.
6. Stability of the Endemic Equilibrium
6.1. Local Stability of the Endemic Equilibrium. Now we consider the case with RV > 1. The stability of the endemic equilibrium is established as follows.
Theorem 10. If RV > 1, E*(S\V\r,B*) is locally asymptotically stable.
Proof. Let
\+alB* 1 + a2I*
(l + «21*)
(1 +a1B*)
Journal of Applied Mathematics The Jacobian matrix at E*(S*,V*,I*,B*) is J(E*)
f-h -4 0 -j2 -j\
_ 0 -9- fa 0 0
_ h 0 h -(d + a + fa) J3 '
\ 0 0 v -fa)
The characteristic polynomial of the matrix J(E*) is given by det (XI - J (E*)) _ b0X4 + b1X3 + b2X2 + b3X + b4, (33) where
b1 _ J1 + 2 fa + fa2 + 9 + $ + w1, b2 _ (9 + 2fa1 + + w2 + fa2 + 2fa1fa2
+ (9 + <p) (fa + fa2) + (a + 2 fa + fa2 + d) J1, b3 _ (9 + fa + fa1w1 + (2 fa + 9 + <p)w2 + fafa (9 + fa + + aj1 (9 + fa + fa2)
+ J1 (fa +(d + 9) (fa + fa2) + 2fafa + 9d), b4 _ + 9 + fa) faw2 + (d + a + 9 + fa) fa faj1
+ (d + a) dfa2J1,
hh2 -b3 {4fa1 (9 + $ +fa +fa) + ($ + 9)2
+ J1 (a + 29 + fa2 + d + ip)j1 + 29fa
+ h + 2fa$}
+ w'2 (9 + 2 fa + + w2 + w2 (fa + J1) + fa (2fa + 9 + J1 +$) + (a + 2 fa + fa2 + d + 9) + 3fa + 9) + (9 + $)2 (fa + fa) + 92h
+ $J1 (9 + a + d + 3fa) + 4fafa (h +9 + $) + 2hfa (9 + <f>) + 4fa (J1 + fa2) + 2fa3 + }1fa1 (d + a + 49). w1 _ d + a + fa - J2, W2 _fa (d + a + fa) - (j2fa + J3I).
Based on (13c)and (13d), we have fa2(d+a+fa) _ (faqS*/(\ + a1B*)) + (fafaS*/(1 + a2I*)). It is then easyto observe that
fa2 (d + a + fa) > faj2 + i.e w2 > 0. (35)
Further,
d + a + fa > J2, i.e w1 > 0. (36)
So b1 > 0, b2 > 0, b3 > 0, b4 > 0, b1b2-b3 > 0, and b1b2b3 > b3 + b^b4 (see Appendix B). Using the well-known Routh-Hurwitz criterion, the proof is thus complete. □
6.2. Global Stability of the Endemic Equilibrium
Theorem 11. When RV > E* (S* ,V* ,1*, B*) is globally asymptotically stable in Q \ Q0 if d + a < fa +9 and (2fa(fa +9)/(fa +9 + $)) + 9<fa.
This approach to global dynamics is developed in the papers of Smith [25] and Li and Muldowney [26-28]. Let f _ (f1,f2,f3,f4)T, where fuf2,f3, and f4 represent the right-hand sides of system (2), respectively. Furthermore, let x _ (S, V, E, I) . Then, the Jacobian matrix for system (2) is
The second additive compound matrix (see Appendix C for details) of J is
\ + a1B l+a2l) r1 r
0 -9 - fa
(l+a2l)2 (l + a1B)2
J^ + J^ 0 -(d + a + fa) 2
1 + a^ 1 + a2I (1 + ^I)2 (1+a1B)2
( 0 0 n -fa )
j[2] =
ß1B 1 ß2I
1 + a1B 1 + a2I 0
(1 + a2I)2 (1 + a1B)2
(1+a1B)2
ßiB + ß2!
