Finite difference method for transmission dynamics of Contagious Bovine Pleuropneumonia

Sait Kıkpınar; Mahmut Modanli; Ali Akgül; Fahd Jarad; Sait Kıkpınar; Mahmut Modanli; Ali Akgül; Fahd Jarad

doi:10.3934/math.2022574

AIMS Mathematics

2022, Volume 7, Issue 6: 10303-10314. doi: 10.3934/math.2022574

Previous Article Next Article

Research article

Finite difference method for transmission dynamics of Contagious Bovine Pleuropneumonia

1.
Harran University, Faculty of Veterinary Medicine, 63300, Sanliurfa/Turkey, ORCID NO: 0000-0001-9065-1904
2.
Harran University, Faculty of Sciences and Arts, Department of Mathematics, 63300, Sanliurfa/Turkey, ORCID NO: 0000-0002-7743-3512
3.
Siirt University, Art and Science Faculty, Department of Mathematics, 56100 Siirt, Turkey
4.
Department of Mathematics, Çankaya University, Etimesgut 06790, Ankara, Turkey
5.
Department of Mathematics, King Abdulaziz University, Jeddah, Saudi Arabia
6.
Department of Medical Research, China Medical University Hospital, China Medical University, Taichung, Taiwan

Received: 30 December 2021 Revised: 28 February 2022 Accepted: 10 March 2022 Published: 24 March 2022
MSC : 34A08; 65N06

In this study, the transmission dynamics of Contagious Bovine Pleuropneumonia (CBPP) by finite difference method are presented. This model is made up of sensitive, exposed, vaccinated, infectious, constantly infected, and treated compartments. The model is studied by the finite difference method. Firstly, the finite difference scheme is constructed. Then the stability estimates are proved for this model. As a result, several simulations are given for this model on the verge of antibiotic therapy. From these figures, the supposition that 50% of infectious cattle take antibiotic therapy or the date of infection decrease to 28 days, 50% of susceptible obtain vaccination within 73 days.

Keywords:

Citation: Sait Kıkpınar, Mahmut Modanli, Ali Akgül, Fahd Jarad. Finite difference method for transmission dynamics of Contagious Bovine Pleuropneumonia[J]. AIMS Mathematics, 2022, 7(6): 10303-10314. doi: 10.3934/math.2022574

Related Papers:

[1]	Din Prathumwan, Thipsuda Khonwai, Narisara Phoochalong, Inthira Chaiya, Kamonchat Trachoo . An improved approximate method for solving two-dimensional time-fractional-order Black-Scholes model: a finite difference approach. AIMS Mathematics, 2024, 9(7): 17205-17233. doi: 10.3934/math.2024836
[2]	Khalid K. Ali, K. R. Raslan, Amira Abd-Elall Ibrahim, Mohamed S. Mohamed . On study the fractional Caputo-Fabrizio integro differential equation including the fractional q-integral of the Riemann-Liouville type. AIMS Mathematics, 2023, 8(8): 18206-18222. doi: 10.3934/math.2023925
[3]	Tahir Khan, Fathalla A. Rihan, Muhammad Bilal Riaz, Mohamed Altanji, Abdullah A. Zaagan, Hijaz Ahmad . Stochastic epidemic model for the dynamics of novel coronavirus transmission. AIMS Mathematics, 2024, 9(5): 12433-12457. doi: 10.3934/math.2024608
[4]	Fawaz K. Alalhareth, Seham M. Al-Mekhlafi, Ahmed Boudaoui, Noura Laksaci, Mohammed H. Alharbi . Numerical treatment for a novel crossover mathematical model of the COVID-19 epidemic. AIMS Mathematics, 2024, 9(3): 5376-5393. doi: 10.3934/math.2024259
[5]	Majeed A. Yousif, Juan L. G. Guirao, Pshtiwan Othman Mohammed, Nejmeddine Chorfi, Dumitru Baleanu . A computational study of time-fractional gas dynamics models by means of conformable finite difference method. AIMS Mathematics, 2024, 9(7): 19843-19858. doi: 10.3934/math.2024969
[6]	Muhammad Asim Khan, Majid Khan Majahar Ali, and Saratha Sathasivam . High-order finite difference method for the two-dimensional variable-order fractional cable equation in complex systems and neuronal dynamics. AIMS Mathematics, 2025, 10(4): 8647-8672. doi: 10.3934/math.2025396
[7]	Parveen Kumar, Sunil Kumar, Badr Saad T Alkahtani, Sara S Alzaid . A mathematical model for simulating the spread of infectious disease using the Caputo-Fabrizio fractional-order operator. AIMS Mathematics, 2024, 9(11): 30864-30897. doi: 10.3934/math.20241490
[8]	Khalid K. Ali, Mohamed A. Abd El Salam, Mohamed S. Mohamed . Chebyshev fifth-kind series approximation for generalized space fractional partial differential equations. AIMS Mathematics, 2022, 7(5): 7759-7780. doi: 10.3934/math.2022436
[9]	Christelle Dleuna Nyoumbi, Antoine Tambue . A fitted finite volume method for stochastic optimal control problems in finance. AIMS Mathematics, 2021, 6(4): 3053-3079. doi: 10.3934/math.2021186
[10]	Ailing Zhu, Yixin Wang, Qiang Xu . A weak Galerkin finite element approximation of two-dimensional sub-diffusion equation with time-fractional derivative. AIMS Mathematics, 2020, 5(5): 4297-4310. doi: 10.3934/math.2020274

