Salmonella enterica subsp. enterica serovar Enteritidis remains one of the most important foodborne pathogens worldwide. To minimise its public health impact when outbreaks of the disease occur, timely investigation to identify and recall the contaminated food source is necessary. Central to this approach is the need for rapid and accurate identification of the bacterial subtype epidemiologically linked to the outbreak. While traditional methods of S. Enteritidis subtyping, such as pulsed field gel electrophoresis (PFGE) and phage typing (PT), have played an important role, the clonal nature of this organism has spurred efforts to improve subtyping resolution and timeliness through molecular based approaches. This study uses a cohort of 92 samples, recovered from a variety of sources, to compare these two traditional methods for S. Enteritidis subtyping with recently developed molecular techniques. These latter methods include the characterisation of two clustered regularly interspaced short palindromic repeats (CRISPR) loci, either in isolation or together with sequence analysis of virulence genes such as fimH. For comparison, another molecular technique developed in this laboratory involved the scoring of 60 informative single nucleotide polymorphisms (SNPs) distributed throughout the genome. Based on both the number of subtypes identified and Simpson's index of diversity, the CRISPR method was the least discriminatory and not significantly improved with the inclusion of fimH gene sequencing. While PT analysis identified the most subtypes, the SNP-PCR process generated the greatest index of diversity value. Combining methods consistently improved the number of subtypes identified, with the SNP/CRISPR typing scheme generating a level of diversity comparable with that of PT/PFGE. While these molecular methods, when combined, may have significant utility in real-world situations, this study suggests that CRISPR analysis alone lacks the discriminatory capability required to support investigations of foodborne disease outbreaks.
Citation: Susan Nadin-Davis, Louise Pope, John Devenish, Ray Allain, Dele Ogunremi. Evaluation of the use of CRISPR loci for discrimination of Salmonella enterica subsp. enterica serovar Enteritidis strains recovered in Canada and comparison with other subtyping methods[J]. AIMS Microbiology, 2022, 8(3): 300-317. doi: 10.3934/microbiol.2022022
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Salmonella enterica subsp. enterica serovar Enteritidis remains one of the most important foodborne pathogens worldwide. To minimise its public health impact when outbreaks of the disease occur, timely investigation to identify and recall the contaminated food source is necessary. Central to this approach is the need for rapid and accurate identification of the bacterial subtype epidemiologically linked to the outbreak. While traditional methods of S. Enteritidis subtyping, such as pulsed field gel electrophoresis (PFGE) and phage typing (PT), have played an important role, the clonal nature of this organism has spurred efforts to improve subtyping resolution and timeliness through molecular based approaches. This study uses a cohort of 92 samples, recovered from a variety of sources, to compare these two traditional methods for S. Enteritidis subtyping with recently developed molecular techniques. These latter methods include the characterisation of two clustered regularly interspaced short palindromic repeats (CRISPR) loci, either in isolation or together with sequence analysis of virulence genes such as fimH. For comparison, another molecular technique developed in this laboratory involved the scoring of 60 informative single nucleotide polymorphisms (SNPs) distributed throughout the genome. Based on both the number of subtypes identified and Simpson's index of diversity, the CRISPR method was the least discriminatory and not significantly improved with the inclusion of fimH gene sequencing. While PT analysis identified the most subtypes, the SNP-PCR process generated the greatest index of diversity value. Combining methods consistently improved the number of subtypes identified, with the SNP/CRISPR typing scheme generating a level of diversity comparable with that of PT/PFGE. While these molecular methods, when combined, may have significant utility in real-world situations, this study suggests that CRISPR analysis alone lacks the discriminatory capability required to support investigations of foodborne disease outbreaks.
In this article, we study the oscillatory behavior of the fourth-order neutral nonlinear differential equation of the form
{(r(t)Φp1[w′′′(t)])′+q(t)Φp2(u(ϑ(t)))=0,r(t)>0, r′(t)≥0, t≥t0>0, | (1.1) |
where w(t):=u(t)+a(t)u(τ(t)) and the first term means the p-Laplace type operator (1<p<∞). The main results are obtained under the following conditions:
L1: Φpi[s]=|s|pi−2s, i=1,2,
L2: r∈C[t0,∞) and under the condition
∫∞t01r1/(p1−1)(s)ds=∞. | (1.2) |
L3: a,q∈C[t0,∞), q(t)>0, 0≤a(t)<a0<∞, τ,ϑ∈C[t0,∞), τ(t)≤t, limt→∞τ(t)=limt→∞ϑ(t)=∞
By a solution of (1.1) we mean a function u ∈C3[tu,∞), tu≥t0, which has the property r(t)(w′′′(t))p1−1∈C1[tu,∞), and satisfies (1.1) on [tu,∞). We assume that (1.1) possesses such a solution. A solution of (1.1) is called oscillatory if it has arbitrarily large zeros on [tu,∞), and otherwise it is called to be nonoscillatory. (1.1) is said to be oscillatory if all its solutions are oscillatory.
