The novel Coronavirus (COVID-19) is spreading and has caused a large-scale infection in China since December 2019. This has led to a significant impact on the lives and economy in China and other countries. Here we develop a discrete-time stochastic epidemic model with binomial distributions to study the transmission of the disease. Model parameters are estimated on the basis of fitting to newly reported data from January 11 to February 13, 2020 in China. The estimates of the contact rate and the effective reproductive number support the efficiency of the control measures that have been implemented so far. Simulations show the newly confirmed cases will continue to decline and the total confirmed cases will reach the peak around the end of February of 2020 under the current control measures. The impact of the timing of returning to work is also evaluated on the disease transmission given different strength of protection and control measures.
Citation: Sha He, Sanyi Tang, Libin Rong. A discrete stochastic model of the COVID-19 outbreak: Forecast and control[J]. Mathematical Biosciences and Engineering, 2020, 17(4): 2792-2804. doi: 10.3934/mbe.2020153
[1] | Saad Ihsan Butt, Artion Kashuri, Muhammad Umar, Adnan Aslam, Wei Gao . Hermite-Jensen-Mercer type inequalities via Ψ-Riemann-Liouville k-fractional integrals. AIMS Mathematics, 2020, 5(5): 5193-5220. doi: 10.3934/math.2020334 |
[2] | Mohammed A. Almalahi, Mohammed S. Abdo, Satish K. Panchal . On the theory of fractional terminal value problem with ψ-Hilfer fractional derivative. AIMS Mathematics, 2020, 5(5): 4889-4908. doi: 10.3934/math.2020312 |
[3] | Naila Mehreen, Matloob Anwar . Some inequalities via Ψ-Riemann-Liouville fractional integrals. AIMS Mathematics, 2019, 4(5): 1403-1415. doi: 10.3934/math.2019.5.1403 |
[4] | Yousaf Khurshid, Muhammad Adil Khan, Yu-Ming Chu . Conformable fractional integral inequalities for GG- and GA-convex functions. AIMS Mathematics, 2020, 5(5): 5012-5030. doi: 10.3934/math.2020322 |
[5] | Mohammad Esmael Samei, Lotfollah Karimi, Mohammed K. A. Kaabar . To investigate a class of multi-singular pointwise defined fractional q–integro-differential equation with applications. AIMS Mathematics, 2022, 7(5): 7781-7816. doi: 10.3934/math.2022437 |
[6] | Feng Qi, Siddra Habib, Shahid Mubeen, Muhammad Nawaz Naeem . Generalized k-fractional conformable integrals and related inequalities. AIMS Mathematics, 2019, 4(3): 343-358. doi: 10.3934/math.2019.3.343 |
[7] | Gou Hu, Hui Lei, Tingsong Du . Some parameterized integral inequalities for p-convex mappings via the right Katugampola fractional integrals. AIMS Mathematics, 2020, 5(2): 1425-1445. doi: 10.3934/math.2020098 |
[8] | Chunhong Li, Dandan Yang, Chuanzhi Bai . Some Opial type inequalities in (p, q)-calculus. AIMS Mathematics, 2020, 5(6): 5893-5902. doi: 10.3934/math.2020377 |
[9] | Xiuzhi Yang, G. Farid, Waqas Nazeer, Muhammad Yussouf, Yu-Ming Chu, Chunfa Dong . Fractional generalized Hadamard and Fejér-Hadamard inequalities for m-convex functions. AIMS Mathematics, 2020, 5(6): 6325-6340. doi: 10.3934/math.2020407 |
[10] | M. Emin Özdemir, Saad I. Butt, Bahtiyar Bayraktar, Jamshed Nasir . Several integral inequalities for (α, s,m)-convex functions. AIMS Mathematics, 2020, 5(4): 3906-3921. doi: 10.3934/math.2020253 |
The novel Coronavirus (COVID-19) is spreading and has caused a large-scale infection in China since December 2019. This has led to a significant impact on the lives and economy in China and other countries. Here we develop a discrete-time stochastic epidemic model with binomial distributions to study the transmission of the disease. Model parameters are estimated on the basis of fitting to newly reported data from January 11 to February 13, 2020 in China. The estimates of the contact rate and the effective reproductive number support the efficiency of the control measures that have been implemented so far. Simulations show the newly confirmed cases will continue to decline and the total confirmed cases will reach the peak around the end of February of 2020 under the current control measures. The impact of the timing of returning to work is also evaluated on the disease transmission given different strength of protection and control measures.
P.L Čebyšev in the year 1882 has proved the following interesting inequality:
|1b−ab∫af(x)g(x)dx−(1b−ab∫af(x)dx)(1b−ab∫ag(x)dx)|≤112(b−a)2‖f′‖∞‖g′‖∞. |
where f,g are absolutely continuous functions defined on [a,b] and f′,g′∈L∞[a,b]. The left hand side of the above equation is denoted by T(f,g) is called Cebysev Functional if the integral exists. The applications of above type of inequalities can be found in the field of coding theory, statistics and other branches of mathematics.
In last few decades many researchers have obtained various extensions and generalizations of above inequalities using various techniques see [1,2]. Study of inequalities have attracted the attention of researchers from various fields due to its wide applications in various fields [3,4].
During last few years the subject of Fractional Calculus has been developed rapidly due to the applications in various fields of science and engineering. Various new definitions of fractional derivatives and integrals have been obtained by various researchers depending on the applications such as Riemann liouville, Caputo, Saigo, Hilfer, Hadmard, Katugampola and others See [5,6,7,8]. Many results on study of mathematical inequalities using various new fractional definitions such as Conformable and generalized fractional integral were obtained in [9,10]. Recently in [11,12,13,14,15] the authors have obtained the results on Cebysev inequalities using various fractional integral and derivatives definitions.