Abstract

1. Introduction

CBPP is a great deal of restriction to cattle augmentation in the vital arcadian territory of Africa (see ^[1,2,3]). In ^[3], mathematical modeling of the transmission dynamics of contagious bovine pleuropneumonia was uncovered aim profiles at a small extent for upgraded vaccines and diagnostic tests. Development of real-time diagnostic analysis specific for Mycoplasma mycoides subspecies mycoides small colony were worked in ^[4]. It brings about high morbidity and fatality rate damages to cattle which causes economic decline (see ^[5,6,7,8] for more information). They worked Contagious Bovine Pleuropneumonia: Challenges and Prospects Regarding Diagnosis and Control Strategies in Africa ^[9]. Charge of restrain of CBPP is a big issue in African regions as well ^[10]. In ^[11], the model was given with no interference, having the purpose of revealing data that have a crucial part in altering the dynamics of the illness.

In the study ^[12], the researchers have examined antibiotic treatment and vaccination as a controlling medium of CBPP and given a segmented model with six parts for the transmission dynamics of the CBPP: sensitive, exposed, vaccinated, infectious, often infected, and treated compartments. Antibiotic therapy was taken into consideration in the model by adding the recovery rate of treated cattle to ensure that the treated moved at a rate from the infectious compartment to cured compartment.

The goal of this study ^[12] was to set up a more efficient handling program out of vaccination, antibiotic care, or both of them. We consider ^[12]:

$\frac{dS\left(t\right)}{dt} = \mu N+\omega V-\frac{\beta SI}{N}-\rho S-\mu S$

(2.1)

$\frac{dV\left(t\right)}{dt} = \rho S-\omega V-\mu V,$

(2.2)

$\frac{dE\left(t\right)}{dt} = \frac{\beta SI}{N}-\gamma E-\mu E,$

(2.3)

$\frac{dI\left(t\right)}{dt} = \gamma E+kQ-\left({\alpha }_{t}+{\alpha }_{r}\right)I-{\alpha }_{q}I-\mu I,$

(2.4)

$\frac{dQ\left(t\right)}{dt} = {\alpha }_{q}I-kQ-\psi Q-\mu Q,$

(2.5)

$\frac{dR\left(t\right)}{dt} = \left({\alpha }_{t}+{\alpha }_{r}\right)I+\psi Q-\mu Q.$

(2.6)

This system has known well-posedness from ^[12]. The contagious bovine pleuropneumonia (CBPP) was a respiratory disease of cattle; CBPP was lead to by Mycoplasma mycoides subsp, mycoides small colony ^[17]. They gave and analyzed a mathematical model of the transmission dynamics of Contagious Bovine Pleuropneumonia (CBPP) in the presence of antibiotic treatment with limited medical supply ^[18]. The equilibrium solutions were studied in detail ^[2]. Using the symbols in this paper are given as:

$N$ : total number
$t$ : time
$S$ : susceptible class
$V$ : vaccinal immune class
$E$ : exposed compartment
$I$ : infectious compartment
$R$ : recovered compartment persistently infected

Local stability analysis and global durability analysis for illness free equipoise the (İFE) were studied in ^[13]. Examining this model, which is defined by the Caputo derivative, with the finite difference method and obtaining simulations for an approximate solution makes this study different from previous studies.

