COVID-19 is caused by SARS-CoV-2, which originated in Wuhan, Hubei province, Central China, in December 2019 and since then has spread rapidly, resulting in a severe pandemic. The infected patient presents with varying non-specific symptoms requiring an accurate and rapid diagnostic tool to detect SARS-CoV-2. This is followed by effective patient isolation and early treatment initiation ranging from supportive therapy to specific drugs such as corticosteroids, antiviral agents, antibiotics, and the recently introduced convalescent plasma. The development of an efficient vaccine has been an on-going challenge by various nations and research companies. A literature search was conducted in early December 2020 in all the major databases such as Medline/PubMed, Web of Science, Scopus and Google Scholar search engines. The findings are discussed in three main thematic areas namely diagnostic approaches, therapeutic options, and potential vaccines in various phases of development. Therefore, an effective and economical vaccine remains the only retort to combat COVID-19 successfully to save millions of lives during this pandemic. However, there is a great scope for further research in discovering cost-effective and safer therapeutics, vaccines and strategies to ensure equitable access to COVID-19 prevention and treatment services.
Citation: Srikanth Umakanthan, Vijay Kumar Chattu, Anu V Ranade, Debasmita Das, Abhishekh Basavarajegowda, Maryann Bukelo. A rapid review of recent advances in diagnosis, treatment and vaccination for COVID-19[J]. AIMS Public Health, 2021, 8(1): 137-153. doi: 10.3934/publichealth.2021011
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COVID-19 is caused by SARS-CoV-2, which originated in Wuhan, Hubei province, Central China, in December 2019 and since then has spread rapidly, resulting in a severe pandemic. The infected patient presents with varying non-specific symptoms requiring an accurate and rapid diagnostic tool to detect SARS-CoV-2. This is followed by effective patient isolation and early treatment initiation ranging from supportive therapy to specific drugs such as corticosteroids, antiviral agents, antibiotics, and the recently introduced convalescent plasma. The development of an efficient vaccine has been an on-going challenge by various nations and research companies. A literature search was conducted in early December 2020 in all the major databases such as Medline/PubMed, Web of Science, Scopus and Google Scholar search engines. The findings are discussed in three main thematic areas namely diagnostic approaches, therapeutic options, and potential vaccines in various phases of development. Therefore, an effective and economical vaccine remains the only retort to combat COVID-19 successfully to save millions of lives during this pandemic. However, there is a great scope for further research in discovering cost-effective and safer therapeutics, vaccines and strategies to ensure equitable access to COVID-19 prevention and treatment services.
We investigate a new class of boundary value problems for generalized Caputo-type fractional differential equations and inclusions supplemented with Katugampola type generalized fractional integral boundary conditions. Precisely, we study the following problems:
{ρcDα0+y(t)=f(t,y(t)), t∈J:=[0,T],y(T)=m∑i=1σiρIβ0+y(ηi)+κ, δy(0)=0, ηi∈(0,T), | (1.1) |
and
{ρcDα0+y(t)∈F(t,y(t)), t∈J:=[0,T],y(T)=m∑i=1σiρIβ0+y(ηi)+κ, δy(0)=0, ηi∈(0,T), | (1.2) |
where ρcDα0+ denotes the generalized Caputo-type fractional derivative of order 1<α≤2,ρ>0, ρIβ0+ is the Katugampola type fractional integral of order β>0,ρ>0, f:J×R→R is a continuous function, σi∈R,i=1,2,…,m,κ∈R, δ=t1−ρddt, and F:J×R→P(R) is a multivalued function (P(R) is the family of all nonempty subjects of R).
Here we emphasize that the problem considered in the present paper is motivated by Laskin's work [1] on the generalization of the Feynman and Wiener path integrals in the context of fractional quantum mechanics and fractional statistical mechanics. One can find more details in the articles [2, 3]. It is expected that the results obtained in this paper will provide more leverage in dealing with Feynman and Wiener path type integrals involving an index like ρ>0 in (1.1), instead of a fixed choice ρ=1 (Caputo fractional derivative case). Moreover, chaos for a fractional order differential equation involving two parameters (α and ρ>0) becomes more complicated than the one containing Caputo fractional derivative of order α (generalized Caputo-type fractional derivative with ρ=1); one can find more details in [4, 5]. It is worthwhile to notice that Katugampola fractional integral unifies the Riemann-Liouville and Hadamard integrals into a single integral [6]. Thus, our results are more general in the context of integral boundary conditions.
The topic of fractional-order differential equations and inclusions attracted significant attention in recent years and several results on fractional differential equations involving Riemann-Liouville, Caputo, Hadamard type derivatives, supplemented with a variety of boundary conditions, can be found in the related literature [7, 8, 9]. The interest in the subject owes to its extensive applications in various disciplines of science and engineering, for instance, see the papers [10, 11, 12, 13, 14, 15, 16, 17, 18], and the references cited therein. In a recent paper [19], the authors studied fractional differential equations involving Caputo-Katugampola derivative. In a more recent work [20], the authors studied a fractional order boundary value problem involving Katugampola-type generalized fractional derivative and generalized fractional integral.
We organize the rest of the paper as follows. Section 2 contains preliminary material related to our work. The existence and uniqueness results for the problem (1.1), obtained with the aid of the standard fixed point theorem, are presented in Section 3. The existence results for the inclusions problem (1.2) are derived in Section 4. Examples are provided to demonstrate the application of the main theorems.
For c∈R,1≤p≤∞, let Xpc(a,b) denote the space of all complex-valued Lebesgue measurable functions ϕ on (a,b) endowed with the norm:
‖ϕ‖Xpc=(∫ba|xcϕ(x)|pdxx)1/p<∞. |
We denote by L1(a,b) the space of all Lebesgue measurable functions φ on (a,b) equipped with the norm:
‖φ‖L1=∫ba|φ(x)|dx<∞. |
Let G=C(J,R) denote the Banach space of all continuous functions from [0,T] to R endowed with the norm defined by ‖y‖=supt∈[0,T]|y(t)|.
Recall that
ACn(J,R)={h:J→R:h,h′,…,h(n−1)∈C(J,R) and h(n−1)is absolutely continuous}. |
For 0≤ϵ<1, we define Cϵ,ρ(J,R)={f:J→R:(tρ−aρ)ϵf(t)∈C(J,R)} equipped with the norm ‖f‖Cϵ,ρ=‖(tρ−aρ)ϵf(t)‖C. Moreover, let us introduce ACnδ(J), which consists of the functions f that have absolutely continuous δn−1-derivative, where δ=t1−ρddt. Thus we define spaces ACnδ(J,R)={f:J→R:δn−1f∈AC(J,R), δ=t1−ρddt}, and Cnδ,ϵ(J,R)={f:J→R:δn−1f∈C(J,R),δnf∈Cϵ,ρ(J,R),δ=t1−ρddt} endowed with the norms ‖f‖Cnδ=∑n−1k=0‖δkf‖C and ‖f‖Cnδ,ϵ=∑n−1k=0‖δkf‖C+‖δnf‖Cϵ,ρ respectively. Here we use the convention Cnδ,0=Cnδ.