We point out that delay differential equations have applications in dynamical systems, optimization, and in the mathematical modeling of engineering problems, such as electrical power systems, control systems, networks, materials, see [1]. The p-Laplace equations have some significant applications in elasticity theory and continuum mechanics.
During the past few years, there has been constant interest to study the asymptotic properties for oscillation of differential equations with p-Laplacian like operator in the canonical case and the noncanonical case, see [2,3,4,11] and the numerical solution of the neutral delay differential equations, see [5,6,7]. The oscillatory properties of differential equations are fairly well studied by authors in [16,17,18,19,20,21,22,23,24,25,26,27]. We collect some relevant facts and auxiliary results from the existing literature.
Liu et al. [4] studied the oscillation of even-order half-linear functional differential equations with damping of the form
{(r(t)Φ(y(n−1)(t)))′+a(t)Φ(y(n−1)(t))+q(t)Φ(y(g(t)))=0,Φ=|s|p−2s, t≥t0>0, |
where n is even. This time, the authors used comparison method with second order equations.
The authors in [9,10] have established sufficient conditions for the oscillation of the solutions of
{(r(t)|y(n−1)(t)|p−2y(n−1)(t))′+∑ji=1qi(t)g(y(ϑi(t)))=0,j≥1, t≥t0>0, |
where n is even and p>1 is a real number, in the case where ϑi(t)≥υ (with r∈C1((0,∞),R), qi∈C([0,∞),R), i=1,2,..,j).
We point out that Li et al. [3] using the Riccati transformation together with integral averaging technique, focuses on the oscillation of equation
{(r(t)|w′′′(t)|p−2w′′′(t))′+∑ji=1qi(t)|y(δi(t))|p−2y(δi(t))=0,1<p<∞, , t≥t0>0. |
Park et al. [8] have obtained sufficient conditions for oscillation of solutions of
{(r(t)|y(n−1)(t)|p−2y(n−1)(t))′+q(t)g(y(δ(t)))=0,1<p<∞, , t≥t0>0. |
As we already mentioned in the Introduction, our aim here is complement results in [8,9,10]. For this purpose we discussed briefly these results.
In this paper, we obtain some new oscillation criteria for (1.1). The paper is organized as follows. In the next sections, we will mention some auxiliary lemmas, also, we will use the generalized Riccati transformation technique to give some sufficient conditions for the oscillation of (1.1), and we will give some examples to illustrate the main results.
For convenience, we denote
A(t)=q(t)(1−a0)p2−1Mp1−p2(ϑ(t)), B(t)=(p1−1)εϑ2(t)ζϑ′(t)r1/(p1−1)(t), ϕ1(t)=∫∞tA(s)ds,R1(t):=(p1−1)μt22r1/(p1−1)(t),ξ(t):=q(t)(1−a0)p2−1Mp2−p11ε1(ϑ(t)t)3(p2−1),η(t):=(1−a0)p2/p1Mp2/(p1−2)2∫∞t(1r(δ)∫∞δq(s)ϑp2−1(s)sp2−1ds)1/(p1−1)dδ,ξ∗(t)=∫∞tξ(s)ds, η∗(t)=∫∞tη(s)ds, |
for some μ∈(0,1) and every M1,M2 are positive constants.
Definition 1. A sequence of functions {δn(t)}∞n=0 and {σn(t)}∞n=0 as
δ0(t)=ξ∗(t), and σ0(t)=η∗(t),δn(t)=δ0(t)+∫∞tR1(t)δp1/(p1−1)n−1(s)ds, n>1σn(t)=σ0(t)+∫∞tσp1/(p1−1)n−1(s)ds, n>1. | (2.1) |
We see by induction that δn(t)≤δn+1(t) and σn(t)≤σn+1(t) for t≥t0, n>1.
In order to discuss our main results, we need the following lemmas:
Lemma 2.1. [12] If the function w satisfies w(i)(ν)>0, i=0,1,...,n, and w(n+1)(ν)<0 eventually. Then, for every ε1∈(0,1), w(ν)/w′(ν)≥ε1ν/n eventually.
Lemma 2.2. [13] Let u(t) be a positive and n-times differentiable function on an interval [T,∞) with its nth derivative u(n)(t) non-positive on [T,∞) and not identically zero on any interval of the form [T′,∞), T′≥T and u(n−1)(t)u(n)(t)≤0, t≥tu then there exist constants θ, 0<θ<1 and ε>0 such that
u′(θt)≥εtn−2u(n−1)(t), |
for all sufficient large t.