This paper is constructed as follows. In Section 2, the dynamics of CBPP for a mathematical model with antibiotic interventions and vaccination are demonstrated with Caputo derivative. In Section 3, finite difference method is constructed and the stability estimates of this model is presented. Numerical simulations have been demonstrated in Section 4. In Section 5, conclusion is proposed.

2. Mathematical model with Caputo derivative

2.1. The Caputo derivative

Definition 2.1: The definition of the Caputo derivative of $\alpha$ order is given as ^[6]:

${D}_{t}^{\alpha }f\left(t\right) = \frac{{\partial }^{\alpha }f\left(t\right)}{\partial {t}^{\alpha }} = \frac{1}{\mathrm{\Gamma }(\mathrm{n}-\alpha )}\underset{a}{\overset{t}{\int }}\frac{1}{{(t-p)}^{\alpha -n+1}}\frac{{\partial }^{\alpha }f\left(p\right)}{\partial {p}^{\alpha }}dp,$

where $n- 1 < \alpha < n$ and $n = \left[\alpha \right] + 1$ . The Caputo derivative has some advantages over the Riemann-Liouville derivative. First, the Caputo derivative is frequently used in the solution of fractional differential equations in the Laplace transform method. The Laplace transform of the Riemann-Liouville derivative requires boundary conditions involving the boundary values of the Riemann-Liouville fractional derivatives at the lower bound at $t = a$ . Although mathematically such problems are solvable, there is no physical interpretation of such conditions. On the other hand the Laplace transform of the Caputo derivative imposes boundary conditions involving integer-order derivatives at the lower point $t = a$ which usually are acceptable physical conditions. The second advantage is that the Caputo derivative of a constant is zero while the Riemann Liouville derivative is nonzero ^[14]. The fractional order partial differential equations were studied by many researchers ^[15,16].

2.2. The constructed finite difference method

In this part, we construct finite difference method for the model of the antibiotic treatment and vaccination as a controlling tool of CBPP and the transmission dynamics of CBPP. The fractional order differential equation model defined by Caputo derivative is given by the following system:

${_0^C}{D}_{t}^{\alpha }S\left(t\right) = \mu N+\omega V-\frac{\beta SI}{N}-\rho S-\mu S$

(2.7)

${_0^C}{D}_{t}^{\alpha }V\left(t\right) = \rho S-\omega V-\mu V,$

(2.8)

${_0^C}{D}_{t}^{\alpha }E\left(t\right) = \frac{\beta SI}{N}-\gamma E-\mu E,$

(2.9)

${_0^C}{D}_{t}^{\alpha }I\left(t\right) = \gamma E+kQ-\left({\alpha }_{t}+{\alpha }_{r}\right)I-{\alpha }_{q}I-\mu I,$

(2.10)

${_0^C}{D}_{t}^{\alpha }Q\left(t\right) = {\alpha }_{q}I-kQ-\psi Q-\mu Q,$

(2.11)

${_0^C}{D}_{t}^{\alpha }R\left(t\right) = \left({\alpha }_{t}+{\alpha }_{r}\right)I+\psi Q-\mu Q,$

(2.12)

with initial conditions

$S\left(0\right) = {S}_{0}, V\left(0\right) = {V}_{0}, E\left(0\right) = {E}_{0}, I\left(0\right) = {I}_{0}, Q\left(0\right) = {Q}_{0}, R\left(0\right) = {R}_{0} .$

3. Finite difference method and stability estimates for mathematical model

We present grids with uniform steps in the domain $[0, \mathrm{T}]$

${\mathrm{W}}^{\rm{ \mathsf{ τ} }} = \left\{{\mathrm{t}}_{\mathrm{n}}:{\mathrm{t}}_{\mathrm{n}} = \mathrm{n}{\rm{ \mathsf{ τ} }}, \mathrm{ }\mathrm{n} = \mathrm{0, 1}, \dots , \mathrm{M}\right\}, {\rm{ \mathsf{ τ} }} = \frac{\mathrm{T}}{\mathrm{M}} .$