Definition 2.1. [6] The generalized fractional integral of order α>0 and ρ>0 of f∈Xpc(a,b) for −∞<a<t<b<∞, is defined by
(ρIαa+f)(t)=ρ1−αΓ(α)∫tasρ−1(tρ−sρ)1−αf(s)ds. | (2.1) |
Note that the integral in (2.1) is called the left-sided fractional integral. Similarly we can define right-sided fractional integral ρIαb−f as
(ρIαb−f)(t)=ρ1−αΓ(α)∫btsρ−1(sρ−tρ)1−αf(s)ds. | (2.2) |
Definition 2.2. [21] The generalized fractional derivative, associated with the generalized fractional integrals (2.1) and (2.2) for 0≤a<t<b<∞, are defined by
(ρDαa+f)(t)=(t1−ρddt)n(ρIn−αa+f)(t)=ρα−n+1Γ(n−α)(t1−ρddt)n∫tasρ−1(tρ−sρ)α−n+1f(s)ds | (2.3) |
and
(ρDαb−f)(t)=(−t1−ρddt)n(ρIn−αb−f)(t)=ρα−n+1Γ(n−α)(−t1−ρddt)n∫btsρ−1(sρ−tρ)α−n+1f(s)ds, | (2.4) |
if the integrals in (2.3) and (2.4) exist.
Definition 2.3. [22] For α≥0 and f∈ACnδ[a,b], the generalized Caputo-type fractional derivatives ρcDαa and ρcDαb are defined via the above generalized fractional derivatives as follows
ρcDαa+f(x)=ρDαa+[f(t)−n−1∑k=0δkf(a)k!(tρ−aρρ)k](x), δ=x1−ρddx, | (2.5) |
ρcDαb−f(x)=ρDαb−[f(t)−n−1∑k=0(−1)kδkf(b)k!(bρ−tρρ)k](x), δ=x1−ρddx, | (2.6) |
where n=[α]+1.
Lemma 2.1. [22] Let α≥0,n=[α]+1 and f∈ACnδ[a,b], where 0<a<b<∞. Then,
(1) for α∉N,
ρcDαa+f(t)=1Γ(n−α)∫ta(tρ−sρρ)n−α−1(δnf)(s)dss1−ρ=ρIn−αa+(δnf)(t), | (2.7) |
ρcDαb−f(t)=1Γ(n−α)∫bt(sρ−tρρ)n−α−1(−1)n(δnf)(s)dss1−ρ=ρIn−αb−(δnf)(t); | (2.8) |
(2) for α∈N,
ρcDαa+f=δnf, ρcDαb−f=(−1)nδnf. | (2.9) |
Lemma 2.2. [22] Let f∈ACnδ[a,b] or Cnδ[a,b] and α∈R. Then
ρIαa+ρcDαa+f(x)=f(x)−n−1∑k=0(δkf)(a)k!(xρ−aρρ)k, |
ρIαb−ρcDαb−f(x)=f(x)−n−1∑k=0(−1)k(δkf)(a)k!(bρ−xρρ)k. |
In particular, for 0<α≤1, we have
ρIαa+ρcDαa+f(x)=f(x)−f(a), |
ρIαb−ρcDαb−f(x)=f(x)−f(b). |
Next we define a solution for the problem (1.1).
Definition 2.4. A function y∈AC2δ([0,T],R) is said to be a solution of (1.1) if y satisfies the equation ρcDαy(t)=f(t,y(t)) on [0,T], and the conditions y(T)=∑mi=1σi ρIβy(ηi)+κ, δy(0)=0.
Relative to the problem (1.1), we consider the following lemma.
Lemma 2.3. Let h∈C(0,T)∩L1(0,T), y∈AC2δ(J) and
Ω=1−m∑i=1σiηρβiρβΓ(β+1)≠0. | (2.10) |
Then the integral solution of the linear boundary value problem (BVP):
{ρcDα0+y(t)=h(t), t∈J:=[0,T],y(T)=m∑i=1σiρIβ0+y(ηi)+κ, δy(0)=0, ηi∈(0,T), | (2.11) |
is given by
y(t)=ρIα0+h(t)+1Ω{−ρIα0+h(T)+m∑i=1σiρIα+β0+h(ηi)+κ}. | (2.12) |
Proof. Applying ρIα0+ on both sides of the fractional differential equation in (2.11) and using Lemma 2.2, we get
y(t)=ρIα0+h(t)+c1+c2tρρ=ρ1−αΓ(α)∫t0sρ−1(tρ−sρ)α−1h(s)ds+c1+c2tρρ, | (2.13) |
for some c1,c2∈R. Taking δ-derivative of (2.13), we get
δy(t)=ρIα−10+h(t)+c2=ρ2−αΓ(α−1)∫t0sρ−1(tρ−sρ)α−2h(s)ds+c2. | (2.14) |
Using the boundary condition δy(0)=0 in (2.14), we get c2=0. Applying the generalized integral ρIβ0+ to (2.13) after inserting the value of c2 in it, we get
ρIβ0+y(t)=ρIα+β0+h(t)+c1tρβρβΓ(β+1). | (2.15) |
Making use of the first boundary condition y(T)=∑mi=1σi ρIβy(ηi)+κ in (2.15), we get
ρIα0+h(T)+c1=m∑i=1σi ρIα+β0+h(ηi)+m∑i=1σic1ηρβiρβΓ(β+1)+κ, |
which, on solving for c1 together with (2.10), yields
c1=1Ω{−ρIα0+h(T)+m∑i=1σi ρIα+β0+h(ηi)+κ}. |
Substituting the values of c1 and c2 in (2.13), we obtain the solution (2.12). The converse follows by direct computation. The proof is completed.
Using Lemma 2.3, we define an operator N:G→G by
Ny(t)=ρIα0+f(t,y(t))+1Ω{−ρIα0+f(T,y(T))+m∑i=1σiρIα+β0+f(ηi,y(ηi))+κ}. | (3.1) |
In the following, for brevity, we set the notation:
Λ=TραραΓ(α+1)+1|Ω|{TραραΓ(α+1)+m∑i=1|σi|ηρ(α+β)iρα+βΓ(α+β+1)}. | (3.2) |
Our first existence result for the problem (1.1) relies on Leray-Schauder nonlinear alternative [23].
Theorem 3.1. Assume that
(A1) |f(t,y)|≤p(t)ψ(‖y‖),∀(t,y)∈[0,T]×R, where p∈L1([0,T],R+) and ψ:R+→ R+ is a nondecreasing function;
(A2) we can find a positive constant W satisfying the inequality:
Wψ(W)(ρIα0+p(T)+1|Ω|{ρIα0+p(T)+m∑i=1|σi| ρIα+β0+p(ηi)+|κ|})>1. |
Then there exists at least one solution for the problem (1.1) on [0,T].
Proof. Consider the operator N:G→G defined by (3.1) and show that it is continuous and completely continuous. We establish in four steps.
(ⅰ) N is continuous. Let {yn} be a sequence such that yn→y in G. Then
|N(yn)(t)−N(y)(t)|≤ρIα0+|f(t,yn(t))−f(t,y(t))|+1Ω{ρIα0+|f(T,yn(T))−f(T,y(T))|+m∑i=1|σi| ρIα+β0+|f(ηi,yn(ηi))−f(ηi,y(ηi))|}≤Λ‖f(⋅,yn)−f(⋅,y)‖. |
Since f is a continuous function, therefore, we have
‖N(yn)−N(y)‖≤Λ‖f(⋅,yn)−f(⋅,y)‖→0,asn→∞. |
(ⅱ) The operator N maps bounded sets into bounded sets in G.