In [7] authors have given definations of fractional derivative and integrals of a functions with respect to another functions. Recently in [16,17] authors have studied the ψ Caputo and ψ Hilfer fractional derivative of a function with respect to another functions and its applications. The ψ fractional and integral definations are more generalized and it reduces to Riemann Liouville, Hadmard and Erdelyi-Kober fractional definitions for different values of ψ.
Motivated from the above mentioned literature the aim of this paper is to obtain ψ Caputo fractional Čebyšev inequalities involving functions of two and three variables.
Now in this section we give some basic definitions and properties which are useful in our subsequent discussions. In [7,8] the authors have defined the fractional integrals and fractional derivative of a function with respect to another function as follows.
Definition 2.1 [7,16]. Let I=[a,b] be an interval, α>0, f is an integrable function defined on I and ψ∈C1(I) an increasing function such that ψ′(x)≠0 for all x∈I then fractional derivative and integral of f is given by
Iα,ψa+f(x)=1Γ(α)x∫aψ′(t)(ψ(x)−ψ(t))α−1f(t)dt |
and
Dα,ψa+f(x)=(1ψ′(x)ddx)nIn−α,ψa+f(x)=1Γ(n−α)(1ψ′(x)ddx)nx∫aψ′(t)(ψ(x)−ψ(t))n−α−1f(t)dt, |
respectively. Similarly right fractional integral and right fractional derivative are given by
Iα,ψb−f(x)=1Γ(α)x∫aψ′(t)(ψ(t)−ψ(x))α−1f(t)dt |
and
Dα,ψb−f(x)=(−1ψ′(x)ddx)nIn−α,ψb−f(x)=1Γ(n−α)(1ψ′(x)ddx)nx∫aψ′(t)(ψ(t)−ψ(x))n−α−1f(t)dt. |
In [16] Almedia has considered a Caputo type fractional derivative with respect to another function.
Definition 2.2 [16] Let α>0, n∈N, I is the interval −∞≤a<b≤∞, f,ψ∈Cn(I) two functions such that ψ is increasing and ψ′(x)≠0 for all x∈I. The left ψ-Caputo fractional derivative of f of order α is given by
CDα,ψa+f(x)=In−α,ψa+(1ψ′(x)ddx)nf(x), |
and the right ψ-Caputo fractional derivative of f is given by
CDα,ψb−f(x)=In−α,ψb−(−1ψ′(x)ddx)nf(x). |
For given α∉N
CDα,ψa+f(x)=1Γ(n−α)x∫aψ′(t)(ψ(x)−ψ(t))n−α−1f[n]ψ(t)dt |
and
CDα,ψb−f(x)=1Γ(n−α)x∫aψ′(t)(ψ(t)−ψ(x))n−α−1(−1)nf[n]ψ(t)dt. |
In particular when α∈(0,1) then
CDα,ψa+f(x)=1Γ(1−α)x∫a(ψ(x)−ψ(t))−αf′(t)dt |
and
CDα,ψb−f(x)=1Γ(1−α)x∫a(ψ(t)−ψ(x))−αf′(t)dt. |
In [18] the author has defined the ψ fractional partial integral with respect to another functions as
Definition 2.3 Let θ=(a,b) and α=(α1,α2) where 0≤α1,α2≤1. Also put I=[a,k]×[b,m] where a,b and k,m are positive constants. Also let ψ(.) be an increasing positive monotone function on (a,k]×(b,m] having continuous derivative ψ′(.) on (a,k]×(b,m]. Then the fractional partial integral is
Iα;ψθu(x,y)=1Γ(α1)Γ(α2)x∫ay∫bψ′(s)ψ′(t)(ψ(x)−ψ(s))α1−1(ψ(y)−ψ(t))α2−1f(s,t)dtds. |
The Caputo fractional partial derivative is defined as follows
Definition 2.4 Let θ=(a,b) and α=(α1,α2) where 0≤α1,α2≤1. Also put I=[a,k]×[b,m] where a,b and a,b are positive constants. Also let ψ(.) be an increasing function on (a,k]×(b,m] and ψ′(.)≠0 on (a,k]×(b,m]. The ψ Caputo fractional partial derivative of functions of two variables of order α is given by
CDα;ψθu(x,y)=I2−α;ψθ(1ψ′(s)ψ′(t)∂2α∂y∂x)u(x,y). |
We use the following notation:
CDα;ψθu(x,y)=∂2αψu∂ψyα∂ψxα(x,y). |
We define the norm for a function of two variables as follows
‖CDα;ψθf‖∞=sup|CDα;ψθf(x,y)|. |
Similarly as in Definition (2.3) and (2.4) we define the ψ fractional partial integral with respect to another functions and ψ Caputo fractional partial derivative of functions of three variables as follows:
Definition 2.5 Let Θ=(a,b,c) and α=(α1,α2,α3) where 0≤α1,α2,α3≤1. Also put I=[a,k]×[b,m]×[c,n] where a,b,c and k,m,n are positive constants. Also let ψ(.) be an increasing positive monotone function on (a,k]×(b,m]×[c,n] having continuous derivative ψ′(.) on (a,k]×(b,m]×(c,n].