Lemma 2.3 [14] Let u∈Cn([t0,∞),(0,∞)). Assume that u(n)(t) is of fixed sign and not identically zero on [t0,∞) and that there exists a t1≥t0 such that u(n−1)(t)u(n)(t)≤0 for all t≥t1. If limt→∞u(t)≠0, then for every μ∈(0,1) there exists tμ≥t1 such that
u(t)≥μ(n−1)!tn−1|u(n−1)(t)| for t≥tμ. |
Lemma 2.4. [15] Assume that (1.2) holds and u is an eventually positive solution of (1.1). Then, (r(t)(w′′′(t))p1−1)′<0 and there are the following two possible cases eventually:
(G1) w(k)(t)>0, k=1,2,3,(G2) w(k)(t)>0, k=1,3, and w′′(t)<0. |
Theorem 2.1. Assume that
liminft→∞1ϕ1(t)∫∞tB(s)ϕp1(p1−1)1(s)ds>p1−1pp1(p1−1)1. | (2.2) |
Then (1.1) is oscillatory.
proof. Assume that u be an eventually positive solution of (1.1). Then, there exists a t1≥t0 such that u(t)>0, u(τ(t))>0 and u(ϑ(t))>0 for t≥t1. Since r′(t)>0, we have
w(t)>0, w′(t)>0, w′′′(t)>0, w(4)(t)<0 and (r(t)(w′′′(t))p1−1)′≤0, | (2.3) |
for t≥t1. From definition of w, we get
u(t)≥w(t)−a0u(τ(t))≥w(t)−a0w(τ(t))≥(1−a0)w(t), |
which with (1.1) gives
(r(t)(w′′′(t))p1−1)′≤−q(t)(1−a0)p2−1wp2−1(ϑ(t)). | (2.4) |
Define
ϖ(t):=r(t)(w′′′(t))p1−1wp1−1(ζϑ(t)). | (2.5) |
for some a constant ζ∈(0,1). By differentiating and using (2.4), we obtain
ϖ′(t)≤−q(t)(1−a0)p2−1wp2−1(ϑ(t)).wp1−1(ζϑ(t))−(p1−1)r(t)(w′′′(t))p1−1w′(ζϑ(t))ζϑ′(t)wp1(ζϑ(t)). |
From Lemma 2.2, there exist constant ε>0, we have
ϖ′(t)≤−q(t)(1−a0)p2−1wp2−p1(ϑ(t))−(p1−1)r(t)(w′′′(t))p1−1εϑ2(t)w′′′(ϑ(t))ζϑ′(t)wp1(ζϑ(t)). |
Which is
ϖ′(t)≤−q(t)(1−a0)p2−1wp2−p1(ϑ(t))−(p1−1)εr(t)ϑ2(t)ζϑ′(t)(w′′′(t))p1wp1(ζϑ(t)), |
by using (2.5) we have
ϖ′(t)≤−q(t)(1−a0)p2−1wp2−p1(ϑ(t))−(p1−1)εϑ2(t)ζϑ′(t)r1/(p1−1)(t)ϖp1/(p1−1)(t). | (2.6) |
Since w′(t)>0, there exist a t2≥t1 and a constant M>0 such that
w(t)>M. |
Then, (2.6), turns to
ϖ′(t)≤−q(t)(1−a0)p2−1Mp2−p1(ϑ(t))−(p1−1)εϑ2(t)ζϑ′(t)r1/(p1−1)(t)ϖp1/(p1−1)(t), |
that is
ϖ′(t)+A(t)+B(t)ϖp1/(p1−1)(t)≤0. |
Integrating the above inequality from t to l, we get
ϖ(l)−ϖ(t)+∫ltA(s)ds+∫ltB(s)ϖp1/(p1−1)(s)ds≤0. |
Letting l→∞ and using ϖ>0 and ϖ′<0, we have
ϖ(t)≥ϕ1(t)+∫∞tB(s)ϖp1/(p1−1)(s)ds. |
This implies
ϖ(t)ϕ1(t)≥1+1ϕ1(t)∫∞tB(s)ϕp1/(p1−1)1(s)(ϖ(s)ϕ1(s))p1/(p1−1)ds. | (2.7) |
Let λ=inft≥Tϖ(t)/ϕ1(t) then obviously λ≥1. Thus, from (2.2) and (2.7) we see that
λ≥1+(p1−1)(λp1)p1/(p1−1) |
or
λp1≥1p1+(p1−1)p1(λp1)p1/(p1−1), |
which contradicts the admissible value of λ≥1 and (p1−1)>0.
Therefore, the proof is complete.
Theorem 2.2. Assume that
liminft→∞1ξ∗(t)∫∞tR1(s)ξp1/(p1−1)∗(s)ds>(p1−1)pp1/(p1−1)1 | (2.8) |
and
liminft→∞1η∗(t)∫∞t0η2∗(s)ds>14. | (2.9) |
Then (1.1) is oscillatory.
proof. Assume to the contrary that (1.1) has a nonoscillatory solution in [t0,∞). Without loss of generality, we let u be an eventually positive solution of (1.1). Then, there exists a t1≥t0 such that u(t)>0, u(τ(t))>0 and u(ϑ(t))>0 for t≥t1. From Lemma 2.4 there is two cases (G1) and (G2).