We use the notation ${u}_{n} = \mathrm{u}\left({\mathrm{t}}_{\mathrm{n}}\right)$ for functions defined on the grid (or parts of this grid) ${\mathrm{W}}^{\rm{ \mathsf{ τ} }}.$

For the fractional Caputo derivative operator, difference scheme is known as ^[13]:

$\begin{array}{*{20}{c}} {{_0^C}DS\left({t}_{n}\right) = \frac{{\partial }^{\alpha }s\left({t}_{n}\right)}{\partial {t}^{\alpha }}\cong \frac{{\tau }^{-\alpha }}{\mathrm{\Gamma }\left(2-\alpha \right)}\sum _{j = 0}^{n}{w}_{j}^{\left(\alpha \right)}({u}_{n-j+1}-{u}_{n-j}) = \frac{{\tau }^{-\alpha }}{\mathrm{\Gamma }\left(2-\alpha \right)}[{S}_{n+1}-{S}_{n}+}\\ {\sum _{j = 1}^{n}{w}_{j}^{\left(\alpha \right)}({S}_{n-j+1}-{S}_{n-j}), } \end{array}$

(2.13)

here ${w}_{j}^{\left(\alpha \right)} = ({j+1)}^{1-\alpha }-({j)}^{1-\alpha }$ , $S\left({t}_{n}\right) = {S}_{n}, {t}_{n} = n\tau.$

Using the formula (2.13), we can obtain the finite difference method for the formulas (2.7)–(2.12)

$\frac{{\tau }^{-\alpha }}{\mathrm{\Gamma }\left(2-\alpha \right)}\left[{S}_{n+1}-{S}_{n}+\sum _{j = 1}^{n}{w}_{j}^{\left(\alpha \right)}({S}_{n-j+1}-{S}_{n-j})\right] = \mu N+\omega {V}_{n}-\frac{\beta {S}_{n}{I}_{n}}{N}-\rho {S}_{n}-\mu {S}_{n},$

(2.14)

$\frac{{\tau }^{-\alpha }}{\mathrm{\Gamma }\left(2-\alpha \right)}\left[{V}_{n+1}-{V}_{n}+\sum _{j = 1}^{k}{w}_{j}^{\left(\alpha \right)}({V}_{n-j+1}-{V}_{n-j})\right] = \rho {S}_{n}-\omega {V}_{n}-\mu {V}_{n},$

(2.15)

$\frac{{\tau }^{-\alpha }}{\mathrm{\Gamma }\left(2-\alpha \right)}\left[{E}_{n+1}-{E}_{n}+\sum _{j = 1}^{n}{w}_{j}^{\left(\alpha \right)}({E}_{n-j+1}-{E}_{n-j})\right] = \frac{\beta {S}_{n}{I}_{n}}{N}-\gamma {E}_{n}-\mu {E}_{n},$

(2.16)

$\frac{{\tau }^{-\alpha }}{\mathrm{\Gamma }\left(2-\alpha \right)}\left[{I}_{n+1}-{I}_{n}+\sum _{j = 1}^{n}{w}_{j}^{\left(\alpha \right)}({I}_{n-j+1}-{I}_{n-j})\right] = \gamma {E}_{n}+k{Q}_{n}-\left({\alpha }_{t}+{\alpha }_{r}\right){I}_{n}-{\alpha }_{q}{I}_{n}-\mu {I}_{n, }$

(2.17)

$\frac{{\tau }^{-\alpha }}{\mathrm{\Gamma }\left(2-\alpha \right)}\left[{Q}_{n+1}-{Q}_{n}+\sum _{j = 1}^{n}{w}_{j}^{\left(\alpha \right)}({Q}_{n-j+1}-{Q}_{n-j})\right] = {\alpha }_{q}{I}_{n}-k{Q}_{n}-\psi {Q}_{n}-\mu {Q}_{n, }$

(2.18)