Then the fractional partial integral is
Iα;ψΘu(x,y,z)=1Γ(α1)Γ(α2)x∫ay∫bz∫cψ′(s)ψ′(t)ψ′(r)×(ψ(x)−ψ(s))α1−1(ψ(y)−ψ(t))α2−1(ψ(z)−ψ(r))α3−1f(s,t,r)drdtds. |
Definition 2.6 Let θ=(a,b,c) and α=(α1,α2,α3) where 0≤α1,α2,α3≤1. Also put I=[a,k]×[b,m]×[c,n] where a,b,c and k,m,n are positive constants. Also let ψ(.) be an increasing function on (a,k]×(b,m]×(c,n] and ψ′(.)≠0 on (a,k]×(b,m]×(c,n]. The ψ Caputo fractional partial derivative of functions of three variables of order α is given by
CDα;ψΘu(x,y,z)=I3−α;ψΘ(1ψ′(s)ψ′(t)ψ′(r)∂3∂z∂y∂x)u(x,y,z). |
We use the following notation:
CDα;ψΘu(x,y,z)=∂3αψu∂ψzα∂ψyαxα(x,y,z). |
We define the norm for a function of three variables as follows
‖CDα;ψΘf‖∞=sup|CDα;ψΘf(x,y,z)|. |
Now we give the ψ Caputo fractional Čebyšev inequality involving functions of two variables as follows:
Theorem 3.1 Let f,g:[a,l]×[b,m]→R be a continuous function on [a,l]×[b,m] and ∂2αf∂ψyα∂ψxα, ∂2αg∂ψyα∂ψxα exists continuous and bounded on [a,l]×[b,m] and α=(α1,α2). Then
|l∫am∫b[f(x,y)g(x,y)−12[G(f(x,y))g(x,y)+G(g(x,y))f(x,y)]dydx]|≤18(ψ(l)−ψ(a))(ψ(m)−ψ(b))l∫am∫b[|g(x,y)|‖Dα;ψθf‖∞+g(x,y)‖Dα;ψθg‖∞]dydx, | (3.1) |
where
G(f(x,y))=12[f(a,y)+f(x,m)+f(x,b)+f(l,y)]−14[f(a,b)+f(a,m)+f(l,b)+f(l,m)] |
and
H(∂2αf∂ψyα∂ψxα(x,y))=1Γ(α1)Γ(α2)××[x∫ay∫bψ′(t)ψ′(s)(ψ(x)−ψ(t))α1−1(ψ(y)−ψ(s))α2−1∂2αf∂ψsα∂ψtα(t,s)dsdt−x∫am∫yψ′(t)ψ′(s)(ψ(x)−ψ(t))α1−1(ψ(m)−ψ(s))α2−1∂2αf∂ψsα∂ψtα(t,s)dsdt−l∫xy∫bψ′(t)ψ′(s)(ψ(l)−ψ(t))α1−1(ψ(y)−ψ(s))α2−1∂2αf∂ψsα∂ψtα(t,s)dsdt+l∫xm∫yψ′(t)ψ′(s)(ψ(l)−ψ(t))α1−1(ψ(m)−ψ(s))α2−1∂2αf∂ψsα∂ψtα(t,s)dsdt]. |
Proof. From the given hypotheses for (x,y)∈[a,l]×[b,m] we have
1Γ(α1)Γ(α2)x∫ay∫bψ′(t)ψ′(s)×(ψ(x)−ψ(t))α1−1(ψ(y)−ψ(s))α2−1∂2αf∂ψsα∂ψtα(t,s)dsdt=1Γ(α1)x∫aψ′(s)(ψ(x)−ψ(t))α1−1[∂αf∂ψsα(s,t)|yc]=1Γ(α1)x∫aψ′(s)(ψ(y)−ψ(t))α1−1[∂αf∂ψsα(t,y)−∂αf∂ψsα(t,b)]=f(t,y)|xa−f(t,b)|xa=f(x,y)−f(a,y)−f(x,b)+f(a,b). | (3.2) |
Similarly we have
1Γ(α1)Γ(α2)x∫am∫yψ′(t)ψ′(s)×(ψ(x)−ψ(t))α1−1(ψ(m)−ψ(s))α2−1∂2αf∂ψsα∂ψtα(t,s)dsdt=−f(x,y)−f(a,m)+f(x,m)+f(a,y), | (3.3) |
1Γ(α1)Γ(α2)l∫xy∫bψ′(t)ψ′(s)×(ψ(l)−ψ(t))α1−1(ψ(y)−ψ(s))α2−1∂2αf∂ψsα∂ψtα(t,s)dsdt=−f(x,y)−f(l,b)+f(x,b)+f(l,y), | (3.4) |
1Γ(α1)Γ(α2)l∫xm∫yψ′(t)ψ′(s)×(ψ(l)−ψ(t))α1−1(ψ(m)−ψ(s))α2−1∂2αf∂ψsα∂ψtα(s,t)dsdt=f(x,y)+f(l,b)−f(x,b)−f(l,y). | (3.5) |
Adding the above identities we have
4f(x,y)−2[f(a,y)+f(x,m)+f(x,b)+f(l,y)]+[f(a,b)+f(a,m)+f(l,b)+f(l,m)]=1Γ(α1)Γ(α2)[x∫ay∫bψ′(t)ψ′(s)(ψ(x)−ψ(t))α1−1(ψ(y)−ψ(s))α2−1∂2αf∂ψsα∂ψtα(t,s)dsdt−x∫ad∫yψ′(t)ψ′(s)(ψ(x)−ψ(t))α1−1(ψ(m)−ψ(s))α2−1∂2αf∂ψsα∂ψtα(t,s)dsdt−l∫xy∫bψ′(t)ψ′(s)(ψ(l)−ψ(t))α1−1(ψ(y)−ψ(s))α2−1∂2αf∂ψsα∂ψtα(t,s)dsdt+l∫xm∫yψ′(t)ψ′(s)(ψ(l)−ψ(t))α1−1(ψ(m)−ψ(s))α2−1∂2αf∂ψsα∂ψtα(t,s)dsdt]. | (3.6) |
From (3.6) we have
f(x,y)−G(f(x,y))=14H(∂2αf∂ψyα∂ψxα(x,y)), | (3.7) |
for (x,y)∈[a,l]×[b,m]. Similarly we have
g(x,y)−G(g(x,y))=14H(∂2αg∂ψyα∂ψxα(x,y)), | (3.8) |
for (x,y)∈[a,l]×[b,m].