For case (G1). Define
ω(t):=r(t)(w′′′(t))p1−1wp1−1(t). |
By differentiating ω and using (2.4), we obtain
ω′(t)≤−q(t)(1−a0)p2−1wp2−1(ϑ(t))wp1−1(t)−(p1−1)r(t)(w′′′(t))p1−1wp1(t)w′(t). | (2.10) |
From Lemma 2.1, we get
w′(t)w(t)≤3ε1t. |
Integrating again from t to ϑ(t), we find
w(ϑ(t))w(t)≥ε1ϑ3(t)t3. | (2.11) |
It follows from Lemma 2.3 that
w′(t)≥μ12t2w′′′(t), | (2.12) |
for all μ1∈(0,1) and every sufficiently large t. Since w′(t)>0, there exist a t2≥t1 and a constant M>0 such that
w(t)>M, | (2.13) |
for t≥t2. Thus, by (2.10), (2.11), (2.12) and (2.13), we get
ω′(t)+q(t)(1−a0)p2−1Mp2−p11ε1(ϑ(t)t)3(p2−1)+(p1−1)μt22r1/(p1−1)(t)ωp1/(p1−1)(t)≤0, |
that is
ω′(t)+ξ(t)+R1(t)ωp1/(p1−1)(t)≤0. | (2.14) |
Integrating (2.14) from t to l, we get
ω(l)−ω(t)+∫ltξ(s)ds+∫ltR1(s)ωp1/(p1−1)(s)ds≤0. |
Letting l→∞ and using ω>0 and ω′<0, we have
ω(t)≥ξ∗(t)+∫∞tR1(s)ωp1/(p1−1)(s)ds. | (2.15) |
This implies
ω(t)ξ∗(t)≥1+1ξ∗(t)∫∞tR1(s)ξp1/(p1−1)∗(s)(ω(s)ξ∗(s))p1/(p1−1)ds. | (2.16) |
Let λ=inft≥Tω(t)/ξ∗(t) then obviously λ≥1. Thus, from (2.8) and (2.16) we see that
λ≥1+(p1−1)(λp1)p1/(p1−1) |
or
λp1≥1p1+(p1−1)p1(λp1)p1/(p1−1), |
which contradicts the admissible value of λ≥1 and (p1−1)>0.
For case (G2). Integrating (2.4) from t to m, we obtain
r(m)(w′′′(m))p1−1−r(t)(w′′′(t))p1−1≤−∫mtq(s)(1−a0)p2−1wp2−1(ϑ(s))ds. | (2.17) |
From Lemma 2.1, we get that
w(t)≥ε1tw′(t) and hence w(ϑ(t))≥ε1ϑ(t)tw(t). | (2.18) |
For (2.17), letting m→∞and using (2.18), we see that
r(t)(w′′′(t))p1−1≥ε1(1−a0)p2−1wp2−1(t)∫∞tq(s)ϑp2−1(s)sp2−1ds. |
Integrating this inequality again from t to ∞, we get
w′′(t)≤−ε1(1−a0)p2/p1wp2/p1(t)∫∞t(1r(δ)∫∞δq(s)ϑp2−1(s)sp2−1ds)1/(p1−1)dδ, | (2.19) |
for all ε1∈(0,1). Define
y(t)=w′(t)w(t). |
By differentiating y and using (2.13) and (2.19), we find
y′(t)=w′′(t)w(t)−(w′(t)w(t))2≤−y2(t)−(1−a0)p2/p1M(p2/p1)−1∫∞t(1r(δ)∫∞δq(s)ϑp2−1(s)sp2−1ds)1/(p1−1)dδ, | (2.20) |
hence
y′(t)+η(t)+y2(t)≤0. | (2.21) |
The proof of the case where (G2) holds is the same as that of case (G1). Therefore, the proof is complete.
Theorem 2.3. Let δn(t) and σn(t) be defined as in (2.1). If
limsupt→∞(μ1t36r1/(p1−1)(t))p1−1δn(t)>1 | (2.22) |
and
limsupt→∞λtσn(t)>1, | (2.23) |
for some n, then (1.1)is oscillatory.
proof. Assume to the contrary that (1.1) has a nonoscillatory solution in [t0,∞). Without loss of generality, we let u be an eventually positive solution of (1.1). Then, there exists a t1≥t0 such that u(t)>0, u(τ(t))>0 and u(ϑ(t))>0 for t≥t1. From Lemma 2.4 there is two cases.
In the case (G1), proceeding as in the proof of Theorem 2.2, we get that (2.12) holds. It follows from Lemma 2.3 that
w(t)≥μ16t3w′′′(t). | (2.24) |
From definition of ω(t) and (2.24), we have
1ω(t)=1r(t)(w(t)w′′′(t))p1−1≥1r(t)(μ16t3)p1−1. |
Thus,
ω(t)(μ1t36r1/(p1−1)(t))p1−1≤1. |
Therefore,
limsupt→∞ω(t)(μ1t36r1/(p1−1)(t))p1−1≤1, |
which contradicts (2.22).