$\frac{{\tau }^{-\alpha }}{\mathrm{\Gamma }\left(2-\alpha \right)}\left[{R}_{n+1}-{R}_{n}+\sum _{j = 1}^{n}{w}_{j}^{\left(\alpha \right)}({R}_{n-j+1}-{R}_{n-j})\right] = \left({\alpha }_{t}+{\alpha }_{r}\right){I}_{n}+\psi {Q}_{n}-\mu {Q}_{n}.$

(2.19)

Now, we shall prove that these systems are satisfied the stability estimates. For this, the Von-Neuman analysis method will be used as follow:

${S}_{n} = {V}_{n} = {E}_{n} = {I}_{n} = {Q}_{n} = {R}_{n} = {r}^{n}.$

(2.20)

Taking $\alpha \to 1, n = 1,$ the formulas (2.14)–(2.19) can be written as:

$\left(\frac{1}{\tau }+\frac{\beta }{N}\right){r}^{2}+\left(\rho +\mu -\frac{1}{\tau }-\omega \right)r-\mu N = 0,$

(2.21)

$\frac{1}{\tau }{r}^{2}+\left(\omega +\mu -\frac{1}{\tau }-\rho \right)r = 0,$

(2.22)

$\left(\frac{1}{\tau }-\frac{\beta }{N}\right){r}^{2}+\left(\gamma +\mu -\frac{1}{\tau }\right)r = 0,$

(2.23)

$\left(\frac{1}{\tau }\right){r}^{2}+\left({\alpha }_{t}+{\alpha }_{r}+{\alpha }_{q}+\mu -k-\gamma -\frac{1}{\tau }\right)r = 0,$

(2.24)

$\left(\frac{1}{\tau }\right){r}^{2}+\left(\mu +\psi +k-{\alpha }_{q}-\frac{1}{\tau }\right)r = 0,$

(2.25)

$\left(\frac{1}{\tau }\right){r}^{2}+(\mu -\psi -{(\alpha }_{t}+{\alpha }_{r})-\frac{1}{\tau })r = 0.$

(2.26)

These formulas are quadratic equations. For the stability estimates the following conditions have to be satisfied:

ⅰ) $a)\omega < \gamma +\mu +\frac{\beta }{N}, b)\frac{1}{\tau }+\frac{\beta }{N} > -\mu N,$

ⅱ) $\rho < \omega +\mu,$

ⅲ) $\beta < N\left(\gamma +\mu \right),$

ⅳ) ${k+\gamma < \alpha }_{t}+{\alpha }_{r}+{\alpha }_{q}+\mu,$

ⅴ) ${\alpha }_{q} < \mu +\psi +k,$

ⅵ) $\psi +{\alpha }_{t}+{\alpha }_{r} < \mu$ .

From the Von-Neumann analysis method, it can be seen that the (2.14)–(2.19) system is stable if the conditions (ⅰ)–(ⅵ) are satisfied. Because the roots of the quadratic equation satisfying the system are $0$ or less than 1.

Table 1. Explanations of the data.

Variables	Definitions	Baseline	references
$\omega$	vaccinal immunity loss rate	$\frac{1}{3\times 365}$	0.00078–0.0011 ^[11]
${P}_{e}$	Vaccination success rate	0.65	0.5–0.8 ^[11]
${P}_{v}$	Vaccination rate	0.5	^[11]
$\in$	Vaccination efficacy	0.8	^[11]
$p$	Ratio of immunization	${P}_{e}\times {P}_{v}\times \in$	^[11]
$\beta$	Rate of contact efficiency	0.126	0.07–0.13 ^[10]
$\rho$	Ratio of vaccination	$\frac{p}{73}$	assumed
$\gamma$	Transition rate from exposed to contagious compartment	0.0238	0.0179–0.0357 ^[11]
${a}_{r}$	Natural recuperation rate of contagious cattle	0.0045	0.0060–0.0036 ^[11]
${a}_{q}$	Rate of sequestrum formation of contagious cattle	${3a}_{r}$	^[11]
${a}_{t}$	Rate of recovery of treated cattle	0.0179	0.0119–0.0214 assumed
$k$	Rate of sequestrum re-initiate	0.00009	0.00007–0.00011 ^[11]
$\psi$	Rate of sequestrum resolution	0.0075	0.0068 to 0.0079 ^[11]
$\mu$	death rate	$\frac{1}{5\times 365}$	$\frac{1}{6\times 365}$ to $\frac{1}{20\times 365}$ ^[11]
B	Birth rate	$\frac{1}{5\times 365}$	$\frac{1}{6\times 365}$ to $\frac{1}{20\times 365}$ ^[11] and estimated