1 + a1B 1 + a2I I
(1+a1B)2
(1+a1B)2 J55 0
J11 = J33 + 2 -fa1
J22 = J44 + d-ß1B
1 + a1B 1 + a2I
___ßj±_
1 + a1B 1 + a2I
-fa -$-fa >
(1+«2l) J55 = -fa1 -fa2
- 2 fa - d - d - a,
0 0 0 0 0 1
0 - 0 0 0 0 I
0 0 0 1
0 0 0 0 -0 B
0 0 0 0 0
J 66 =
(1+«2l)
2 -fa -fa - a- a.
Then we have
df [2]
b = QfQ-1
ß1B ß2I
(1+«2l)
022 ö
1 + a1B 1 + a2I
l(1 + a1B)2 0
l(1+a1B)2 0
ß1 BS
ß1 B , ß2l
1 + a1B 1 + a2I I
l(1+a1B)
l(1+a1B)2 0
(1+«2l) 0
where 011 =
. . -9 + d + a+ — + a44,
l(1+a1B) 1 + a21 B y44
ß2tX2SI , I1
I(1+<X1B) (1 + a21)2 + B+044'
ß2tX2SI
-fa - 0,
l(1 + aiB) (1 + a2l)'
ß1 B ß2l nl
1 + a1B 1+a2I B fa
#55 = - -J - fai - 0>
^66 = --TJ - - Fl - d - «-
In (41), Qy is the directional derivative of Q in the direction of the vector field / in system (2).
Here, we will use the theorem in [28] to give a sufficient condition on the parameters, which when satisfied, implies that the endemic equilibrium is globally asymptotically stable.
Lemma 12 (see [28]). If Qj is a compact absorbing subset in the interior of Q and there exists v >0 and a Lozinskii measure ^(8) < -v for all x e Q^ then every omega point of system (2) in int Q is an equilibrium in Q.
For RV > 1, the disease-free equilibrium is repelling towards the interior. In fact, there is a compact absorbing set in int Q which attracts all orbits that intersect int Q. This gives the following results.
Corollary 13. If RV > 1 and there exists a Lozinskii measure F(B) such that ^(8) < 0 for all x e int Q, then each orbit of system (2) which intersects int Q limits to the endemic equilibrium.
For a norm || • || on R", the Lozinskii measure ^ associated with || • || can be evaluated for an nxn matrix T as
= inf {
o : D+ ||z|| < o ||z||, for all solutions of z = Tz \,
where D+ is the right-hand derivative [29]. Hence, if we can find a norm on R6 for which the associated Lozinskii measure satisfied ~p(B) < 0 for all x e int Q then the endemic equilibrium is globally asymptotically stable for RV > 1.
We now define a norm on R6 [30] for which the definition varies from one orthant to another. Let
jjzjj = max
M ,Ü2
where z e R6, with components z; (i = 1,2, 3,4, 5, 6) and
max { | z^ , | z2 | + | z3 | },
if sgn = sgn (z2) = sgn (z3) ,
max {|z2|, | 2— + |z3|},
if sgn = sgn (z2) = - sgn (z3) , max { | 1 , | ^2 | , | ^3 | } ,
if sgn (zi ) = - sgn (z2) = sgn (z3^
max{|z—| + |z3|, |z2| + |z3|},
if - sgn (zj = sgn (Z2) = sgn (Z3) ,
Ul (Zi,Z2,Z3) =
^2 (^4,Z5,Z6) =
|Z4| + |Z5| + |Z6| ,
if sgn (z4 ) = sgn (z5 ) = sgn (z6) ,
max { | z4 | + | z5 |, | z4 | + | z6 | },
if sgn (^4) = sgn (z5) = - sgn (z6) , max { | ^5 | , | ^4 | + | ^6 | } ,
if sgn (z4 ) = - sgn (z5) = sgn (z6) , max { | ^4 | + | ^6 | , | ^5 | + | ^6 | } ,
if - sgn (Z4) = sgn (Z5) = sgn (Z6) .
Theorem 14. Assume that
d + a < fa + 0,
2fa (Fi +0)
+ 0 < fa.