Multiplying (3.7) by g(x,y), (3.8) by f(x,y) adding them and Integrating over (x,y)∈[a,l]×[b,m] we get
l∫am∫b[2f(x,y)g(x,y)−g(x,y)G(f(x,y))−f(x,y)G(g(x,y))]dydx=18l∫am∫b[H(∂2αf∂ψyα∂ψxα(x,y))g(x,y)+14f(x,y)H(∂2αg∂ψyα∂ψxα(x,y))]. | (3.9) |
From the properties of modulus we have
|H(∂2αf∂ψyα∂ψxα(x,y))|≤1Γ(α1)Γ(α2)l∫am∫bψ′(t)ψ′(s)(ψ(l)−ψ(t))α1−1(ψ(m)−ψ(s))α2−1|∂2αf∂ψsα∂ψtα(t,s)|dsdt≤(ψ(l)−ψ(a))α1(ψ(m)−ψ(b))α2‖cDα;ψθf‖∞, | (3.10) |
|H(∂2αg∂ψyα∂ψxα(x,y))|≤1Γ(α1)Γ(α2)l∫am∫bψ′(t)ψ′(s)(ψ(l)−ψ(t))α1−1(ψ(m)−ψ(s))α2−1|∂2αg∂ψsα∂ψtα(t,s)|dsdt≤(ψ(l)−ψ(a))α1(ψ(m)−ψ(b))α2‖cDα;ψθg‖∞. | (3.11) |
From (3.9), (3.10) and (3.11) we have
|l∫am∫b[f(x,y)g(x,y)−12[G(f(x,y))g(x,y)+G(g(x,y))f(x,y)]]dydx|≤18l∫am∫b[|H(∂2αf∂ψyα∂ψxα(x,y))||g(x,y)|+|H(∂2αg∂ψyα∂ψxα(x,y))||f(x,y)|]≤18l∫am∫b{|g(x,y)|[1Γ(α1)Γ(α2)×[l∫am∫bψ′(t)ψ′(s)(ψ(l)−ψ(t))α1−1(ψ(m)−ψ(s))α2−1|∂2αf∂ψsα∂ψtα(t,s)|dsdt]+|f(x,y)|×[l∫am∫bψ′(t)ψ′(s)(ψ(l)−ψ(t))α1−1(ψ(m)−ψ(s))α2−1|∂2αg∂ψsα∂ψtα(t,s)|dsdt]}dydx≤18(ψ(l)−ψ(a))α1(ψ(m)−ψ(b))α2×l∫am∫b[|g(x,y)|‖cDα;ψθf‖∞+|f(x,y)|‖cDα;ψθg‖∞]dydx, | (3.12) |
which is required inequality.
Theorem 3.2 Let f,g,G(f(x,y)),G(g(f(x,y)),∂2αf∂ψyα∂ψxα,∂2αg∂ψyα∂ψxα be as in Theorem 3.1 then
|l∫am∫b{f(x,y)g(x,y)−[G(f(x,y))g(x,y)+G(g(x,y))f(x,y)−G(f(x,y))G(g(x,y))]}dydx≤116{(ψ(l)−ψ(a))α1(ψ(m)−ψ(b))α2}2‖cDα;ψθf‖∞‖cDα;ψθg‖∞, | (3.13) |
for (x,y)∈[a,l]×[b,m].
Proof. Multiplying left hand side and right hand side of (3.7) and (3.8) we have
f(x,y)g(x,y)−[f(x,y)G(g(x,y))+g(x,y)G(f(x,y))]=116H(∂2αf∂ψyα∂ψxα(x,y))H(∂2αg∂ψyα∂ψxα(x,y)). | (3.14) |
Integrating (3.14) over [a,l]×[b,m] and from the properties of modulus we get
|l∫am∫b{f(x,y)g(x,y)−[G(g(x,y))f(x,y)+G(f(x,y))g(x,y)]−G(f(x,y))G(g(x,y))}dydx|≤116l∫am∫b|H(∂2αf∂ψyα∂ψxα(x,y))||H(∂2αg∂ψyα∂ψxα(x,y))|dydx. | (3.15) |
Now using (3.13),(3.14) in (3.19) we get required inequality (3.13).