| Show Table

DownLoad: CSV

4. Numerical simulations

Firstly, we investigated 500 bovine populations consisting of an infectious cattle and 499 susceptible cattle with individual animals as epidemiological units. Consistent with the conclusion of ^[7], we suggested that the best way to control the disease is vaccination with antibiotic therapy. Because the proportion to be vaccinated ${p}_{v}$ and $t$ are not dependent variables of $\rho$ , a given value of $\rho$ can have many practical interpretation. Therefore, practical application of the value of $\rho$ can be adjusted based on cost of control, availability, and time value.

Numerical simulations are obtained using MATLAB in Figures 1–7.

Figure 1. Using the supposition that 50% of infectious cattle take antibiotic therapy or the interval of infection is decreased to 28 days (

${a}_{t}$ = 1/28−1/56), 50% of credulous obtain vaccination within 73 days (𝜌 = (0.5×0.8×0.65)/73),

${I}_{0} = 1$ ,

${S}_{0} = 499$ and

${V}_{0} = {E}_{0} = {Q}_{0} = {R}_{0} = 0$ .

DownLoad: Full-Size Img PowerPoint

Figure 2. Using the supposition that 80% of sensitive cattle are vaccinated within 49 days (𝜌 = (0.65 ∗ 0.8 ∗ 0.8)/49) with no treating infectious cattle (

${a}_{t} = 0$ ),

${I}_{0} = 1$ ,

${S}_{0} = 499$ and

${V}_{0} = {E}_{0} = {Q}_{0} = {R}_{0} = 0$ .

DownLoad: Full-Size Img PowerPoint

Figure 3. Using the Table 1 with the supposition that 85.7% of infectious cattle give antibiotic treatment within 8 days (

${a}_{t} =$ 1/8−1/56) with no vaccinating healthy cattle (

$\rho = 0$ ),

${I}_{0} = 1$ ,

${S}_{0} = 499$ and

${V}_{0} = {E}_{0} = {Q}_{0} = {R}_{0} = 0$ .

DownLoad: Full-Size Img PowerPoint

Figure 4. Using the Table 1 with

$\rho = {a}_{t} = 0$ ,

${I}_{0} = 1$ ,

${S}_{0} = 499$ and

${V}_{0} = {E}_{0} = {Q}_{0} = {R}_{0} = 0$ .

DownLoad: Full-Size Img PowerPoint

Figure 5. Using the Table 1 with

${a}_{r}$ = 1/56 (as in ^[14]) and

${I}_{0} = 1$ ,

${S}_{0} = 499$ and

${V}_{0} = {E}_{0} = {Q}_{0} = {R}_{0} = 0$ .

DownLoad: Full-Size Img PowerPoint

Figure 6. Using the Table 1 with

$\alpha = 0.01$

${a}_{r}$ = 1/56,

${a}_{t} = 0.1049$ and

${S}_{0} = 499$ and

${V}_{0} = {E}_{0} = 0$ .

DownLoad: Full-Size Img PowerPoint

Figure 7. Using the Table 1 with

$\alpha = 0.50$

${a}_{r}$ = 1/56,

${a}_{t} = 0.1049$ and

${S}_{0} = 499$ and

${V}_{0} = {E}_{0} = 0$ .

DownLoad: Full-Size Img PowerPoint

We did not treat any of the infected cattle in the 49-day period. We can control by vaccinating 80% of susceptible cattle in see Figure 2.

On the condition that the other values in and are the same, the values of $\alpha = 0.01$ and $\alpha = 0.50$ are compared.

Finally, for the parametric values in Table 1, assuming 50% of susceptible people are vaccinated a period of 73 days and 50% of infected cattle appear to be cured.