The proof of the case where (G2) holds is the same as that of case (G1). Therefore, the proof is complete.
Corollary 2.1. Let δn(t) and σn(t) be defined as in (2.1). If
∫∞t0ξ(t)exp(∫tt0R1(s)δ1/(p1−1)n(s)ds)dt=∞ | (2.25) |
and
∫∞t0η(t)exp(∫tt0σ1/(p1−1)n(s)ds)dt=∞, | (2.26) |
for some n, then (1.1) is oscillatory.
proof. Assume to the contrary that (1.1) has a nonoscillatory solution in [t0,∞). Without loss of generality, we let u be an eventually positive solution of (1.1). Then, there exists a t1≥t0 such that u(t)>0, u(τ(t))>0 and u(ϑ(t))>0 for t≥t1. From Lemma 2.4 there is two cases (G1) and (G2).
In the case (G1), proceeding as in the proof of Theorem 2, we get that (2.15) holds. It follows from (2.15) that ω(t)≥δ0(t). Moreover, by induction we can also see that ω(t)≥δn(t) for t≥t0, n>1. Since the sequence {δn(t)}∞n=0 monotone increasing and bounded above, it converges to δ(t). Thus, by using Lebesgue's monotone convergence theorem, we see that
δ(t)=limn→∞δn(t)=∫∞tR1(t)δp1/(p1−1)(s)ds+δ0(t) |
and
δ′(t)=−R1(t)δp1/(p1−1)(t)−ξ(t). | (2.27) |
Since δn(t)≤δ(t), it follows from (2.27) that
δ′(t)≤−R1(t)δ1/(p1−1)n(t)δ(t)−ξ(t). |
Hence, we get
δ(t)≤exp(−∫tTR1(s)δ1/(p1−1)n(s)ds)(δ(T)−∫tTξ(s)exp(∫sTR1(δ)δ1/(p1−1)n(δ)dδ)ds). |
This implies
∫tTξ(s)exp(∫sTR1(δ)δ1/(p1−1)n(δ)dδ)ds≤δ(T)<∞, |
which contradicts (2.25). The proof of the case where (G2) holds is the same as that of case (G1). Therefore, the proof is complete.
Example 2.1. Consider the differential equation
(u(t)+12u(t2))(4)+q0t4u(t3)=0, | (2.28) |
where q0>0 is a constant. Let p1=p2=2, r(t)=1, a(t)=1/2, τ(t)=t/2, ϑ(t)=t/3 and q(t)=q0/t4. Hence, it is easy to see that
A(t)=q(t)(1−a0)(p2−1)Mp2−p1(ϑ(t))=q02t4, B(t)=(p1−1)εϑ2(t)ζϑ′(t)r1/(p1−1)(t)=εt227 |
and
ϕ1(t)=q06t3, |
also, for some ε>0, we find
liminft→∞1ϕ1(t)∫∞tB(s)ϕp1/(p1−1)1(s)ds>(p1−1)pp1/(p1−1)1.liminft→∞6εq0t3972∫∞tdss4>14q0>121.5ε. |
Hence, by Theorem 2.1, every solution of Eq (2.28) is oscillatory if q0>121.5ε.
Example 2.2. Consider a differential equation
(u(t)+a0u(τ0t))(n)+q0tnu(ϑ0t)=0, | (2.29) |
where q0>0 is a constant. Note that p=2, t0=1, r(t)=1, a(t)=a0, τ(t)=τ0t, ϑ(t)=ϑ0t and q(t)=q0/tn.
Easily, we see that condition (2.8) holds and condition (2.9) satisfied.
Hence, by Theorem 2.2, every solution of Eq (2.29) is oscillatory.
Remark 2.1. Finally, we point out that continuing this line of work, we can have oscillatory results for a fourth order equation of the type:
{(r(t)|y′′′(t)|p1−2y′′′(t))′+a(t)f(y′′′(t))+∑ji=1qi(t)|y(σi(t))|p2−2y(σi(t))=0,t≥t0, σi(t)≤t, j≥1,, 1<p2≤p1<∞. |
The paper is devoted to the study of oscillation of fourth-order differential equations with p-Laplacian like operators. New oscillation criteria are established by using a Riccati transformations, and they essentially improves the related contributions to the subject.
Further, in the future work we get some Hille and Nehari type and Philos type oscillation criteria of (1.1) under the condition ∫∞υ01r1/(p1−1)(s)ds<∞.
The authors express their debt of gratitude to the editors and the anonymous referee for accurate reading of the manuscript and beneficial comments.
The author declares that there is no competing interest.