Then there exits a = maxj-fa, d + a-fa -0, (2jS2(fa +^)/(fi + 0 + 0)) + 0 - fa} < 0, such that D+||z|| < ct||z|| for all z e R6.
Proof. We should show that
Ö+ jjzjj
2fa2 (fa + 0)
-fa^+a-fa --—- + 0-fa
(47) □
Case 1 (U:(z) > U2(z) and zj,z2,z3 > 0). In this case, ||z|| =
max{|Zl|,|z2l + |Z3|}.
Subcase 1.1 (|zj > |z2| + |z3|). Then ||z|| = |zj = zx and U2(z) < |zj. Taking the right-hand derivative of ||z||, we obtain
D+ jjzjj = zj
faBS ftS faB ^
7(l+^ß) 1 + a27 1 + ^B 1 + a27
+ d + a - fa - 0 - zx
fts abs
Z* + ■
(l + a2;)2 7(1+
Since |z5| < U2(z) < |^|,|z3| < |^|,wehave
n IMI < ( faB
jj^N < I--;-r------
^ 1 7(1+^£) 1 + a2/ 1 + 1 + a2/
+ d + - 0- 0)
| , ftfiS
(1 + a2;)2 7(1+
ß1a1B2S ß2a2SI ß1B
l(\ + a1B)2 (l + a2l)2 l+«iB ßil
1 + a2I
+ d + a - fa -d -<p ¡{z^.
Hence,
ß1a1B2S ß2a2SI ß1B
l(l + a1B)2 (l + a2l)2 1 + aiB
1 + a2I
+ d + a - fa -9 - <p
It is easy to see that (50) also holds for U1 > U2 and z1,z2,z3 < 0 when \z1\ > \z2\ + \z3\, which canbe obtained by linearity.
Subcase 1.2 (\z1 \ < \z2\ + \z3\). Then \\z\\ = \z2 \ + \z3 \ = z2 + z3 andU2(z) < \z2\ + \z3\.
Since \z4 + z5 + z6\ <U2(z) < \z2 \ + \z3\, taking the right-hand derivative of \\z\\,we obtain
ß1B ß2l
1 + a1 B 1 + a2I,
ß1BS ß2«2SI
I(1 + «1B) (1
ß1B ß2l
1 + a1B 1 + a2I ß1BS ß2«2SI
fa1 )Z2
I(1 + «1B) (1 + «2-0
2 -fa1 )Z3
-2 (z4 + z5 + z6)
i(1 + a1B)2 5
ß1BS ß2 a-,SI
I(1+<*1B) (1
-fa1 ) (\ Z2 I + I Z3 I )
l(1+a1B)
2 (\ Z4 + Z5 + Z6
ß1a1B2S ß2a2SI
Therefore, D+ INI
l(1+a1B)2 (1 + a2l)2
< ( ß^B'S ß2a2^I ~ l(1+a1B)2 (1 + a2l)2 1
-fa1 )( I z2 | + | z3 I )-
I - (52)
It is easy to see that (52) also holds for U1 > U2 and z1,z2,z3 < 0 when \z1 \ < \z2 \ + \z3 \,which canbe obtainedbylinearity.
Case 2 (U1(z) > U2(z) and z1 < 0 < z2,z3). In this case, \\z\\ = max^zj + \Z3\,\Z2\ + NH.
Subcase 2.1 (\z1\ > \z2\). Then \\z\\ = \z1\ + \z3\ = -z1 +z3 and U2(z) < \z1\ + \z3\. Taking the right-hand derivative of \\z\\, we obtain
ß1BS ß2S , „
+ --; - d - a + fa1 + 9 + 0 )z1
1(1 + a1B) 1 + a2I
ß2S ß1BS
+ $z2 + ( -
1 + a2I I (1 + a1 B) ß1BS ß2S
-fa1 -d)z3
7(1 + a1ß) 1 + a2I ß2S ß1BS
1 + 021 /(1 + a1ß) ß1BS ß2S
+ d + a - fa1 - 9 ) ^^
-fa1 -d ) N
7(1 + a1ß) 1 + a2I
+ d + a -fa1 -9
Therefore,
D+ \\z\\<\-^ß1BS ^--¿2L + d + a-fa1 -e
7(1+a1ß) 1 + a2I
It is easy to see that (54) also holds for U1 > U2 and z2, z3 < 0 < z1 when \z1\ > \z2\,which canbe obtainedbylinearity.