For any ˉr>0, it is indeed enough to show that there exists a positive constant ℓ such that ‖N(y)‖≤ℓ for y∈Bˉr={y∈G:‖y‖≤ˉr}. By the assumption (A1), for each t∈J, we have
|N(y)(t)|≤ρIα0+|f(t,y(t))|+1|Ω|{ρIα0+|f(T,y(T))|+m∑i=1|σi| ρIα+β0+|f(ηi,y(ηi))|+|κ|}≤ρIα0+p(T)Ω(‖y‖)+1|Ω|{ρIα0+p(T)ψ(‖y‖)+m∑i=1|σi| ρIα+β0+p(ηi)ψ(‖y‖)+|κ|}≤ψ(‖y‖)(ρIα0+p(T)+1|Ω|{ρIα0+p(T)+m∑i=1|σi| ρIα+β0+p(ηi)+|κ|}). |
Thus
‖N(y)‖≤ψ(ˉr)(ρIα0+p(T)+1|Ω|{ρIα0+p(T)+m∑i=1|σi| ρIα+β0+p(ηi)+|κ|}):=ℓ. |
(ⅲ) N maps bounded sets into equicontinuous sets of G.
Let t1,t2∈(0,T],t1<t2, Bˉr be a bounded set of G as in (ii) and let y∈Bˉr. Then
|N(y)(t2)−N(y)(t1)|≤|ρIα0+f(t2,y(t2))−ρIα0+f(t1,y(t1))|≤ρ1−αψ(ˉr)Γ(α)|∫t10[sρ−1(tρ2−sρ)1−α−sρ−1(tρ1−sρ)1−α]p(s)ds+∫t2t1sρ−1(tρ2−sρ)1−αp(s)ds|→0 as t1⟶t2, independent ~of y. |
From the steps (i)−(iii), we deduce by the Arzelá-Ascoli theorem that N:G⟶G is completely continuous.
(ⅳ) There exists an open set V⊆G with y≠νN(y) for ν∈(0,1) and y∈∂V.
Let y∈G be a solution of y−νNy=0 for ν∈[0,1]. Then, for t∈[0,T], we obtain
|y(t)|=|ν(Ny)(t)|≤ρIα0+|f(t,y(t))|+1|Ω|{ρIα0+|f(T,y(T))|+m∑i=1|σi| ρIα+β0+|f(ηi,y(ηi))|+|κ|}≤ρIα0+p(T)ψ(‖y‖)+1|Ω|{ρIα0+p(T)ψ(‖y‖)+m∑i=1|σi| ρIα+β0+p(ηi)ψ(‖y‖)+|κ|}≤ψ(‖y‖)(ρIα0+p(T)+1|Ω|{ρIα0+p(T)+m∑i=1|σi| ρIα+β0+p(ηi)+|κ|}), |
which, on taking the norm for t∈[0,T], implies that
‖y‖ψ(‖y‖)(ρIα0+p(T)+1|Ω|{ρIα0+p(T)+m∑i=1|σi| ρIα+β0+p(ηi)+|κ|})≤1. |
By the assumption (A2), there exists a positive constant W such that ‖y‖≠W. Next we define V={y∈G:‖y‖<W} and note that the operator N:¯V→G is continuous and completely continuous. By the choice of V, there does not exist any y∈∂V satisfying y=νN(y) for some ν∈(0,1). In consequence, by the nonlinear alternative of Leray-Schauder type [23], we deduce that there exists a fixed point y∈¯V for the operator N, which is a solution of the problem (1.1).
In our next result, we make use of Banach contraction mapping principle to establish the uniqueness of solutions for the problem (1.1).
Theorem 3.2. Suppose that
(A3) there exists a nonnegative constant L such that
|f(t,u)−f(t,v)|≤L‖u−v‖,fort∈[0,T]and everyu,v∈R. |
Then the problem (1.1) has a unique solution on [0,T] if
LΛ<1, | (3.3) |
where Λ is defined by (3.2).
Proof. Consider the operator N:G→G associated with the problem (1.1) defined by (3.1). With Λ given by (3.2), we fix
r≥Λf0+|κ|/|Ω|1−LΛ,f0=supt∈[0,T]|f(t,0)|, |
and show that FBr⊂Br, where Br={y∈G:‖y‖≤r}. For y∈Br, using (A3), we get
|N(y)(t)|≤ρIα0+[|f(t,y(t))−f(t,0)|+|f(t,0)|]+1|Ω|{ρIα0+[|f(T,y(T))−f(T,0)|+|f(T,0)|]+m∑i=1|σi| ρIα+β0+[|f(ηi,y(ηi))−f(ηi,0)|+|f(ηi,0)|]+|κ|}≤(L‖y‖+f0)[TραραΓ(α+1)+1|Ω|{TραραΓ(α+1)+m∑i=1|σi|ηρ(α+β)iρα+βΓ(α+β+1)}]+|κ||Ω|≤(Lr+f0)Λ+|κ||Ω|≤r, |
which, on taking the norm for t∈[0,T], yields ‖N(y)‖≤r. This shows that N maps Br into itself. Now we show that the operator N is a contraction. Let y,u∈G. Then we get
|N(y)(t)−N(u)(t)|≤ρIα0+|f(t,y(t))−f(t,u(t))|+1|Ω|{ρIα0+|f(T,y(T))−f(T,u(T))|+m∑i=1|σi| ρIα+β0+|f(ηi,y(ηi))−f(ηi,u(ηi))|}≤LΛ‖y−u‖. |
Consequently we obtain ‖N(y)−N(u)‖≤LΛ‖y−u‖, which shows that N is a contraction by means of (3.3). Thus the contraction mapping principle applies and the operator N has a unique fixed point. This shows that there exists a unique solution for the problem (1.1) on [0,T].
Now we prove the uniqueness of solutions for the problem (1.1) by applying a fixed point theorem for nonlinear contractions due to Boyd and Wong [24].
Definition 3.1. A mapping H:E→E is called a nonlinear contraction if we can find a continuous nondecreasing function ϕ:R+→R+ such that ϕ(0)=0, ϕ(ξ)<ξ for all ξ>0 and ‖Hy−Hu‖≤ϕ(‖y−u‖),∀y,u∈E (E is a Banach space).
Lemma 3.1. (Boyd and Wong) [24] Let E be a Banach space and let N:E→E be a nonlinear contraction. Then N has a unique fixed point in E.
Theorem 3.3. Assume that
(A4) |f(t,y)−f(t,u)|≤g(t)|y−u|G∗+|y−u|,t∈[0,T],y,u≥0, where g:[0,T]→R+ is continuous and
G∗=ρIα0+g(T)+1|Ω|{ρIα0+g(T)+m∑i=1|σi|ρIα+β0+g(ηi)}. | (3.4) |
Then the problem (1.1) has a unique solution on [0,T].
Proof. Let Ψ:R+→R+ be a continuous nondecreasing function such that Ψ(0)=0 and Ψ(ξ)<ξ for all ξ>0, defined by
Ψ(ξ)=G∗ξG∗+ξ,∀ξ≥0. |
Let y,u∈G. Then
|f(s,y(s))−f(s,u(s))|≤g(s)G∗Ψ(‖y−u‖), |
so that
|N(y)(t)−N(u)(t)|≤ρIα0+(g(t)|y(t)−u(t)|G∗+|y(t)−u(t)|)+1|Ω|{ρIα0+(g(T)|y(T)−u(T)|G∗+|y(T)−u(T)|)+m∑i=1|σi|ρIα+β0+(g(ηi)|y(ηi)−u(ηi)|G∗+|y(ηi)−u(ηi)|)}≤|y(t)−u(t)|G∗+|y(t)−u(t)|{ρIα0+g(T)+1|Ω|{ρIα0+g(T)+m∑i=1|σi|ρIα+β0+g(ηi)}}, |
for t∈[0,T]. By the condition (3.4), we deduce that ‖N(y)−N(u)‖≤Ψ(‖y−u‖) and hence N is a nonlinear contraction. Thus it follows from the fixed point theorem due to Boyd and Wong [24] that the operator N has a unique fixed point in G, which is indeed a unique solution of problem (1.1).