[1] |
Carstens CK, Salazar JK, Darkoh C (2019) Multistate outbreaks of foodborne illness in the United States associated with fresh produce from 2010 to 2017. Front Microbiol 10: 2667-2667. https://doi.org/10.3389/fmicb.2019.02667 ![]() |
[2] |
Majowicz S, Musto J, Scallan E, et al. (2010) The global burden of nontyphoidal Salmonella. Clin Infect Dis 50: 882-889. https://doi.org/10.1086/650733 ![]() |
[3] |
Liu H, Whitehouse CA, Li B (2018) Presence and persistence of Salmonella in water: the impact on microbial quality of water and food safety. Front Public Health 6: 159-159. https://doi.org/10.3389/fpubh.2018.00159 ![]() |
[4] | (2018) PHACNational Enteric Surveillance Program Annual Summary 2016. Guelph, Ontario: Public Health Agency, Government of Canada. |
[5] |
Chai S, White P, Lathrop S, et al. (2012) Salmonella enterica serotype Enteritidis: increasing incidence of domestically acquired infections. Clin Infect Dis 54: S488-497. https://doi.org/10.1093/cid/cis231 ![]() |
[6] |
Lane C, LeBaigue S, Esan O, et al. (2014) Salmonella enterica serovar Enteritidis, England and Wales, 1945–2011. Emerging Infect Dis 20: 1097-1104. https://doi.org/10.3201/eid2007.121850 ![]() |
[7] |
Kozak GK, MacDonald D, Landry L, et al. (2013) Foodborne outbreaks in Canada linked to produce: 2001 through 2009. J Food Prot 76: 173-183. https://doi.org/10.4315/0362-028X.JFP-12-126 ![]() |
[8] |
Chai SJ, Cole D, Nisler A, et al. (2017) Poultry: the most common food in outbreaks with known pathogens, United States, 1998–2012. Epidemiol Infect 145: 316-325. https://doi.org/10.1017/S0950268816002375 ![]() |
[9] |
Middleton D, Savage R, Tighe M, et al. (2014) Risk factors for sporadic domestically acquired Salmonella serovar Enteritidis infections: a case-control study in Ontario, Canada, 2011. Epidemiol Infect 142: 1411-1421. https://doi.org/10.1017/S0950268813001945 ![]() |
[10] |
Nesbitt A, Ravel A, Murray R, et al. (2012) Integrated surveillance and potential sources of Salmonella Enteritidis in human cases in Canada from 2003 to 2009. Epidemiol Infect 140: 1757-1772. https://doi.org/10.1017/S0950268811002548 ![]() |
[11] |
Taylor M, Leslie M, Ritson M, et al. (2012) Investigation of the concurrent emergence of Salmonella enteritidis in humans and poultry in British Columbia, Canada, 2008-2010. Zoonoses Public Health 59: 584-892. https://doi.org/10.1111/j.1863-2378.2012.01500.x ![]() |
[12] |
Kuehn B (2010) Salmonella cases traced to egg producers. J Am Medl Assoc 304: 1316. https://doi.org/10.1001/jama.2010.1330 ![]() |
[13] |
Chousalkar K, Gast R, Martelli F, et al. (2018) Review of egg-related salmonellosis and reduction strategies in United States, Australia, United Kingdom and New Zealand. Crit Rev Microbiol 44: 290-303. https://doi.org/10.1080/1040841X.2017.1368998 ![]() |
[14] |
Tang S, Orsi RH, Luo H, et al. (2019) Assessment and comparison of molecular subtyping and characterization methods for Salmonella. Front Microbiol 10: 1591. https://doi.org/10.3389/fmicb.2019.01591 ![]() |
[15] |
Ward LR, de Sa JDH, Rowe B (1987) A phage-typing scheme for Salmonella enteritidis. Epidemiol Infect 99: 291-294. https://doi.org/10.1017/S0950268800067765 ![]() |
[16] |
Carrique-Mas JJ, Papadopoulou C, Evans SJ, et al. (2008) Trends in phage types and antimicrobial resistance of Salmonella enterica serovar Enteritidis isolated from animals in Great Britain from 1990 to 2005. Vet Rec 162: 541-546. https://doi.org/10.1136/vr.162.17.541 ![]() |
[17] |
Peters TM, Berghold C, Brown D, et al. (2007) Relationship of pulsed-field profiles with key phage types of Salmonella enterica serotype Enteritidis in Europe: results of an international multi-centre study. Epidemiol Infect 135: 1274-1281. https://doi.org/10.1017/S0950268807008102 ![]() |
[18] |
Ribot EM, Fair MA, Gautom R, et al. (2006) Standardization of pulsed-field gel electrophoresis protocols for the subtyping of Escherichia coli O157:H7, Salmonella, and Shigella for PulseNet. Foodborne Pathog Dis 3: 59-67. https://doi.org/10.1089/fpd.2006.3.59 ![]() |
[19] |