Subcase 2.2 (\z1\ < \z2\). Then \\z\\ = \z2\ + \z3\ = z2 + z3 and U2(z) < \z2\ + \z3\. Taking the right-hand derivative of \\z\\, since \z4 + z5 + z6\ < U2(z) < \z2\ + \z3\, we obtain
= Z2 +Z3
ß1B ß2l
1 + a1B 1 + a2I
ß1BS ß2a2SI ß1B
I(1 + «1B) (1+a2l)2 1+«1l ßil
1 + a2I
fa1 )Z2
= -Z1 + Z3
ß,BS ß2a2SI + ( --+1-- - '2 2 ,2 -Vi
I(1 + a^B) (1 + a2I)2
ßi B S
(z4 + z5 + z6)
I(1 + aiB)
1 + a1B 1 + a2I
ßiBS ß2«2SI ßiB
I(1 + «1B) (\ + a2l)2 1+<*1
1 + a2I
Vi llZ2l
ßiBS ß2«2SI u ,z
+ ( ---~--~2 -fa )z3
I(l + aiB) (l + a2I) ßi B S
-2 (\z4 +z5 +z6
I(l + alB)2 ' 4 5 6
ßiaiB2S ß2a2SI
I(l + «iB) (1
-Vi )(Ы + Ы)-
Therefore,
< ^Ы + кзР
^--ß^-ß^-nl-^ i, IZ4I
1+aiB 1 + a2I B ri ' 1 41
+ \-B-Vi)\z5\+ ,л ^ л 2 lz61 B ' (1+«2I)
(1+*2I)2
-Vi1 (Ы + hd-
Recalling that \\z\\ = \z4\ + \z5\ yields
D+ \\z\\ <
(1+*2I)2
It is easy to see that (58) also holds for U2 > Uy and z4, z5 < 0 < z6 when \z5\ > |z6|,which canbe obtainedbylinearity.
Subcase 3.2 (\z5\ < \z6\). Then \\z\\ = \z4\ + \z6\ = z4 - z6 and U1(z) < \z4\ + \z6\. Taking the right-hand derivative of \\z\\, we obtain
(55) D+
_ I I _ ,
= z4 - z6 = Bz2
1 + aiB 1 + a2I B ri r ' 4
D+ w^-J^-J*^-fa
I(1 + aiB) (1 + a2I)2
It is easy to see that (56) also holds for Uy > U2 and z2,z3 < 0 < zy when \z1\ < \z2\, which can be obtained by linearity.
Case 3 (U2(z) > U1(z) and z6 < 0 < z4,z5). In this case, \\z\\ = max{\z4\ + \z5\, \z4\ +
Subcase 3.1 (\z5\ > \z6\). Then \\z\\ = \z4\ + \z5\ = z4 + z5 and Uy(z) < \z4\ + \z5\. Taking the right-hand derivative of \\z\\, we obtain
= z4 + z5 = — z2
ßiB ß2I nI
1 + aiB 1 + a2I B
-fa - ф] z4 + 0z5
(1+«2I) nI
z6 + -¿z3 + Фz4
+ (-~B-Vi -0>z5
a I 2ßiS nI , + 9z5 -( --~ -B-Vi -d-a lz6
< — IzJ + ( -
(1 + CX2I)
2ßiB 2ß2I -£ 1 + aiB 1 + a2I B
+ в \z5\ +
- fa -<fr]\z
2ß2S nI
(1+*2I)2
■fa - d - a )\z6
2 ß2 S
(1+a2I)2
+ d-fa )(\z4\ + \z6\).