Example 3.1. Let us consider the following boundary value problem
{1/3 cD5/40+y=f(t,y), t∈[0,2],y(2)=21/3I3/4y(1/2)+1/2 1/3I3/4y(3/2)+1/4, δy(0)=0, | (3.5) |
where ρ=1/3,α=5/4,σ1=2,σ2=1/2,β=3/4,η1=1/2,η2=3/2,κ=1/4, T=2 and f(t,y(t)) will be fixed later.
Using the given data, we find that |Ω|=4.543695998 and Λ=7.572001575, where Ω and Λ are given by (2.10) and (3.2) respectively.
For illustrating Theorem 3.1, we take
f(t,y)=(1+t)30(|y||y|+1+y+18), | (3.6) |
and find that p(t)=(1+t)30 and ψ(‖y‖)=||y||+98. By condition (A2), we have W>0.7066246467. Thus, the hypothesis of Theorem 3.1 holds true, which implies that the problem (3.5) has at least one solution.
Furthermore, for the uniqueness results, Theorem 3.2 can be illustrated by choosing
f(t,y)=tan−1y+e−t2√81+sint. | (3.7) |
Clearly the condition (A3) is satisfied with L=1/18. Also
LΛ≈0.4206667542<1. |
Obviously all the conditions of Theorem 3.2 hold and consequently the problem (3.5) with f(t,y) given by (3.7) has a unique solution on [0,2] by the conclusion of Theorem 3.2.
Finally, for illustrating Theorem 3.3, we take
f(t,y)=t(|y||y|+11+18). | (3.8) |
Here we choose g(t)=(1+t) and find that
G∗=ρIα0+g(T)+1|Ω|{ρIα0+g(T)+m∑i=1|σi|ρIα+β0+g(ηi)}≈9.923097014, |
and
|f(t,y)−f(t,u)|=t(|y|−|u|11+|y|+|u|+|y||u|11)≤(1+t)|y−u|9.923097014+|y−u|. |
So, the conclusion of Theorem 3.3 applies to the problem (3.5) with f(t,y) given by (3.8).
In this section, we present existence results for the problem (1.2).
Definition 4.1. A function y∈AC2δ([0,T],R) is called a solution of the problem (1.2) if y(T)=∑mi=1σi ρIβ0+y(ηi)+κ,δy(0)=0 and there exists function v∈L1([0,T],R) such that v(t)∈F(t,y(t)) a.e. on [0,T] and
y(t)=ρIα0+v(t)+1Ω{−ρIα0+v(T)+m∑i=1σi ρIα+β0+v(ηi)+κ}. |
Here we prove an existence result for the problem (1.2) by applying nonlinear alternative for Kakutani maps [23] when F has convex values and is of Carathéodory type.
Theorem 4.1. Assume that
(B1) F:[0,T]×R→Pcp,c(R) is L1-Carathéodory, where Pcp,c(R) ={Y∈P(R): Y is compact and convex};
(B2) there exist a continuous nondecreasing function φ:[0,∞)→(0,∞) and a function p∈L1([0,T],R+) such that
‖F(t,y)‖P:=sup{|x|:x∈F(t,y)}≤p(t)φ(‖y‖) for each (t,y)∈[0,T]×R; |
(B3) there exists a constant ˆW>0 satisfying
ˆWφ(ˆW)(ρIα0+p(T)+1Ω( ρIα0+p(T)+m∑i=1|σi| ρIα+β0+p(ηi)+κ))>1. |
Then there exists at least one solution for the problem (1.2) on [0,T].
Proof. Define an operator M:C([0,T],R)⟶P(C([0,T],R)) by
M(y)={h∈C([0,T],R):h(t)=F(y)(t)}, | (4.1) |
where
F(y)(t)=ρIα0+v(t)+1Ω{−ρIα0+v(T)+m∑i=1σi ρIα+β0+v(ηi)+κ}, |
for v∈SF,y. Here SF,y denotes the set of selections of F and is defined by
SF,y:={v∈L1([0,T],R):v(t)∈F(t,y(t)) a.e. on [0,T]}, |
for each y∈C([0,T],R). Notice that the fixed points of the operator M are solutions of the problem (1.2).
To show that M satisfies the assumptions of Leray-Schauder nonlinear alternative [23], we split the proof in several steps.
Step 1. M(y) is convex for each y∈C([0,T],R) as SF,y is convex (F has convex values).
Step 2. Let Br={y∈C([0,T],R):‖y‖≤r} be a bounded ball in C([0,T],R), where r is a positive number. Then, for each h∈M(y),y∈Br, there exists v∈SF,y such that
h(t)=ρIα0+v(t)+1Ω{−ρIα0+v(T)+m∑i=1σi ρIα+β0+v(ηi)+κ} |
with
‖h‖≤φ(r)(ρIα0+p(T)+1|Ω|{ρIα0+p(T)+m∑i=1|σi| ρIα+β0+p(ηi)+|κ|}):=ℓ1. |
This shows that M maps bounded sets (balls) into bounded sets in C([0,T],R).
Step 3. In order to show that M maps bounded sets into equicontinuous sets of C([0,T],R), we take t1,t2∈(0,T],t1<t2, and y∈Br. Then we find that
|h(t2)−h(t1)|≤ρ1−αφ(r)Γ(α)|∫t10[sρ−1(tρ2−sρ)1−α−sρ−1(tρ1−sρ)1−α]p(s)ds+∫t2t1sρ−1(tρ−sρ)1−αp(s)ds|, |
which tends to zero independently of y∈Br as t2−t1→0. In view of the foregoing steps, it follows by the Arzelá-Ascoli theorem that M:C([0,T],R)→P(C([0,T],R)) is completely continuous.
Step 4. In our next step, we show that M is upper semi-continuous (u.s.c.). Since M is completely continuous, it is enough to establish that it has a closed graph (see [25, Proposition 1.2]). For that, let yn→y∗,hn∈M(yn) and hn→h∗. Then we have to show that h∗∈M(y∗). Associated with hn∈M(yn), we can find vn∈SF,yn such that for each t∈[0,T],
hn(t)=ρIα0+vn(t)+1Ω{−ρIα0+vn(s)vn(T)+m∑i=1σi ρIα+β0+vn(ηi)+κ}. |
Next, for each t∈[0,T], we establish that there exists v∗∈SF,y∗ satisfying
h∗(t)=ρIα0+v∗(t)+1Ω{−ρIα0+v∗(T)+m∑i=1σi ρIα+β0+v∗(ηi)+κ}. |
Consider the linear operator Θ:L1([0,T],R)→C([0,T],R) given by
v↦Θv(t)=ρIα0+v(t)+1Ω{−ρIα0+v(T)+m∑i=1σi ρIα+β0+v(ηi)+κ}. |
Notice that ‖hn(t)−h∗(t)‖→0 as n→∞. Thus we deduce by the closed graph theorem [26] that Θ∘SF is a closed graph operator. Furthermore, we have hn(t)∈Θ(SF,yn). As yn→y∗, we have
h∗(t)=ρIα0+v∗(t)+1Ω{−ρIα0+v∗(T)+m∑i=1σi ρIα+β0+v∗(ξ)+κ},for ~some v∗∈SF,y∗. |
Step 5. Finally, we show the existence of an open set U⊆C([0,T],R) such that y∉λM(y) for any λ∈(0,1) and all y∈∂U. For that we take λ∈(0,1) and y∈λM(y). Then there exists v∈L1([0,T],R) with v∈SF,y such that, for t∈[0,T], we have
y(t)=λρIα0+v(t)+λΩ{−ρIα0+v(T)+m∑i=1σi ρIα+β0+v(ηi)+κ}. |
As in the second step, one can obtain
|y(t)|≤ρIα0+|v(T)|+1|Ω|{ρIα0+|v(T)|+m∑i=1|σi| ρIα+β0+|v(ηi)|+|κ|}≤φ(‖y‖)(ρIα0+p(T)+1|Ω|{ρIα0+p(T)+m∑i=1|σi| ρIα+β0+p(ηi)+|κ|}), |
which implies that
‖y‖φ(‖y‖)(ρIα0+p(T)+1|Ω|{ρIα0+p(T)+m∑i=1|σi| ρIα+β0+p(ηi)+|κ|})≤1. |
By the hypothesis (B3), we can find ˆW such that ‖y‖≠ˆW. Setting Y={y∈C(J,R):‖y‖<ˆW}, we notice that the operator M:¯Y→P(C(J,R)) is compact multi-valued, u.s.c. with convex closed values. From the choice of Y, there does not exist any y∈∂Y satisfying y∈λM(y) for some λ∈(0,1). In consequence, we deduce by the nonlinear alternative of Leray-Schauder type [23] that M has a fixed point y∈¯Y which is a solution of the problem (1.2). This completes the proof.