Nadon CA, Trees E, Ng LK, et al. (2013) Development and application of MLVA methods as a tool for inter-laboratory surveillance. Eurosurveillance 18: 20565. https://doi.org/10.2807/1560-7917.ES2013.18.35.20565 ![]() |
[20] |
Bertrand S, De Lamine de Bex G, Wildemauwe C, et al. (2015) Multi Locus Variable-Number Tandem Repeat (MLVA) typing tools improved the surveillance of Salmonella Enteritidis: A 6 Years Retrospective Study. PLoS One 10: e0117950. https://doi.org/10.1371/journal.pone.0117950 ![]() |
[21] |
Maiden MCJ, Bygraves JA, Feil E, et al. (1998) Multilocus sequence typing: A portable approach to the identification of clones within populations of pathogenic microorganisms. Proc Natl Acad Sci 95: 3140-3145. https://doi.org/10.1073/pnas.95.6.3140 ![]() |
[22] |
Achtman M, Wain J, Weill F-X, et al. (2012) Multilocus sequence typing as a replacement for serotyping in Salmonella enterica. PLoS Pathog 8: e1002776. https://doi.org/10.1371/journal.ppat.1002776 ![]() |
[23] |
Yoshida C, Kruczkiewicz P, Laing C, et al. (2016) The Salmonella In Silico Typing Resource (SISTR): an open web-accessible tool for rapidly typing and subtyping draft Salmonella genome assemblies. PLoS One 11: e0147101. https://doi.org/10.1371/journal.pone.0147101 ![]() |
[24] |
Pearce ME, Alikhan NF, Dallman TJ, et al. (2018) Comparative analysis of core genome MLST and SNP typing within a European Salmonella serovar Enteritidis outbreak. Int J Food Microbiol 274: 1-11. https://doi.org/10.1016/j.ijfoodmicro.2018.02.023 ![]() |
[25] |
Ogunremi D, Kelly H, Dupras AA, et al. (2014) Development of a new molecular subtyping tool for Salmonella enterica serovar Enteritidis based on single nucleotide polymorphism genotyping using PCR. J Clin Microbiol 52: 4275-4285. https://doi.org/10.1128/JCM.01410-14 ![]() |
[26] |
Shariat N, Dudley EG (2014) CRISPRs: molecular signatures used for pathogen subtyping. Appl Environ Microbiol 80: 430-439. https://doi.org/10.1128/AEM.02790-13 ![]() |
[27] |
Horvath P, Barrangou R (2010) CRISPR/Cas, the immune system of bacteria and archaea. Science 327: 167-170. https://doi.org/10.1126/science.1179555 ![]() |
[28] |
Burmistrz M, Krakowski K, Krawczyk-Balska A (2020) RNA-targeting CRISPR-Cas systems and their applications. Int J Mol Sci 21: 1122. https://doi.org/10.3390/ijms21031122 ![]() |
[29] |
Fricke WF, Mammel MK, McDermott PF, et al. (2011) Comparative genomics of 28 Salmonella enterica isolates: evidence for CRISPR-mediated adaptive sublineage evolution. J Bacteriol 193: 3556-3568. https://doi.org/10.1128/JB.00297-11 ![]() |
[30] |
Fabre L, Le Hello S, Roux C, et al. (2014) CRISPR Is an optimal target for the design of specific PCR assays for Salmonella enterica serotypes Typhi and Paratyphi A. PLoS Neglected Trop Dis 8: e2671. https://doi.org/10.1371/journal.pntd.0002671 ![]() |
[31] |
Fabre L, Zhang J, Guigon G, et al. (2012) CRISPR typing and subtyping for improved laboratory surveillance of Salmonella infections. PLoS ONE 7: e36995. https://doi.org/10.1371/journal.pone.0036995 ![]() |
[32] |
Liu F, Barrangou R, Gerner-Smidt P, et al. (2011) Novel virulence gene and Clustered Regularly Interspaced Short Palindromic Repeat (CRISPR) multilocus sequence typing scheme for subtyping of the major serovars of Salmonella enterica subsp. enterica. Appl Environ Microbiol 77: 1946-1956. https://doi.org/10.1128/AEM.02625-10 ![]() |
[33] |
Liu F, Kariyawasam S, Jayarao BM, et al. (2011) Subtyping Salmonella enterica serovar Enteritidis isolates from different sources by using sequence typing based on virulence genes and Clustered Regularly Interspaced Short Palindromic Repeats (CRISPRs). Appl Environ Microbiol 77: 4520-4526. https://doi.org/10.1128/AEM.00468-11 ![]() |
[34] |
Shariat N, Sandt CH, DiMarzio MJ, et al. (2013) CRISPR-MVLST subtyping of Salmonella enterica subsp. enterica serovars Typhimurium and Heidelberg and application in identifying outbreak isolates. BMC Microbiol 13: 254-254. https://doi.org/10.1186/1471-2180-13-254 ![]() |
[35] |
Fu S, Hiley L, Octavia S, et al. (2017) Comparative genomics of Australian and international isolates of Salmonella Typhimurium: correlation of core genome evolution with CRISPR and prophage profiles. Sci Rep 7: 9733-9733. https://doi.org/10.1038/s41598-017-06079-1 ![]() |