Therefore,
D+ \\z\\ <
2 ß2 S
(1+a2I)
+ в - fa
It is easy to see that (60) also holds for U2 > Ui and z4, z5 < 0 < z6 when \z6\ > \z5\,which canbe obtainedbylinearity.
Combing the results of the six cases presented here in (50)-(60), as well as the remaining 10 cases, we obtain the result
D+ \\z\\
-fai,d + a - fa -9,
2ßi (fai + 9) fa +9 + <p
+ 9 - fa
when I > 0, S < (fai + 9)/(fa +0 + 0). Hence, fa(B) < max{-fai,d + a- fa1 -9, ((2p2(fai + d))/(fai +9+ <p)) + 9- fax}. Therefore, fa(B) < 0 in int Q if d + a < fai +9 and ((2p2(fai + 9))/(fai + 9 + $)) + 9 < fai. Thus, Theorem 14 is satisfied. From Corollary 13, we can obtain that all solutions intersect the interior of Q and the endemic equilibrium E* is globally asymptotically stable, completing the proof of Theorem 11.
7. Sensitivity Analysis of RV
To facilitate the interpretation of the sensitivity of RV, we now present some numerical simulations by using the set of parameters values in Table 1.
Now, we regard the vaccinated rate 0 and the waning rate 9 as the control parameters, while the other parameters are fixed. From Figures 3 and 4, the effects of various parameters, that is, 0 and 9 on the control reproduction number RV have been shown. It is noted that as the parameter 0 increases, RV decreases and as 9 decreases, RV decreases. In fact, we can obtain the critical values of 0 and 9 that reduce RV to 1:
(fai + d) (faißi + ßit1) - fai (fai +9){d + a + fa) fa2 (d + a + fa)
fai jfaißi + ßi1) - fai (fai +<f)(d + a + fai) (faißi + ßin)-fai (d + a + fa)
= 9„.
In Figure 3, we select 9 = 0.07, 0.03,0.007,0.0001, corresponding = 2.07,0.89,0.21,0.01, respectively. We can see that when the wanning rate 9 has a greater value, then there is no <pv such that RV < 1. Similarly, in Figure 4, we select <p = 0.01, 0.1,0.3,0.6, 0.99, corresponding 9V = 0.0002,0.003,0.01,0.02,0.03, respectively. We can see that when <p is smaller, then there is no 9V such that RV < 1. Thus, the control reproduction number cannot reduce below unity only by increasing 9 or decreasing 0. The critical values <pv and 9V are important in regulating the infection magnitude. In order to reduce RV to 1, a greater vaccinated rate than and a smaller wanning rate than 9V have to be achieved simultaneously. We will deduce RV below 1 by using both <p and 9 at the same time, which can control cholera. Otherwise, the cholera persists (see Figure 5).
8. Conclusion
In this paper, we have conducted global stability analysis of a SVIR-B cholera model. Based on the imperfective vaccine, with the environment component incorporated and multiple
0.4 0.6
Figure 3: The contour diagram of the control reproduction number RV with (p when 0 has some fixed value.
0.4 0.6
Figure 4: The contour diagram of the control reproduction number RV with 0 when (p has some fixed value.
transmission pathways coupled, the cholera models distinguish themselves from regular SIR epidemiological models. The mathematical results show that the control reproduction number RV satisfies a threshold property with threshold value 1. When RV < 1, it has been proved that the disease-free equilibrium E0 is globally asymptotically stable under some sufficient conditions. And, when RV > 1, the unique endemic equilibrium E* is globally asymptotically stable. This shows that cholera can be eliminated from the community if the imperfect vaccine brings RV to a value less than unity.
Table 1: Estimation of parameters.