Now in our result we give the ψ Caputo fractional Čebyšev inequality involving functions of three variables. We use some notations as follows:
A(p(u,v,w))=18[p(a,b,c)+p(k,m,n)]−14[p(u,b,c)+p(u,m,n)+p(u,m,c)+p(u,b,n)]−14[p(a,v,c)+p(k,v,n)+p(a,v,n)+p(k,v,c)]−14[p(a,b,w)+p(k,m,w)+p(k,b,w)+p(a,m,w)]+12[p(a,v,w)+p(k,v,w)]+12[p(u,b,w)+p(u,m,w)]+12[p(u,v,c)+p(u,v,n)] | (4.1) |
and
B(∂3αp∂ψwα∂ψvα∂ψuα(u,v,w))=1Γ(α1)Γ(α2)Γ(α3)u∫av∫bw∫cψ′(r)ψ′(s)ψ′(t)(ψ(u)−ψ(r))α1−1×(ψ(v)−ψ(s))α2−1(ψ(w)−ψ(t))α3−1∂3αp∂ψtα∂ψsα∂ψrα(r,s,t)dtdsdr−1Γ(α1)Γ(α2)Γ(α3)u∫av∫bn∫cψ′(r)ψ′(s)ψ′(t)(ψ(u)−ψ(r))α1−1×(ψ(v)−ψ(s))α2−1(ψ(n)−ψ(t))α3−1∂3αp∂ψtα∂ψsα∂ψrα(r,s,t)dtdsdr−1Γ(α1)Γ(α2)Γ(α3)u∫am∫vw∫cψ′(r)ψ′(s)ψ′(t)(ψ(u)−ψ(r))α1−1×(ψ(m)−ψ(s))α2−1(ψ(w)−ψ(t))α3−1∂3αp∂ψtα∂ψsα∂ψrα(r,s,t)dtdsdr−1Γ(α1)Γ(α2)Γ(α3)k∫uv∫bw∫cψ′(r)ψ′(s)ψ′(t)(ψ(k)−ψ(r))α1−1×(ψ(u)−ψ(s))α2−1(ψ(w)−ψ(t))α3−1∂3αp∂ψtα∂ψsα∂ψrα(r,s,t)dtdsdr+1Γ(α1)Γ(α2)Γ(α3)u∫am∫rn∫wψ′(r)ψ′(s)ψ′(t)(ψ(u)−ψ(r))α1−1×(ψ(m)−ψ(s))α2−1(ψ(n)−ψ(t))α3−1∂3αp∂ψtα∂ψsα∂ψrα(r,s,t)dtdsdr+1Γ(α1)Γ(α2)Γ(α3)k∫um∫vw∫cψ′(r)ψ′(s)ψ′(t)(ψ(k)−ψ(r))α1−1×(ψ(m)−ψ(s))α2−1(ψ(w)−ψ(t))α3−1∂3αp∂ψtα∂ψsα∂ψrα(r,s,t)dtdsdr+1Γ(α1)Γ(α2)Γ(α3)k∫uv∫bn∫wψ′(r)ψ′(s)ψ′(t)(ψ(k)−ψ(r))α1−1×(ψ(v)−ψ(s))α2−1(ψ(n)−ψ(t))α3−1∂3αp∂ψtα∂ψsα∂ψrα(r,s,t)dtdsdr−1Γ(α1)Γ(α2)Γ(α3)k∫um∫vn∫wψ′(r)ψ′(s)ψ′(t)(ψ(k)−ψ(r))α1−1×(ψ(m)−ψ(s))α2−1(ψ(n)−ψ(t))α3−1∂3αp∂ψtα∂ψsα∂ψrα(r,s,t)dtdsdr. | (4.2) |
Now we give our next result as
Theorem 4.1 Let f,g:[a,k]×[b,m]×[c,n]→R be a continuous function on [a,l]×[b,m] and ∂3αf∂ψtα∂ψsα∂ψrα, ∂3αg∂ψtα∂ψsα∂ψrα exists and continuous and bounded on [a,k]×[b,m]×[c,n]. Then
k∫am∫bn∫c[f(u,v,w)g(u,v,w)−12[f(u,v,w)A(g(u,v,w))+g(u,v,w)A(f(u,v,w))]]dwdvdu≤116(ψ(k)−ψ(a))α1(ψ(m)−ψ(b))α2(ψ(n)−ψ(c))α3×k∫am∫bn∫c[|g(u,v,w)|‖cDα;ψΘf‖∞+|f(u,v,w)|‖cDα;ψΘg‖∞]dwdvdu, | (4.3) |
where A,B are as given in (4.1),(4.2).
Proof. From the hypotheses we have for u,v,w∈[a,k]×[b,m]×[c,n]
1Γ(α1)Γ(α2)Γ(α3)u∫av∫bw∫cψ′(r)ψ′(s)ψ′(t)(ψ(u)−ψ(r))α1−1(ψ(v)−ψ(s))α2−1(ψ(w)−ψ(t))α3−1∂3αf∂ψtα∂ψsα∂ψrα(r,s,t)dtdsdr=1Γ(α1)Γ(α2)u∫av∫bψ′(r)ψ′(s)(ψ(u)−ψ(r))α1−1(ψ(v)−ψ(s))α2−1∂2αf∂ψsα∂ψrα(r,s,t)|wcdsdr=1Γ(α1)Γ(α2)u∫av∫bψ′(r)ψ′(s)(ψ(u)−ψ(r))α1−1(ψ(v)−ψ(s))α2−1∂2αf∂ψsα∂ψrα(r,s,w)dsdr−1Γ(α1)Γ(α2)u∫av∫bψ′(r)ψ′(s)(ψ(u)−ψ(r))α1−1(ψ(v)−ψ(s))α2−1∂2αf∂ψsα∂ψrα(r,s,c)dsdr=1Γ(α1)u∫aψ′(r)(ψ(u)−ψ(r))α1−1∂αf∂ψrα(r,s,w)|vbdr−1Γ(α1)u∫aψ′(r)(ψ(u)−ψ(r))α1−1∂αf∂ψrα(r,s,c)|vbdr=1Γ(α1)u∫aψ′(r)(ψ(u)−ψ(r))α1−1∂αf∂ψrα(r,v,w)dr−1Γ(α1)u∫aψ′(r)(ψ(u)−ψ(r))α1−1∂αf∂ψrα(r,b,w)dr−1Γ(α1)u∫aψ′(r)(ψ(u)−ψ(r))α1−1∂αf∂ψrα(r,v,c)dr+1Γ(α1)u∫aψ′(r)(ψ(u)−ψ(r))α1−1∂αf∂ψrα(r,b,c)dr=f(r,v,w)|ua−f(r,b,w)|ua−f(r,v,c)|ua+f(r,b,c)|ua=f(u,v,w)−f(a,v,w)−f(u,b,w)+f(a,b,w)−f(u,v,c)+f(a,v,c)+f(u,b,c)+f(a,b,c). |
Thus we have