[36] |
Fei X, He X, Guo R, et al. (2017) Analysis of prevalence and CRISPR typing reveals persistent antimicrobial-resistant Salmonella infection across chicken breeder farm production stages. Food Control 77: 102-109. https://doi.org/10.1016/j.foodcont.2017.01.023 ![]() |
[37] |
Li Q, Wang X, Yin K, et al. (2018) Genetic analysis and CRISPR typing of Salmonella enterica serovar Enteritidis from different sources revealed potential transmission from poultry and pig to human. Int J Food Microbiol 266: 119-125. https://doi.org/10.1016/j.ijfoodmicro.2017.11.025 ![]() |
[38] |
Shariat N, Timme RE, Pettengill JB, et al. (2015) Characterization and evolution of Salmonella CRISPR-Cas systems. Microbiology 161: 374-386. https://doi.org/10.1099/mic.0.000005 ![]() |
[39] |
Touchon M, Rocha EPC (2010) The small, slow and specialized CRISPR and anti-CRISPR of Escherichia and Salmonella. PLoS ONE 5: e11126. https://doi.org/10.1371/journal.pone.0011126 ![]() |
[40] |
Nadin-Davis S, Pope L, Ogunremi D, et al. (2019) A real-time PCR regimen for testing environmental samples for Salmonella enterica subsp. enterica serovars of concern to the poultry industry, with special focus on Salmonella Enteritidis. Can J Microbiol 65: 162-173. https://doi.org/10.1139/cjm-2018-0417 ![]() |
[41] | Reid A Isolation and identification of Salmonella from food and environmental samples: Standard operating procedure MFHPB 20. Microbiological Methods Committee, Bureau of Microbial Hazards, Health Canada (2009). |
[42] |
Kumar S, Stecher G, Li M, et al. (2018) MEGA X: Molecular Evolutionary Genetics Analysis across Computing Platforms. Mol Biol Evol 35: 1547-1549. https://doi.org/10.1093/molbev/msy096 ![]() |
[43] |
Couvin D, Bernheim A, Toffano-Nioche C, et al. (2018) CRISPRCasFinder, an update of CRISRFinder, includes a portable version, enhanced performance and integrates search for Cas proteins. Nucleic Acids Res 46: W246-W251. https://doi.org/10.1093/nar/gky425 ![]() |
[44] | PulseNetInternationalStandard Operating Procedure for PulseNet PFGE of Escherichia coli O157:H7, Escherichia coli non-O157 (STEC), Salmonella serotypes, Shigella sonnei and Shigella flexneri. (2013). |
[45] |
Hunter PR, Gaston MA (1988) Numerical index of the discriminatory ability of typing systems: an application of Simpson's index of diversity. J Clin Microbiol 26: 2465-2466. https://doi.org/10.1128/jcm.26.11.2465-2466.1988 ![]() |
[46] |
Nadin-Davis S, Pope L, Chmara J, et al. (2020) An unusual Salmonella Enteritidis strain carrying a modified virulence plasmid lacking the prot6e gene represents a geographically widely distributed lineage. Front Microbiol 11: 1322. https://doi.org/10.3389/fmicb.2020.01322 ![]() |
[47] |
Deng X, Shariat N, Driebe EM, et al. (2015) Comparative analysis of subtyping methods against a whole-genome-sequencing standard for Salmonella enterica serotype Enteritidis. J Clin Microbiol 53: 212-218. https://doi.org/10.1128/JCM.02332-14 ![]() |
[48] |
Ogunremi D, Devenish J, Amoako K, et al. (2014) High resolution assembly and characterization of genomes of Canadian isolates of Salmonella Enteritidis. BMC Genomics 15: 713. https://doi.org/10.1186/1471-2164-15-713 ![]() |
[49] |
Shariat N, DiMarzio MJ, Yin S, et al. (2013) The combination of CRISPR-MVLST and PFGE provides increased discriminatory power for differentiating human clinical isolates of Salmonella enterica subsp. enterica serovar Enteritidis. Food Microbiol 34: 164-173. https://doi.org/10.1016/j.fm.2012.11.012 ![]() |
[50] |
Hiley L, Graham RMA, Jennison AV (2021) Characterisation of IncI1 plasmids associated with change of phage type in isolates of Salmonella enterica serovar Typhimurium. BMC Microbiol 21: 92-92. https://doi.org/10.1186/s12866-021-02151-z ![]() |
[51] |
Tankouo-Sandjong B, Kinde H, Wallace I (2012) Development of a sequence typing scheme for differentiation of Salmonella Enteritidis strains. FEMS Microbiol Lett 331: 165-175. https://doi.org/10.1111/j.1574-6968.2012.02568.x ![]() |
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