Consider a mapping Hd:P(X)×P(X)→R∪{∞} defined by
Hd(S,V)=max{sups∈Sd(a,V),supv∈Vd(S,v)}, |
where d(S,v)=infs∈Sd(s;v), d(s,V)=infv∈Vd(s;v) and (X,d) is a metric space induced from the normed space (X;‖⋅‖). Note that (Pcl,b(X),Hd) is a metric space (see [27]), where Pcl,b(X)={Y∈P(X):Y is closed and bounded}.
The following result, dealing with the existence of solutions for the problem (1.2) with nonconvex valued right hand side of the inclusion, relies on Covitz and Nadler's fixed point theorem for multivalued maps [28].
Theorem 4.2. Assume that
(C1) F:[0,T]×R→Pcp(R) is such that F(⋅,y):[0,T]→Pcp(R) is measurable for each y∈R, where Pcp(R)={Y∈P(R):Y is compact};
(C2) Hd(F(t,y),F(t,ˉy))≤μ(t)|y−ˉy| for almost all t∈[0,T] and y,ˉy∈R with μ∈C([0,T],R+) and d(0,F(t,0))≤μ(t) for almost all t∈[0,T].
Then the problem (1.2) has at least one solution on [0,T] provided that
ϑ=‖μ‖Λ<1, | (4.2) |
where Λ is given by (3.2).
Proof. By the assumption (C1), the set SF,y is nonempty for each y∈C([0,T],R) and F has a measurable selection by Theorem Ⅲ.6 in [29]. Now we proceed to show that the operator M:(C[0,T],R)→Pcl(C([0,T],R)) (Pcl(C([0,T],R))={Y∈P(C([0,T],R)):Y is closed}) is a contraction so that Covitz and Nadler's Theorem [28] is applicable.
In the first step, we show that M(y)∈Pcl((C[0,T],R)) for each y∈C([0,T],R). Let {un}n≥0∈M(y) with un→u (n→∞) in C([0,T],R). Then u∈C([0,T],R) and there exists vn∈SF,yn satisfying
un(t)=ρIα0+vn(t)+1Ω{−ρIα0+vn(T)+m∑i=1σi ρIα+β0+vn(ηi)+κ}for each t∈[0,T]. |
In view of the compact values of F, we pass onto a subsequence (if necessary) to find that vn converges to v in L1([0,T],R). For v∈SF,y and for each t∈[0,T], we have
un(t)→u(t)=ρIα0+v(t)+1Ω{−ρIα0+v(T)+m∑i=1σi ρIα+β0+v(ηi)+κ}. |
Thus u∈M(y).
Now, for each y,ˉy∈C([0,T],R), we establish that there exists ϑ<1 (defined by (4.2)) satisfying
Hd(M(y),M(ˉy))≤ϑ‖y−ˉy‖. |
Let y,ˉy∈C([0,T],R) and h1∈M(y). Then there exists v1(t)∈F(t,y(t)) satisfying
h1(t)=ρIα0+v1(t)+1Ω{−ρIα0+v1(T)+m∑i=1σi ρIα+β0+v1(ηi)+κ}, |
for each t∈[0,T]. By (C2), we have
Hd(F(t,y),F(t,ˉy))≤μ(t)|y(t)−ˉy(t)|. |
Therefore, we can find w∈F(t,ˉy(t)) satisfying
|v1(t)−w|≤μ(t)|y(t)−ˉy(t)|, t∈[0,T]. |
Introduce U:[0,T]→P(R) by
U(t)={w∈R:|v1(t)−w|≤μ(t)|y(t)−ˉy(t)|}. |
As U(t)∩F(t,ˉy(t)) is measurable (Proposition Ⅲ.4 [29]), we can find a measurable selection v2(t) for U such that v2(t)∈F(t,ˉy(t)) satisfying |v1(t)−v2(t)|≤μ(t)|y(t)−ˉy(t)| for each t∈[0,T].
Define
h2(t)=ρIα0+v2(t)+1Ω{−ρIα0+v2(T)+m∑i=1σi ρIα+β0+v2(ηi)+κ}, |
for each t∈[0,T]. Then
|h1(t)−h2(t)|≤ρIα0+|v1(t)−v2(t)|+1|Ω|{ρIα0+|v1(T)−v2(T)|+m∑i=1|σi| ρIα+β0+|v1(ηi)−v2(ηi)|}≤‖μ‖[TραραΓ(α+1)+1|Ω|{TραραΓ(α+1)+m∑i=1|σi|ηρ(α+β)iρα+βΓ(α+β+1)}]‖y−ˉy‖. |
Hence
‖h1−h2‖≤‖μ‖[TραραΓ(α+1)+1|Ω|{TραραΓ(α+1)+m∑i=1|σi|ηρ(α+β)iρα+βΓ(α+β+1)}]‖y−ˉy‖. |
Analogously, switching the roles of y and ¯y, we can obtain
Hd(M(y),M(ˉy))≤‖μ‖[TραραΓ(α+1)+1|Ω|{TραραΓ(α+1)+m∑i=1|σi|ηρ(α+β)iρα+βΓ(α+β+1)}]‖y−ˉy‖. |
So M is a contraction. Thus, by Covitz and Nadler's fixed point theorem [28], the operator M has a fixed point y, which corresponds to a solution of (1.2).
Example 4.1. Consider the following boundary value problem
{1/3 cD5/40+y∈F(t,y), t∈[0,2],y(2)=21/3I3/4y(1/2)+1/2 1/3I3/4y(3/2)+1/4, δy(0)=0, | (4.3) |
where F(t,y) will be fixed later.
For illustrating Theorem 4.1, we take
F(t,y)=[e−t√900+t(siny+12) ,(1+t)30(|y||y|+1+y+18)]. | (4.4) |
Using the given data, we find p(t)=(1+t)30,φ(‖y‖)=||y||+98, and by condition (B3), we have ˆW>0.7066246467. Thus all conditions of Theorem 4.1 are satisfied and consequently, there exists at least one solution for the problem (4.3) with F(t,y) given by (4.4) on [0,2].