f(u,v,w)=f(a,v,w)+f(u,b,w)−f(a,b,w)+f(u,v,c)−f(a,v,c)−f(u,b,c)−f(a,b,c)1Γ(α1)Γ(α2)Γ(α3)u∫av∫bw∫cψ′(r)ψ′(s)ψ′(t)(ψ(u)−ψ(r))α1−1(ψ(v)−ψ(s))α2−1(ψ(w)−ψ(t))α3−1∂3αf∂ψtα∂ψsα∂ψrα(r,s,t)dtdsdr, | (4.4) |
Similarly we have
f(u,v,w)=f(u,v,n)+f(a,v,w)+f(u,b,w)+f(a,b,n)−f(a,b,w)−f(a,v,n)−f(v,b,n)−1Γ(α1)Γ(α2)Γ(α3)u∫av∫bn∫wψ′(r)ψ′(s)ψ′(t)(ψ(u)−ψ(r))α1−1(ψ(v)−ψ(s))α2−1(ψ(n)−ψ(t))α3−1∂3αf∂ψtα∂ψsα∂ψrα(r,s,t)dtdsdr, | (4.5) |
f(u,v,w)=f(u,m,w)+f(u,v,c)+f(a,m,c)+f(a,v,w)−f(u,m,c)−f(a,m,w)−f(a,v,c)−1Γ(α1)Γ(α2)Γ(α3)u∫am∫vw∫cψ′(r)ψ′(s)ψ′(t)(ψ(u)−ψ(r))α1−1(ψ(m)−ψ(s))α2−1(ψ(w)−ψ(t))α3−1∂3αf∂ψtα∂ψsα∂ψrα(r,s,t)dtdsdr, | (4.6) |
f(u,v,w)=f(k,s,t)+f(k,b,c)+f(u,v,c)+f(u,b,w)−f(k,v,c)−f(k,b,w)−f(u,b,c)−1Γ(α1)Γ(α2)Γ(α3)k∫uv∫bw∫cψ′(r)ψ′(s)ψ′(t)(ψ(k)−ψ(r))α1−1(ψ(v)−ψ(s))α2−1(ψ(w)−ψ(t))α3−1∂3αf∂ψtα∂ψsα∂ψrα(r,s,t)dtdsdr, | (4.7) |
f(u,v,w)=f(u,m,w)+f(u,v,n)+f(a,m,n)+f(a,v,w)−f(u,m,n)−f(a,m,w)−f(a,v,n)+1Γ(α1)Γ(α2)Γ(α3)u∫am∫vn∫wψ′(r)ψ′(s)ψ′(t)(ψ(u)−ψ(r))α1−1(ψ(m)−ψ(s))α2−1(ψ(n)−ψ(t))α3−1∂3αf∂ψtα∂ψsα∂ψrα(r,s,t)dtdsdr, | (4.8) |
f(u,v,w)=f(r,m,t)+f(u,v,c)+f(k,s,t)+f(k,m,c)−f(k,m,w)−f(k,v,c)−f(u,m,c)+1Γ(α1)Γ(α2)Γ(α3)k∫um∫vw∫cψ′(r)ψ′(s)ψ′(t)(ψ(k)−ψ(r))α1−1(ψ(m)−ψ(s))α2−1(ψ(w)−ψ(t))α3−1∂3αf∂ψtα∂ψsα∂ψrα(r,s,t)dtdsdr, | (4.9) |
f(u,v,w)=f(k,v,w)+f(k,b,n)+f(u,v,n)+f(u,b,t)−f(k,v,n)−f(k,b,w)−f(u,b,n)+1Γ(α1)Γ(α2)Γ(α3)k∫uv∫bn∫wψ′(r)ψ′(s)ψ′(t)(ψ(k)−ψ(r))α1−1(ψ(v)−ψ(s))α2−1(ψ(n)−ψ(t))α3−1∂3αf∂ψtα∂ψsα∂ψrα(r,s,t)dtdsdr | (4.10) |
and
f(u,v,w)=f(k,m,n)+f(k,v,w)+f(u,m,w)+f(u,v,n)−f(k,m,w)−f(k,v,n)−f(u,m,n)+1Γ(α1)Γ(α2)Γ(α3)k∫um∫vn∫wψ′(r)ψ′(s)ψ′(t)(ψ(k)−ψ(r))α1−1(ψ(m)−ψ(s))α2−1(ψ(n)−ψ(t))α3−1∂3αf∂ψtα∂ψsα∂ψrα(r,s,t)dtdsdr. | (4.11) |
Adding the above identities we have
f(u,v,w)−A(f(u,v,w))=18B(∂3αf∂ψwα∂ψvα∂ψuα(u,v,w)), | (4.12) |
for (u,v,w)∈[a,k]×[b,m]×[c,n].
Similarly we have
g(u,v,w)−A(g(u,v,w))=18B(∂3αg∂ψwα∂ψvα∂ψuα(u,v,w)), | (4.13) |
for (u,v,w)∈[a,k]×[b,m]×[c,n].
Now multiplying (4.12) and (4.13) by g(u,v,w) and f(u,v,w) respectively, adding them and Integrating over [a,k]×[b,m]×[c,n] we have
k∫am∫bn∫c[f(u,v,w)g(u,v,w)−12[g(u,v,w)A(f(u,v,w))g(u,v,w)A(f(u,v,w))]]dwdvdu=116k∫am∫bn∫c[g(u,v,w)B(∂3αf∂ψwα∂ψvα∂ψuα(u,v,w))+f(u,v,w)B(∂3αg∂ψwα∂ψvα∂ψuα(u,v,w))]. | (4.14) |
From the properties of modulus we have
|B(∂3αf∂ψwα∂ψvα∂ψuα(u,v,w))|≤k∫am∫bn∫cψ′(r)ψ′(s)ψ′(t)(ψ(k)−ψ(r))α1−1(ψ(m)−ψ(s))α2−1×(ψ(n)−ψ(t))α3−1∂3αf∂ψtα∂ψsα∂ψrα(r,s,t)dtdsdr≤(ψ(k)−ψ(a))α1(ψ(m)−ψ(b))α2(ψ(n)−ψ(c))α3‖CDα;ψΘf‖∞, | (4.15) |
|B(∂3αg∂ψwα∂ψvα∂ψuα(u,v,w))|≤k∫am∫bn∫cψ′(r)ψ′(s)ψ′(t)(ψ(k)−ψ(r))α1−1(ψ(m)−ψ(s))α2−1×(ψ(n)−ψ(t))α3−1∂3αg∂ψtα∂ψsα∂ψrα(r,s,t)dtdsdr≤(ψ(k)−ψ(a))α1(ψ(m)−ψ(b))α2(ψ(n)−ψ(c))α3‖CDα;ψΘg‖∞. | (4.16) |
Now by substituting the values from equation (4.15) and (4.16) in (4.14) we get the required inequality (4.3).