Parameters Meaning Values Reference
Natural human birth and death rate 9.13 x 10-5/day [3]
ßl Contact rates for the human-environment interaction 0.214/day [3]
ß2 Contact rates for the human-human interaction 0.02/day [3]
d Disease-induced death rate 0.013/day [4]
a Recovery rate at which people recover from environment 0.2/day [3]
n Contribution of infected individuals to the 10 [7]
population of Vibrio cholera
Net death rate of Vibrio cholera 0.33/day [7]
0.4 '^t^rnw
e 0 0.2 0.4 0.6 0.8 1
Figure 5: The contour diagram of the control reproduction number RV with (¡>, 0 variables. All the other parameter values are the same as those in Figure 3.
Now, if we consider that the case there is no vaccination. Model (1) can be written as follows:
dS (t) dt
ßx S(t)B(t) ß2S(t)I(t) 1 + axB (t) 1+a2I(t)
-faS(t),
dl(t) ß1S(t)B(t) ß2S(t)I(t)
+ --r— - (d + a + fa) I (t),
dt 1 + axB(t)
dR(t) dt dB (t) dt
1+«2l(t) = al (t) - faR (t),
= nI(t)-faB(t).
According to Theorem 2 in [31], the basic reproduction number of model (*) is R0 = (fi2fa + p1^)/(fa2(d + a + fa)). We can express the control reproduction number RV as RV = R0((fa + 9)/(fa + 9 + 0)). Note that RV < R0 with equality only if 0 = 0. That is, despite being imperfect,
the vaccine will always reduce the reproduction number of the disease, and in the absence of the vaccination, the disease transmission will be high. Further simulations in Section 7 also show that the vaccinated rate (0) and the wanning rate (0) play equal important roles in reducing the control reproduction number. The disease can be controlled if and only if the reproduction number is reduced to values less than unityifthe vaccinatedrate(0) exceed the threshold 0V and the wanning rate (0) less than the threshold 0V simultaneously. Therefore, vaccination is a good way to control cholera.
However, there are inherent disadvantages towards the vaccination modeling. For cholera with incubation period, it is hard to rapidly identify those with ambiguous symptoms [4]. Moreover, the vaccination does not always work well due to the limitations of medical development level and financial budget (some vaccines are very expensive and some portions of people cannot be covered) [32]. Nonetheless, [33] indicated that during cholera outbreaks (periodic control) vaccination campaigns can be a good strategy to control cholera epidemics. Besides, they pointed out that vaccination and improvement in the sanitation system and food/personal hygiene are the most efficient control strategies to prevent cholera transmission and outbreaks. Hence, incorporating some other control strategies, we may consider the more realistic ordinary differential equation model. The theoretical study of cholera models has been in progress and is an exciting area of future research.
Appendices
A. The Proof of a1a2a3 > a^+a^o4
The proof of axa2a3 - a^ - a^a4 = ^faKd + <p)2 + fa(3Q + (*) 2fa + 3$)} + ï2{(4fa(e + fa+$) + (d + $)2 + $2 + fa(d + $)3 +
3fafa2(<p + d)2 + fa2(8fai + 9fa)(9 + $) + + 29) + fa2(4fa2 +
5d2 + 6fafa )} + ^ {(4fa3 +fa(8fa + 6fa)(d + + 4fa(d + $)2 + $fa (2$+l+4d) + (d + $)3 +fa2 (2d2 + 8fa2))$2 + ((p + d + 2fa)^:2 + fa2(4fa + l0fa )(9+4>)2+7d$dfa+fafa(2d3 +602^+2^>3 + 5d^) + fa\(6fa2 + 2fa2) + 3fafal(d2 + $2) + + I6fa\fa + 5fa4)(9 +
$) + fa2(492 + $3 + 39$2 + 93 + 8fa2 fa2 + 392$)} + i,lfa{$ + 2 fa + 9} + ^{fa2(d + $)3 + faMfa + fa)(9 + $)2 + fafa(4fa + 6fa)(d + $) + Afa^fa^} + fafa(fa + fa)(9 + $)3 + 4fa3fa(4fa2 + 5fafa + fa2)(9 +<p)2 +fa2fa(5fa2 + 8 fa fa + 3fa2)(9 + <p) + fa3fa2(2fa + 4 fa).