In order to demonstrate the application of Theorem 4.2, let us choose
F(t,y)=[e−t√900+t(tan−1y+12) ,(1+t)30(|y||y|+1+18)]. | (4.5) |
Clearly
Hd(F(t,y),F(t,ˉy))≤(t+1)30‖y−ˉy‖. |
Letting μ(t)=(t+1)30, it is easy to check that d(0,F(t,0))≤μ(t) holds for almost all t∈[0,2] and ϑ≈0.7572001575<1 (ϑ is given by 4.2). As the hypotheses of Theorem 4.2 are satisfied, we conclude that the problem (4.3) with F(t,y) given by (4.5) has at least one solution on [0,2].
We have developed the existence theory for fractional differential equations and inclusions involving Caputo-type generalized fractional derivative equipped with generalized fractional integral boundary conditions (in the sense of Katugampola). Standard fixed point theorems for single-valued and multi-valued maps are employed to obtain the desired results, which are well illustrated with the aid of examples. Our results are new in the given configuration and contribute significantly to the existing literature on the topic.
The authors gratefully acknowledge the referees for their useful comments on their paper.
The authors declare that they have no conflict of interests.
[1] |
Pooladanda V, Thatikonda S, Godugu C (2020) The current understanding and potential therapeutic options to combat COVID-19. Life Sci 254: 117765. doi: 10.1016/j.lfs.2020.117765
![]() |
[2] |
Wang H, Li X, Li T, et al. (2020) The genetic sequence, origin, and diagnosis of SARS-CoV-2. Eur J Clin Microbiol Infect Dis 39: 1629-1635. doi: 10.1007/s10096-020-03899-4
![]() |
[3] |
Li H, Liu SM, Yu XH, et al. (2020) Coronavirus disease 2019 (COVID-19): current status and future perspectives. Int J Antimicrob Agents 55: 105951. doi: 10.1016/j.ijantimicag.2020.105951
![]() |
[4] | Wu J, Deng W, Li S, et al. (2020) Advances in research on ACE2 as a receptor for 2019-nCoV. Cell Mol Life Sci 1-14. |
[5] | Umakanthan S, Sahu P, Ranade AV, et al. (2020) Origin, transmission, diagnosis and management of coronavirus disease 2019 (COVID-19). Postgrad Med J 96: 753-758. |
[6] |
Fu L, Wang B, Yuan T, et al. (2020) Clinical characteristics of coronavirus disease 2019 (COVID-19) in China: A systematic review and meta-analysis. J Infect 80: 656-665. doi: 10.1016/j.jinf.2020.03.041
![]() |
[7] |
Iyer M, Jayaramayya K, Subramaniam MD, et al. (2020) COVID-19: an update on diagnostic and therapeutic approaches. BMB Rep 53: 191-205. doi: 10.5483/BMBRep.2020.53.4.080
![]() |
[8] |
Touma M (2020) COVID-19: molecular diagnostics overview. J Mol Med (Berl) 98: 947-954. doi: 10.1007/s00109-020-01931-w
![]() |
[9] |
Huang C, Wang Y, Li X, et al. (2020) Clinical features of patients infected with 2019 novel coronavirus in Wuhan, China. Lancet 395: 497-506. doi: 10.1016/S0140-6736(20)30183-5
![]() |
[10] |
Zhou F, Yu T, Du R, et al. (2020) Clinical course and risk factors for mortality of adult inpatients with COVID-19 in Wuhan, China: a retrospective cohort study. Lancet 395: 1054-1062. doi: 10.1016/S0140-6736(20)30566-3
![]() |
[11] | Zheng F, Tang W, Li H, et al. (2020) Clinical characteristics of 161 cases of corona virus disease 2019 (COVID-19) in Changsha. Eur Rev Med Pharmacol Sci 24: 3404-3410. |
[12] |
Chu DKW, Pan Y, Cheng SMS, et al. (2020) Molecular diagnosis of a novel coronavirus (2019-nCoV) causing an outbreak of pneumonia. Clin Chem 66: 549-555. doi: 10.1093/clinchem/hvaa029
![]() |
[13] |
Loeffelholz MJ, Tang YW (2020) Laboratory diagnosis of emerging human coronavirus infections—the state of the art. Emerg Microbes Infect 9: 747-756. doi: 10.1080/22221751.2020.1745095
![]() |
[14] |
Emery SL, Erdman DD, Bowen MD, et al. (2004) Real-time reverse transcription-polymerase chain reaction assay for SARS-associated Coronavirus. Emerg Infect Dis 10: 311-316. doi: 10.3201/eid1002.030759
![]() |
[15] |
Wölfel R, Corman VM, Guggemos W, et al. (2020) Virological assessment of hospitalized patients with COVID-2019. Nature 581: 465-469. doi: 10.1038/s41586-020-2196-x
![]() |
[16] |
Hans R, Marwaha N (2014) Nucleic acid testing-benefits and constraints. Asian J Transfus Sci 8: 2-3. doi: 10.4103/0973-6247.126679
![]() |
[17] | La Marca A, Capuzzo M, Paglia T, et al. (2020) Testing for SARS-CoV-2 (COVID-19): a systematic review and clinical guide to molecular and serological in-vitro diagnostic assays. Reprod Biomed Online . |
[18] | Doi A, Iwata K, Kuroda H, et al. (2020) Estimation of seroprevalence of novel coronavirus disease (COVID-19) using preserved serum at an outpatient setting in Kobe, Japan: A cross-sectional study. medRxiv . |
[19] | Li Z, Yi Y, Luo X, et al. (2020) Development and Clinical Application of A Rapid IgM-IgG Combined Antibody Test for SARS-CoV-2 Infection Diagnosis. J Med Virol 27: 25727. |
[20] | Liu R, Liu X, Han H, et al. The comparative superiority of IgM-IgG antibody test to real-time reverse transcriptase PCR detection for SARS-CoV-2 infection diagnosis 2020 (2020) .Available from: https://doi.org/10.1101/2020.03.28.20045765. |
[21] |
Grant BD, Anderson CE, Williford JR, et al. (2020) SARS-CoV-2 Coronavirus Nucleocapsid Antigen-Detecting Half-Strip Lateral Flow Assay Toward the Development of Point of Care Tests Using Commercially Available Reagents. Anal Chem 92: 11305-11309. doi: 10.1021/acs.analchem.0c01975
![]() |
[22] |
Burki TK (2020) Testing for COVID-19. Lancet Respir Med 8: e63-e64. doi: 10.1016/S2213-2600(20)30247-2
![]() |
[23] | Chen H, Ai L, Lu H, et al. (2020) Clinical and imaging features of COVID-19. Radiol Infect Dis . |
[24] |
Singh AK, Majumdar S, Singh R, et al. (2020) Role of corticosteroid in the management of COVID-19: A systemic review and a Clinician's perspective. Diabetes Metab Syndr 14: 971-978. doi: 10.1016/j.dsx.2020.06.054
![]() |
[25] |
Perez A, Jansen-Chaparro S, Saigi I, et al. (2014) Glucocorticoid-induced hyperglycemia. J Diabetes 6: 9-20. doi: 10.1111/1753-0407.12090
![]() |
[26] |
Singh AK, Singh A, Singh R, et al. (2020) Remdesivir in COVID-19: A critical review of pharmacology, pre-clinical and clinical studies. Diabetes Metab Syndr 14: 641-648. doi: 10.1016/j.dsx.2020.05.018
![]() |
[27] |