Theorem 4.2 Let f,g, ∂3αf∂ψtα∂ψsα∂ψrα and ∂3αg∂ψtα∂ψsα∂ψrα be as in Theorem 4.1. Then
|k∫am∫bn∫c[f(u,v,w)g(u,v,w)−[A(f(u,v,w))g(u,v,w)A(g(u,v,w))f(u,v,w)−A(f(u,v,w))A(g(u,v,w))|dwdvdu≤164{(ψ(k)−ψ(a))α1(ψ(m)−ψ(b))α2(ψ(n)−ψ(c))α3}2‖CDα;ψΘf‖∞‖CDα;ψΘg‖∞, | (4.17) |
for (r,s,t)∈[a,k]×[b,m]×[c,n] and A,B are as given in (4.1),(4.2).
Proof. Multiplying left hand and right hand side of equation (4.12) and (4.13) we have
f(u,v,w)g(u,v,w)−[f(u,v,w)A(g(u,v,w))+g(u,v,w)A(f(u,v,w))−A(f(u,v,w))A(g(u,v,w))]=164B(∂3αf∂ψwα∂ψvα∂ψuα(u,v,w))B(∂3αg∂ψwα∂ψvα∂ψuα(u,v,w)). | (4.18) |
Integrating over [a,k]×[b,m]×[c,n] and from the properties of modulus we have
|k∫am∫bn∫c[f(u,v,w)g(u,v,w)−[f(u,v,w)A(g(u,v,w))+g(u,v,w)A(f(u,v,w))−A(f(u,v,w))A(g(u,v,w))]]|dwdvdu≤164k∫am∫bn∫c|B(∂3αf∂ψwα∂ψvα∂ψuα(u,v,w))B(∂3αf∂ψwα∂ψvα∂ψuα(u,v,w))|dwdvdu. | (4.19) |
Using (4.15) and (4.16) in (4.19) we get the required inequality (4.17).
Remark: If we put different values for ψ(x) as x,lnx,xσthen it reduces to various types of fractional Čebyšev inequalities such as Riemann Liouville fractional, Hadmard Fractional and Erdelyi-Kober fractional inequalities respectively.
In this paper, we studied Čebyšev like inequalities. We proved some new ψ Caputo fractional Čebyšev type inequalities involving functions of two and three variables.
All authors declare no conflict of interest in this paper.
[1] | World Health Organization (WHO). Coronavirus. Available from: https://www.who.int/health-topics/coronavirus (accessed on January 23, 2020). |
[2] | Wuhan Municipal Health Commission. Available from: http://wjw.wuhan.gov.cn/front/web/showDetail/2019123108989 (accessed on December 31, 2019). |
[3] | World Health Organization (WHO). Disease Outbreak News. Available from: https://www.who.int/csr/don/archive/disease/novelcoronavirus/en/ (accessed on January 14, 2020). |
[4] | World Health Organization (WHO). Situation reports. Available from: http://who.maps.arcgis.com/apps/opsdashboard/index.html#/c88e37cfc43b4ed3baf977d77e4a0667 (accessed on January 23, 2020). |
[5] | National Health Commission of the People's Republic of China. Available from: http://www.nhc.gov.cn/xcs/xxgzbd/gzbdindex.shtml (accessed on February 14, 2020). |
[6] |
Y. Zhou, Z. Ma, F. Brauer, A Discrete Epidemic Model for SARS Transmission and Control in China, Math. Comput. Model., 40 (2004), 1491-1506. doi: 10.1016/j.mcm.2005.01.007
![]() |
[7] |
G. Chowell, C. Castillo-Chavez, P. Fenimore, M. Christopher, C. Kribs-Zaleta, L. Arriola, et al., Model Parameters and Outbreak Control for SARS, Emerg. Infect. Dis., 10 (2004), 1258-1263. doi: 10.3201/eid1007.030647
![]() |
[8] |
P. Lekone, B. Finkenstädt, Statistical Inference in a Stochastic Epidemic SEIR Model with Control Intervention: Ebola as a Case Study, Biometrics, 62 (2006), 1170-1177. doi: 10.1111/j.1541-0420.2006.00609.x
![]() |
[9] | J. Wu, K. Leung, G. Leung, Nowcasting and forecasting the potential domestic and international spread of the 2019-nCoV outbreak originating in Wuhan, China: a modelling study, Lancet (2020). |
[10] |
S. Zhao, S. Musa, Q. Lin, J. Ran, G. Yang, W. Wang, et al., Estimating the unreported number of novel coronavirus (2019-nCoV) vases in China in the first half of January 2020: a data-driven modelling analysis of the early outbreak, J. Clin. Med., 9 (2020), 388. doi: 10.3390/jcm9020388
![]() |
[11] | B. Prasse, M. Achterberg, L. Ma, P. Mieghem, Network-Based Prediction of the 2019-nCoV Epidemic Outbreak in the Chinese Province Hubei, arXiv preprint arXiv (2002), 2002.04482. |
[12] | C. Anastassopoulou, L. Russo, A. Tsakris, C. Siettos, Data-Based Analysis, Modelling and Forecasting of the novel Coronavirus (2019-nCoV) outbreak, medRxiv (2020). |
[13] | Y. Yang, Q. Lu, M. Liu, Y. Wang, A. Zhang, N. Jalali, et al., Epidemiological and clinical features of the 2019 novel coronavirus outbreak in China, medRxiv (2020). |
[14] | C. You, Y. Deng, W. Hu, J. Sun, Q. Lin, F. Zhou, et al., Estimation of the Time-Varying Reproduction Number of COVID-19 Outbreak in China, medRxiv (2020). |
[15] | S. Hermanowicz, Forecasting the Wuhan coronavirus (2019-nCoV) epidemics using a simple (simplistic) model, medRxiv (2020). |