From Section 4, we know that ^ > 0, £,2 > 0. After some algebraic manipulations, we have a1a2a3-a^-a^a4 > 0. Thus, a1a2a3 > a^ + a^, when RV < 1.
B. The Proof of b1b2b3 > b^+bfy
The proof of blb2b3 - bl - b^b4 = w3fa {(0 + $)2 + 3fa (9 + $) + 2^} + w2^ {&2 ((9 + $)2 + 4 fa (9 + $ + fa)) + fafa((4fa + a + 49 + 3$+d)]1 +3(9+$)2 + (6+9$)fa)+fa(9+$)3 + (d+a+d)-$J1fa + 5^2(9 + $)2 + 99fa2(fa + J1) + (<p + 2d)cpJ1fa + 92J1 (fa2 + 3fa) + fa3(4fa +89)+3J1fai(d+a+2$)+9J1(9+$+4fa1)(d+a)+49$fa1 + 6]1faL +2a$faJ1 +8$^}+^ ■{w22($+2fa +9)+tM2(4fa($+9)2 + 4fafa(2fa+2Q+2<p+J1)+<pfa (2$+J1 +4d)+6^2(d+4>)+3d4>J1 + 2dJ1(d + a + d + fa) + (3a + 3d + 5$)faj1 + J1$($ + a + d) + 292 ■ fa2 + 8dfaJ1 + (9+$)3 +4fa^(2J1 +fa)+J1fa(d+a))+fa(fa+fa + 2J1)(9+$)3 + (4^3 + (d+a+9)fa J1 +fafa(10fa +3fa) + (d92 + 4fa2+a9)J1 )(9+$)2 + fai(9fa2 + 4fa2)(e+$) + 8fai(fa + j1) + 2fa'l + fai(5d+169 + 6]1 + 5a+11$)J1 + 3fa2h((2aJ1 + 6d+$)J1 +2d2 + 39(d+a+J1)+2(d+a+9)$)+fa (((a+d)2+8(d+a)d+2(d+a+ 9)$+392)]2 + (3dd2 + 5ad2 + $(d+a)(2$+79))J1)+fafa ((5a+ 89 + 3$ + 5d)j1 + 5((d + a)(d + (p) + 3d(p + (p2)J1) + fa2((d + a + d)(J1 + 2$) J1 + 2d2J1) + faj2((d + 9)2 + (a + d)2 + (d + a + 9)$ + 2ad)+2fafa1J1 (dj1 + 49+3$+a)+fa fa2(8J^ + 16fa(9+$+J1) + (5a+5dfa +24e+19<p)J1)+fa2fa1(6fa +8]1)+]2 (d(a+d)2+0(d+ a)(29+ifi))}+ui22{ifij1 +9 fa+2faJ1(pfa+dj1 +2 fa fa }+w2 {(fa(d+ $)3 + (2fa2J1 +4fafa+9J1)(9+$)2 +2 fa J1(3d+d+a+2fa +2J1 + 2$) + fa1((49 + 3a + 3d + $)f^ + ((9 + $)2 + 3$(d + a) + 9($ + a + d))J1)+fa2((a+d+<p+2d)J1+(<p+2d)<p)+fa faJ1((d+a+89+7$+ 4J1)+4(f2 + 8d(p)+fa J1((a+d+$+29)J1 + (d+a)9) + 6fa2fa(4>+ d)+4fa1fa2(<p+d+J1 +fa)+(d<fi+d9+a<fi)ifij1+((a+d+9)$+(2d+ 9 + 2a)9)j1} + (fafa(29J1 +fa2 +4fa2 + 5fafa) + dJ1fa2)(d + 4>)2 + fafa(fa2+fa)(9+$)3 +fa fa{4J3i(d+a+9)+jl(6(d+a)(9+$i) + (d+a)2 +d(7d+6(p))+(d+a)(d+(p)(d+2(p)J1}+fa31fa{4J31 +(7a+ 7d+15d + 8^)j1 + (5$(a + $ + d) + (16+15$+109 + 4d)9)J1 +
4da}+fa1{J1(2a+2fa+59+3$+2 ■ d)+4]2}+fai{2f^ +f^(5d+5a+
1 1 1 1 1 1 12 2
5a<p)} + 4^92(a + d))J1 + $2J1(d + 9 + a) + fa{(d(4d + d + 4a) + (a + d)2)jl + (9(d + 9)2 + a(9 + $)(a + 2d) + ($ + 3d + 5a)d2 + 5$9(d+a)+$d2)J2^ + (9(4>2 + e2)(d+a) + 2$e2a)Jl}+fa2{f^(a+
e+d)+j1(d2+$(d+a+e))}+fa^,{j3^(a+d+e)+j12((2$+e)(e+
a+d) + e)+J1$(d+a)(e+$)}+fa2{J3^((d+a)2+2d(a+d) + d2) + f^(d2d + (a + d)d2 +$(d + 9)2 + $a(a + 2d + 2d))} + fa^Ml + J'2 (3a + 3d+5ifi + 89)+J1(7d2 +119$+ 4$2 + (9 + 3$)(d + a))} + fa1fa3,(J1(d + 3$ + 49 + a + 2J1)) + fa\fa(10j2 + J1(13$ + 3a + 3d)) + fa2fa2{J1(8J1 +2d + 4fa +10fa + 13$ + 2a+159) + (3fa + 8fa)(9 + $)+4fa2 + 2fafa} + faAlfa{8J1 + 5$ + 59 + 2fa} + J3 {9(a+ d)2 + 92a} + J^{9$d2 + 92(d + a)(9 + $)} + 2d9$(a + 9J1).