5. Conclusions

The simulations obtained using the Matlab program and the results obtained from these simulations were given in the main text. In this paper, we presented differential equations defined by Caputo derivative for the transmission aspects of CBPP with intercession. We constructed finite difference scheme for this equation. The stability estimates are proved for this difference method. Consequently, the verge of the antibiotic therapy is ${\bf{a}}_{\bf{t}} {\bf{= 0.1049}}.$ The values of $\mathit{\boldsymbol{\alpha }}{{\bf = 0.01}}$ and $\mathit{\boldsymbol{\alpha }}{\bf = 0.50}$ are compared and showed by Figures 6 and 7. This fractional order model defined by the Atangana-Baleanu derivative can be compared with the Caputo derivative by applying the finite difference method.

Conflict of interest

There is no conflict of interest declared by the authors.

References

[1]	A. A. Aligaz, J. M. W. Munganga, Analysis of a mathematical model of the dynamics of contagious bovine pleuropneumonia, Texts Biomath., 1 (2017), 64–80. https://doi.org/10.11145/texts.2017.12.253 doi: 10.11145/texts.2017.12.253
[2]	A. A. Aligaz, J. M. W. Munganga, Mathematical modelling of the transmission dynamics of contagious bovine pleuropneumonia with vaccination and antibiotic treatment, J. Appl. Math., 2019. https://doi.org/10.1155/2019/2490313 doi: 10.1155/2019/2490313
[3]	A. Ssematimba, J. Jores, J. C. Mariner, Mathematical modelling of the transmission dynamics of contagious bovine pleuropneumonia reveals minimal target profiles for improved vaccines and diagnostic assays, PLoS One, 10 (2015), e0116730. https://doi.org/10.1371/journal.pone.0116730 doi: 10.1371/journal.pone.0116730
[4]	N. B. Alhaji, P. I. Ankeli, L. T. Ikpa, O. O. Babalobi, Contagious Bovine Pleuropneumonia: Challenges and prospects regarding diagnosis and control strategies in Africa, Vet. Med.: Res. Rep., 11 (2020), 71. https://doi.org/10.2147/VMRR.S180025 doi: 10.2147/VMRR.S180025
[5]	E. M. Vilei, J. Frey, Detection of Mycoplasma mycoides subsp. mycoides SC in bronchoalveolar lavage fluids of cows based on a TaqMan real-time PCR discriminating wild type strains from an lppQ-mutant vaccine strain used for DIVA-strategies, J. Microbiol. Meth., 81 (2010), 211–218. https://doi.org/10.1016/j.mimet.2010.03.025 doi: 10.1016/j.mimet.2010.03.025
[6]	I. Podlubny, Fractional differential equations: An introduction to fractional derivatives, fractional differential equations, to methods of their solution and some of their applications. Elsevier, 1998.
[7]	J. C. Mariner, J. McDermott, J. A. P. Heesterbeek, G. Thomson, P. L. Roeder, S. W. Martin, A heterogeneous population model for contagious bovine pleuropneumonia transmission and control in pastoral communities of East Africa, Prev. Vet. Med., 73 (2006), 75–91. https://doi.org/10.1016/j.prevetmed.2005.09.002 doi: 10.1016/j.prevetmed.2005.09.002
[8]	J. Jores, J. C. Mariner, J. Naessens, Development of an improved vaccine for contagious bovine pleuropneumonia: An African perspective on challenges and proposed actions, Vet. Res., 44 (2013), 1–5. https://doi.org/10.1186/1297-9716-44-122 doi: 10.1186/1297-9716-44-122
[9]	J. O. Onono, B. Wieland, J. Rushton, Estimation of impact of contagious bovine pleuropneumonia on pastoralists in Kenya, Pre. Vet. Med., 115 (2014), 122–129. https://doi.org/10.1016/j.prevetmed.2014.03.022 doi: 10.1016/j.prevetmed.2014.03.022
[10]	M. J. Otte, R. Nugent, A. McLeod, Transboundary animal diseases: Assessment of socio-economic impacts and institutional responses, Rome, Italy: Food Agriculture Organization, (2004), 119–126.
[11]	N. Abdela, N. Yune, Seroprevalence and distribution of contagious bovine pleuropneumonia in Ethiopia: update and critical analysis of 20 years (1996–2016) reports, Front. Vet. Sci., 4 (2017), 100. https://doi.org/10.3389/fvets.2017.00100 doi: 10.3389/fvets.2017.00100
[12]	N. E. Tambi, W. O. Maina, C. Ndi, An estimation of the economic impact of contagious bovine pleuropneumonia in Africa, Rev. Sci. Tech. OIE., 25 (2006), 999–1011. https://doi.org/10.20506/rst.25.3.1710 doi: 10.20506/rst.25.3.1710
[13]	T. S. Gorton, M. M. Barnett, T. Gull, R. A. French, Z. Lu, G. F. Kutish, et al., Development of real-time diagnostic assays specific for Mycoplasma mycoides subspecies mycoides Small Colony, Vet. Microbiol., 111 (2005), 51–58. https://doi.org/10.1016/j.vetmic.2005.09.013 doi: 10.1016/j.vetmic.2005.09.013
[14]	E. Sousa, How to approximate the fractional derivative of order 1 < α≤ 2, Int. J. Bifurcat. Chaos, 22 (2012), 1250075. https://doi.org/10.1142/S0218127412500757 doi: 10.1142/S0218127412500757
[15]	Q. M. Al-Mdallal, M. A. Hajji, T. Abdeljawad, On the iterative methods for solving fractional initial value problems: New perspective, J. Fractional Calculus Nonlinear Syst, 2 (2021), 76–81. https://doi.org/10.48185/jfcns.v2i1.297 doi: 10.48185/jfcns.v2i1.297
[16]	A. Khan, H. M. Alshehri, T. Abdeljawad, Q. M. Al-Mdallal, H. Khan, Stability analysis of fractional nabla difference COVID-19 model, Results Phys., 22 (2021), 103888. https://doi.org/10.1016/j.rinp.2021.103888 doi: 10.1016/j.rinp.2021.103888
[17]	M. Lesnoff, G. Laval, P. Bonnet, K. Chalvet-Monfray, R. Lancelot, F. Thiaucourt, A mathematical model of the effects of chronic carriers on the within-herd spread of contagious bovine pleuropneumonia in an African mixed crop–livestock system, Pre. Vet. Med., 62 (2004), 101–117. https://doi.org/10.1016/j.prevetmed.2003.11.009 doi: 10.1016/j.prevetmed.2003.11.009
[18]	A. A. Aligaz, J. M. Munganga, Modelling the transmission dynamics of Contagious Bovine Pleuropneumonia in the presence of antibiotic treatment with limited medical supply, Math. Model. Anal., 26 (2021), 1–20. https://doi.org/10.3846/mma.2021.11795 doi: 10.3846/mma.2021.11795