[16] | K. Mizumoto, K. Kagaga, G. Chowell, Early epidemiological assessment of the transmission potential and virulence of 2019 Novel Coronavirus in Wuhan City: China, 20192020. medRxiv (2020). |
[17] |
B. Tang, X. Wang, Q. Li, N. L. Bragazzi, S. Tang, Y. Xiao, et al., Estimation of the transmission risk of the 2019-nCoV and its implication for public health interventions, J. Clin. Med., 9 (2020), 462. doi: 10.3390/jcm9020462
![]() |
[18] |
B. Tang, N. Bragazzi, Q. Li, S. Tang, Y. Xiao, J. Wu, An updated estimation of the risk of transmission of the novel coronavirus (2019-nCov), Infect. Disease Model., 5 (2020), 248-255. doi: 10.1016/j.idm.2020.02.001
![]() |
[19] | Health Commission of Hubei Province. Available from: http://wjw.hubei.gov.cn/bmdt/ztzl/fkxxgzbdgr f yyq/ (accessed on February 15, 2020). |
[20] |
L. Bettencourt, R. Ribeiro, Real Time Bayesian Estimation of the Epidemic Potential of Emerging Infectious Diseases, PLoS One, 3 (2008), e2185. doi: 10.1371/journal.pone.0002185
![]() |
[21] |
A. Morton, B. Finkenstädt, Discrete time modelling of disease incidence time series by using Markov chain Monte Carlo methods, J. R. Stat. Soc., 54 (2005), 575-594. doi: 10.1111/j.1467-9876.2005.05366.x
![]() |
[22] |
S. Tang, Y. Xiao, Y. Yang, Y. zhou, J. Wu, Z. Ma, Community-based measures for mitigating the 2009 H1N1 pandemic in China, PLoS One, 5 (2010), e10911. doi: 10.1371/journal.pone.0010911
![]() |
[23] | World Health Organization (WHO). Available from: https://www.who.int/news-room/detail/23-01-2020-statement-on-the-meeting-of-the-international-health-regulations-(2005)-emergency-committee-regarding-the-outbreak-of-novel-coronavirus-(2019-ncov) (accessed on January 23, 2020). |
[24] |
G. Chowell, N. Hengartner, C. Castillo-Chavez, P. Fenimore, J. Hyman, The basic reproductive number of Ebola and the effects of public health measures: the cases of Congo and Uganda, J. Theor. Biol., 229 (2004), 119-126. doi: 10.1016/j.jtbi.2004.03.006
![]() |
[25] |
C. Favier, N. Degallier, M. Rosa-Freitas, J. Boulanger, J. R. Costa Lima, J. Luitgards-Moura, et al., Early determination of the reproductive number for vector-borne diseases: the case of dengue in Brazil, Trop. Med. Int. Health., 11 (2006), 332-340. doi: 10.1111/j.1365-3156.2006.01560.x
![]() |
[26] | C. Althaus, Estimating the Reproduction Number of Ebola Virus (EBOV) During the 2014 Outbreak in West Africa, PLoS Curr., 6 (2014). |
[27] |
G. Chowell, H. Nishiura, L. Bettencourt, Comparative estimation of the reproduction number for pandemic influenza from daily case notification data, J. R. Soc. Interface., 4 (2007), 155-166. doi: 10.1098/rsif.2006.0161
![]() |
[28] | S. Paine, G. Mercer, P. Kelly, D. Bandaranayake, M. Baker, W. Huang, et al., Transmissibility of 2009 pandemic influenza A(H1N1) in New Zealand: effective reproduction number and influence of age, ethnicity and importations, Euro. Surveill., 15 (2010), pii = 19591. |
[29] | S. Park, B. Bolker, D. Champredon, D. Earn, M. Li, J. Weitz, et al., Reconciling early-outbreak estimates of the basic reproductive number and its uncertainty: framework and applications to the novel coronavirus (2019-nCoV) outbreak, medRxiv (2020). |
[30] | A. Kucharski, T. Russell, C. Diamond, Y. Liu, J. Edmunds, S. Funk, et al., Early dynamics of transmission and control of COVID-19: a mathematical modelling study, medRxiv (2020). |
[31] | Y. Liu, A. Gayle, A. Wilder-Smith, J. Rocklöv, The reproductive number of COVID-19 is higher compared to SARS coronavirus, J. Travel. Med., (2020), 1-4. |
1. | Tamer Nabil, Ulam stabilities of nonlinear coupled system of fractional differential equations including generalized Caputo fractional derivative, 2021, 6, 2473-6988, 5088, 10.3934/math.2021301 | |
2. | MAYSAA AL-QURASHI, SAIMA RASHID, YELIZ KARACA, ZAKIA HAMMOUCH, DUMITRU BALEANU, YU-MING CHU, ACHIEVING MORE PRECISE BOUNDS BASED ON DOUBLE AND TRIPLE INTEGRAL AS PROPOSED BY GENERALIZED PROPORTIONAL FRACTIONAL OPERATORS IN THE HILFER SENSE, 2021, 29, 0218-348X, 2140027, 10.1142/S0218348X21400272 |