Bhatraju PK, Ghassemieh BJ, Nichols M, et al. (2020) Covid-19 in Critically Ill Patients in the Seattle Region—Case Series. N Engl J Med 382: 2012-2022. doi: 10.1056/NEJMoa2004500
![]() |
[28] |
Singh AK, Singh A, Shaikh A, et al. (2020) Chloroquine and hydroxychloroquine in the treatment of COVID-19 with or without diabetes: A systematic search and a narrative review with a special reference to India and other developing countries. Diabetes Metab Syndr 14: 241-246. doi: 10.1016/j.dsx.2020.03.011
![]() |
[29] |
Moore N (2020) Chloroquine for COVID-19 Infection. Drug Saf 43: 393-394. doi: 10.1007/s40264-020-00933-4
![]() |
[30] | Owa AB, Owa OT (2020) Lopinavir/ritonavir use in Covid-19 infection: is it completely non-beneficial? J Microbiol Immunol Infect . |
[31] |
Uzunova K, Filipova E, Pavlova V, et al. (2020) Insights into antiviral mechanisms of remdesivir, lopinavir/ritonavir and chloroquine/hydroxychloroquine affecting the new SARS-CoV-2. Biomed Pharmacother 131: 110668. doi: 10.1016/j.biopha.2020.110668
![]() |
[32] | Heidary F, Gharebaghi R, et al. (2020) Ivermectin: a systematic review from antiviral effects to COVID-19 complementary regimen. J Antibiot (Tokyo) 1-10. |
[33] | Gupta D, Sahoo AK, Singh A (2020) Ivermectin: potential candidate for the treatment of Covid 19. Braz J Infect Dis . |
[34] |
Coomes EA, Haghbayan H (2020) Favipiravir, an antiviral for COVID-19? J Antimicrob Chemother 75: 2013-2014. doi: 10.1093/jac/dkaa171
![]() |
[35] | Cai Q, Yang M, Liu D, et al. (2020) Experimental Treatment with Favipiravir for COVID-19: An Open-Label Control Study. Engineering (Beijing) . |
[36] | Wu R, Wang L, Kuo HD, et al. (2020) An Update on Current Therapeutic Drugs Treating COVID-19. Curr Pharmacol Rep 1-15. |
[37] |
Huttner BD, Catho G, Pano-Pardo JR, et al. (2020) COVID-19: don't neglect anti-microbial stewardship principles!. Clin Microbiol Infect 26: 808-810. doi: 10.1016/j.cmi.2020.04.024
![]() |
[38] |
Rawson TM, Ming D, Ahmad R, et al. (2020) Anti-microbial use, drug-resistant infections and COVID-19. Nat Rev Microbiol 18: 409-410. doi: 10.1038/s41579-020-0395-y
![]() |
[39] | Cai X, Ren M, Chen F, et al. (2020) Blood transfusion during the COVID-19 outbreak. Blood Transfus 18: 79-82. |
[40] | Kumar S, Sharma V, Priya K (2020) Battle against COVID-19: Efficacy of Convalescent Plasma as an emergency therapy. Am J Emerg Med . |
[41] |
Mair-Jenkins J, Saavedra-Campos M, Baillie JK (2015) The effectiveness of convalescent plasma and hyperimmune immunoglobulin for the treatment of severe acute respiratory infections of viral etiology: a systematic review and exploratory meta-analysis. J Infect Dis 211: 80-90. doi: 10.1093/infdis/jiu396
![]() |
[42] | Valk SJ, Piechotta V, Chai KL, et al. (2020) Convalescent plasma or hyperimmune immunoglobulin for people with COVID-19: a rapid review. Cochrane Db Syst Rev 5: CD013600. |
[43] | Hartman WR, Hess AS, Connor JP Hospitalized COVID-19 Patients treated with Convalescent Plasma in a Mid-size City in the Midwest (2020) .Available from: https://www.researchgate.net/publication/346772157_Hospitalized_COVID-19_Patients_treated_with_Convalescent_Plasma_in_a_Mid-size_City_in_the_Midwest. |
[44] |
Li L, Zhang W, Hu Y, et al. (2020) Effect of Convalescent Plasma Therapy on Time to Clinical Improvement in Patients With Severe and Life-threatening COVID-19: A Randomized Clinical Trial. JAMA 324: 460-470. doi: 10.1001/jama.2020.10044
![]() |
[45] |
Rajarshi K, Chatterjee A, Ray S (2020) Combating COVID-19 with Mesenchymal Stem Cell therapy. Biotechnol Rep (Amst) 26: e00467. doi: 10.1016/j.btre.2020.e00467
![]() |
[46] |
Golchin A, Seyedjafari E, Ardeshirylajimi A (2020) Mesenchymal Stem Cell Therapy for COVID-19: Present or Future. Stem Cell Rev Rep 16: 427-433. doi: 10.1007/s12015-020-09973-w
![]() |
[47] |
Golchin A, Farahany TZ, Khojasteh A, et al. (2018) The clinical trials of Mesenchymal stem cell therapy in skin diseases: An update and concise review. Curr Stem Cell Res Ther 14: 22-33. doi: 10.2174/1574888X13666180913123424
![]() |
[48] | Kewan T, Covut F, Al-Jaghbeer MJ, et al. (2020) Tocilizumab for treatment of patients with severe COVID–19: A retrospective cohort study. E Clin Med 100418. |
[49] | Guaraldi G, Meschiari M, Cozzi-Lepri A, et al. (2020) Tocilizumab in patients with severe COVID-19: a retrospective cohort study. Lancet Rheumatol 2. |
[50] | Cantini F, Niccoli L, Nannini C, et al. (2020) Beneficial impact of Baricitinib in COVID-19 moderate pneumonia; multicentre study. J Infect . |
[51] |
Cantini F, Niccoli L, Matarrese D, et al. (2020) Baricitinib therapy in COVID-19: A pilot study on safety and clinical impact. J Infect 81: 318-356. doi: 10.1016/j.jinf.2020.04.017
![]() |
[52] |
Maoujoud O, Asserraji M, Ahid S, et al. (2020) Anakinra for patients with COVID-19. Lancet Rheumatol 2: e383. doi: 10.1016/S2665-9913(20)30177-6
![]() |
[53] |
Filocamo G, Mangioni D, Tagliabue P, et al. (2020) Use of anakinra in severe COVID-19: A case report. Int J Infect Dis 96: 607-609. doi: 10.1016/j.ijid.2020.05.026
![]() |
[54] | Kow CS, Hasan SS (2020) Use of low-molecular-weight heparin in COVID-19 patients. J Vasc Surg Venous Lymphat Disord . |
[55] | Costanzo L, Palumbo FP, Ardita G, et al. (2020) Coagulopathy, thromboembolic complications, and the use of heparin in COVID-19 pneumonia. J Vasc Surg Venous Lymphat Disord . |
[56] | Turshudzhyan A (2020) Anticoagulation Options for Coronavirus Disease 2019 (COVID-19)-Induced Coagulopathy. Cureus 12: e8150. |
[57] |
Boretti A, Banik BK (2020) Intravenous Vitamin C for reduction of cytokines storm in Acute Respiratory Distress Syndrome. PharmaNutrition 12: 100190. doi: 10.1016/j.phanu.2020.100190
![]() |
[58] |
Simonson W (2020) Vitamin C and Coronavirus. Geriatr Nurs 41: 331-332. doi: 10.1016/j.gerinurse.2020.05.002
![]() |
[59] |
Hemilä H (2003) Vitamin C and SARS coronavirus. J Antimicrob Chemother 52: 1049-1050. doi: 10.1093/jac/dkh002
![]() |
[60] | World Health Organization Draft landscape of COVID-19 candidate vaccines (2020) .Available from: https://www.who.int/who-documents-detail/draft-landscape-of-covid-19-candidate-vaccines. |
[61] |