This article has been cited by:

Syed Ahmed Pasha, Yasir Nawaz, Muhammad Shoaib Arif, On the nonstandard finite difference method for reaction–diffusion models, 2023, 166, 09600779, 112929, 10.1016/j.chaos.2022.112929

Reader Comments

Your name:*

Email:*
© 2022 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)

通讯作者: 陈斌, bchen63@163.com

1.
沈阳化工大学材料科学与工程学院沈阳 110142

AIMS Mathematics

1.8 3.1

Metrics

Article views(1668) PDF downloads(82) Cited by(1)

Preview PDF

Download XML

Export Citation

Article outline

Show full outline

Figures and Tables

Figures(7) / Tables(1)

AIMS Mathematics

Finite difference method for transmission dynamics of Contagious Bovine Pleuropneumonia

Related Papers:

Abstract

1. Introduction

2. Mathematical model with Caputo derivative

2.1. The Caputo derivative

2.2. The constructed finite difference method

3. Finite difference method and stability estimates for mathematical model

4. Numerical simulations

5. Conclusions

Conflict of interest

References

This article has been cited by:

Reader Comments

通讯作者: 陈斌, bchen63@163.com

Metrics

Figures and Tables

Other Articles By Authors

Catalog

AIMS Mathematics

Finite difference method for transmission dynamics of Contagious Bovine Pleuropneumonia

Related Papers:

Abstract

1. Introduction

2. Mathematical model with Caputo derivative

2.1. The Caputo derivative

2.2. The constructed finite difference method

3. Finite difference method and stability estimates for mathematical model

4. Numerical simulations

5. Conclusions

Conflict of interest

References

This article has been cited by:

Reader Comments

通讯作者: 陈斌, bchen63@163.com

Metrics

Figures and Tables

Other Articles By Authors

Related pages

Tools

Export File

Citation

Format

Content

Catalog