From Section 6, we know that w1 > 0,w2 > 0. After some algebraic manipulations, we have b1b2b3 - b3 - b^bA > 0. Thus, b1b2b3 > b2 + bfy, when RV > 1.
C. The Second Additive Compound Matrix
If A = [atj ] is an nxn matrix, its second additive compound is (2) x (2) matrix defined as follows. For any integer i = 1,..., (2), let (i) = (i1t i2) such that 1 < i1 < i2 < n. Then the element in the «-row and j-column of A[2 is
%h + %i2, if 00 = (i), (-1)r+sat j, if exactly one entry ir of (i) does not occur in (j) and js does not occur in (i), 0, if neither entry from (i) occurs in (j).
In the special case when n = 2, we have A[2
= an + On =
tr A. In general, each entry of A[2 is a linear expression of those of A. For instance, when n = 3, the second additive compound matrix of A = (atj) is
A[2] =
a-1-1 + a
au + a-
3$ + 89) + h (49 +3$(d+a) + $ +59(d + a + $))} + fa^{J^(3h + when n = 4, the second additive compound matrix of A = 4$)(9+a+d) + (99(a+d) + (d+a)2 + 592) + (9(92 +2$9 + 5d$ +
(ai}) is
A[2] =
/ an +fl22 a23 a24 -a13 a14 0
a32 a11 + a33 a34 a12 0 -a14
aA2 aA3 + fl44 0 a12 a13
-a31 a21 0 a22 + a33 a34 -a24
-a41 \ 0 a21 a43 a22 + a44 a23
( 0 -a41 a31 -a42 a32 a33 + a44
For detailed discussions of compound matrices and their properties, readers can refer to [34, 35]. A comprehension survey on compound matrices and their relations to differential equations is given in [34].
Acknowledgments
This work is supported by the National Natural Science Foundation of China (no. 11071011), Funding Project for Academic Human Resources Development in Institutions of Higher Learning under the Jurisdiction of Beijing Municipality (no. PHR201107123), and Basic and Frontier Technology Research Program of Henan Province (nos. 132300410025 and 1323000410364).
Conflict of Interests
The authors declare that they have no financial and personal relationships with other people or organizations that can inappropriately influence their work. There is no professional or other personal interest of any nature or kind in any product, service, or company that could be construed as influencing the position presented in or the review of the paper.
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