Day M (2020) Covid-19: four fifths of cases are asymptomatic, China figures indicate. BMJ 369: 1375. doi: 10.1136/bmj.m1375
![]() |
[62] |
Sutton D, Fuchs K, D'Alton M, et al. (2020) Universal Screening for SARS-CoV-2 in Women Admitted for Delivery. N Engl J Med 382: 2163-2164. doi: 10.1056/NEJMc2009316
![]() |
[63] |
Mizumoto K, Kagaya K, Zarebski A, et al. (2020) Estimating the asymptomatic proportion of coronavirus disease 2019 (COVID-19) cases on board the Diamond Princess cruise ship, Yokohama, Japan, 2020. Euro Surveill 25: 2000180. doi: 10.2807/1560-7917.ES.2020.25.10.2000180
![]() |
[64] |
Wrapp D, Wang N, Corbett KS, et al. (2020) Cryo-EM structure of the 2019-nCoV spike in the prefusion conformation. Science 367: 1260-1263. doi: 10.1126/science.abb2507
![]() |
[65] |
Andersen KG, Rambaut A, Lipkin WI, et al. (2020) The proximal origin of SARS-CoV-2. Nat Med 26: 450-452. doi: 10.1038/s41591-020-0820-9
![]() |
[66] |
Benvenuto D, Giovanetti M, Ciccozzi A, et al. (2020) The 2019-new coronavirus epidemic: Evidence for virus evolution. J Med Virol 92: 455-459. doi: 10.1002/jmv.25688
![]() |
[67] |
Yan R, Zhang Y, Li Y, et al. (2020) Structural basis for the recognition of SARS-CoV-2 by full-length human ACE2. Science 367: 1444-1448. doi: 10.1126/science.abb2762
![]() |
[68] |
Enjuanes L, Zuñiga S, Castaño-Rodriguez C, et al. (2016) Molecular Basis of Coronavirus Virulence and Vaccine Development. Adv Virus Res 96: 245-286. doi: 10.1016/bs.aivir.2016.08.003
![]() |
[69] |
Song Z, Xu Y, Bao L, et al. (2019) From SARS to MERS, Thrusting Coronaviruses into the Spotlight. Viruses 11: 59. doi: 10.3390/v11010059
![]() |
[70] | Wu F, Wang A, Liu M, et al. (2020) Neutralizing antibody responses to SARS-CoV-2 in a COVID-19 recovered patient cohort and their implications. medRxiv . |
[71] |
Thi Nhu Thao T, Labroussaa F, Ebert N, et al. (2020) Rapid reconstruction of SARS-CoV-2 using a synthetic genomics platform. Nature 582: 561-565. doi: 10.1038/s41586-020-2294-9
![]() |
[72] |
Xie X, Muruato A, Lokugamage KG, et al. (2020) An Infectious cDNA Clone of SARS-CoV-2. Cell Host Microbe 27: 841-848. doi: 10.1016/j.chom.2020.04.004
![]() |
[73] |
Dicks MD, Spencer AJ, Edwards NJ, et al. (2012) A novel chimpanzee adenovirus vector with low human seroprevalence: improved systems for vector derivation and comparative immunogenicity. PLoS One 7: e40385. doi: 10.1371/journal.pone.0040385
![]() |
[74] |
Fausther-Bovendo H, Kobinger GP (2014) Pre-existing immunity against Advectors. Hum Vaccines Immunother 10: 2875-2884. doi: 10.4161/hv.29594
![]() |
[75] |
Alberer M, Gnad-Vogt U, Hong HS, et al. (2017) Safety and immunogenicity of a mRNA rabies vaccine in healthy adults: an open-label, non-randomized, prospective, first-in-human phase 1 clinical trial. Lancet 390: 1511-1520. doi: 10.1016/S0140-6736(17)31665-3
![]() |
[76] |
Smith TRF, Patel A, Ramos S, et al. (2020) Immunogenicity of a DNA vaccine candidate for COVID-19. Nat Commun 11: 2601. doi: 10.1038/s41467-020-16505-0
![]() |
[77] | BIONTECH BioNTech and Pfizer announce regulatory approval from German authority Paul-Ehrlich-Institut to commence first clinical trial of COVID-19 vaccine candidates (2020) .Available from: https://investors.biontech.de/node/7431/pdf. |
[78] |
Takashima Y, Osaki M, Ishimaru Y, et al. (2011) Artificial molecular clamp: A novel device for synthetic polymerases. Angew Chem Int Ed 50: 7524-7528. doi: 10.1002/anie.201102834
![]() |
[79] |
Singh K, Mehta S (2016) The clinical development process for a novel preventive vaccine: An overview. J Postgrad Med 62: 4-11. doi: 10.4103/0022-3859.173187
![]() |
[80] |
Wang Q, Zhang L, Kuwahara K, et al. (2016) Immunodominant SARS Coronavirus Epitopes in Humans Elicited both Enhancing and Neutralizing Effects on Infection in Non-human Primates. ACS Infect Dis 2: 361-376. doi: 10.1021/acsinfecdis.6b00006
![]() |
[81] | de Sousa E, Ligeiro D, Lérias JR, et al. (2020) Mortality in COVID-19 disease patients: Correlating Association of Major histocompatibility complex (MHC) with severe acute respiratory syndrome 2 (SARS-CoV-2) variants. Int J Infect Dis . |
[82] |
Li H, Liu SM, Yu XH, et al. (2020) Coronavirus disease 2019 (COVID-19): current status and future perspectives. Int J Antimicrob Agents 55: 105951. doi: 10.1016/j.ijantimicag.2020.105951
![]() |
[83] |
Deb B, Shah H, Goel S (2020) Current global vaccine and drug efforts against COVID-19: Pros and cons of bypassing animal trials. J Biosci 45: 82. doi: 10.1007/s12038-020-00053-2
![]() |
[84] |
Abd El-Aziz TM, Stockand JD (2020) Recent progress and challenges in drug development against COVID-19 Coronavirus (SARS-CoV-2)—an update on the status. Infect Genet Evol 83: 104327. doi: 10.1016/j.meegid.2020.104327
![]() |
[85] |
Polack FP, Thomas SJ, Kitchin N, et al. (2020) C4591001 Clinical Trial Group. Safety and Efficacy of the BNT162b2 mRNA Covid-19 Vaccine. N Engl J Med 383: 2603-2615. doi: 10.1056/NEJMoa2034577
![]() |
[86] |
Walsh EE, Frenck RW, Falsey AR, et al. (2020) Safety and Immunogenicity of Two RNA-Based Covid-19 Vaccine Candidates. N Engl J Med 383: 2439-2450. doi: 10.1056/NEJMoa2027906
![]() |
[87] |
Anderson EJ, Rouphael NG, Widge AT, et al. (2020) mRNA-1273 Study Group. Safety and Immunogenicity of SARS-CoV-2 mRNA-1273 Vaccine in Older Adults. N Engl J Med 383: 2427-2438. doi: 10.1056/NEJMoa2028436
![]() |
[88] |
Giamarellos-Bourboulis EJ, Tsilika M, Moorlag S, et al. (2020) Activate: Randomized Clinical Trial of BCG Vaccination against Infection in the Elderly. Cell 183: 315-323. doi: 10.1016/j.cell.2020.08.051
![]() |
[89] |
Junqueira-Kipnis AP, Dos Anjos LRB, Barbosa LCS, et al. (2020) BCG revaccination of health workers in Brazil to improve innate immune responses against COVID-19: A structured summary of a study protocol for a randomized controlled trial. Trials 21: 881. doi: 10.1186/s13063-020-04822-0
![]() |
[90] | Rivas MN, Ebinger JE, Wu M, et al. (2021) BCG vaccination history associates with decreased SARS-CoV-2 seroprevalence across a diverse cohort of health care workers. J Clin Invest 2021 131. |
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