
Monkeypox (MPX) is a global public health concern. This infectious disease affects people all over the world, not just those in West and Central Africa. Various approaches have been used to study epidemiology, the source of infection, and patterns of transmission of MPX. In this article, we analyze the dynamics of MPX using a fractional mathematical model with a power law kernel. The human-to-animal transmission is considered in the model formulation. The fractional model is further reformulated via a generalized fractal-fractional differential operator in the Caputo sense. The basic mathematical including the existence and uniqueness of both fractional and fractal-fractional problems are provided using fixed points theorems. A numerical scheme for the proposed model is obtained using an efficient iterative method. Moreover, detailed simulation results are shown for different fractional orders in the first stage. Finally, a number of graphical results of fractal-fractional MPX transmission models are presented showing the combined effect of fractal and fractional orders on model dynamics. The resulting simulations conclude that the new fractal-fractional operator added more biological insight into the dynamics of illness.
Citation: Alia M. Alzubaidi, Hakeem A. Othman, Saif Ullah, Nisar Ahmad, Mohammad Mahtab Alam. Analysis of Monkeypox viral infection with human to animal transmission via a fractional and Fractal-fractional operators with power law kernel[J]. Mathematical Biosciences and Engineering, 2023, 20(4): 6666-6690. doi: 10.3934/mbe.2023287
[1] | Rong Chen, Shihang Pan, Baoshuai Zhang . Global conservative solutions for a modified periodic coupled Camassa-Holm system. Electronic Research Archive, 2021, 29(1): 1691-1708. doi: 10.3934/era.2020087 |
[2] | Li Yang, Chunlai Mu, Shouming Zhou, Xinyu Tu . The global conservative solutions for the generalized camassa-holm equation. Electronic Research Archive, 2019, 27(0): 37-67. doi: 10.3934/era.2019009 |
[3] | Cheng He, Changzheng Qu . Global weak solutions for the two-component Novikov equation. Electronic Research Archive, 2020, 28(4): 1545-1562. doi: 10.3934/era.2020081 |
[4] | Xiaochen Mao, Weijie Ding, Xiangyu Zhou, Song Wang, Xingyong Li . Complexity in time-delay networks of multiple interacting neural groups. Electronic Research Archive, 2021, 29(5): 2973-2985. doi: 10.3934/era.2021022 |
[5] | Shuang Wang, Chunlian Liu . Multiplicity of periodic solutions for weakly coupled parametrized systems with singularities. Electronic Research Archive, 2023, 31(6): 3594-3608. doi: 10.3934/era.2023182 |
[6] | Linlin Tan, Bianru Cheng . Global well-posedness of 2D incompressible Navier–Stokes–Darcy flow in a type of generalized time-dependent porosity media. Electronic Research Archive, 2024, 32(10): 5649-5681. doi: 10.3934/era.2024262 |
[7] | Janarthanan Ramadoss, Asma Alharbi, Karthikeyan Rajagopal, Salah Boulaaras . A fractional-order discrete memristor neuron model: Nodal and network dynamics. Electronic Research Archive, 2022, 30(11): 3977-3992. doi: 10.3934/era.2022202 |
[8] | Liju Yu, Jingjun Zhang . Global solution to the complex short pulse equation. Electronic Research Archive, 2024, 32(8): 4809-4827. doi: 10.3934/era.2024220 |
[9] | Zhili Zhang, Aying Wan, Hongyan Lin . Spatiotemporal patterns and multiple bifurcations of a reaction- diffusion model for hair follicle spacing. Electronic Research Archive, 2023, 31(4): 1922-1947. doi: 10.3934/era.2023099 |
[10] | Shiyong Zhang, Qiongfen Zhang . Normalized solution for a kind of coupled Kirchhoff systems. Electronic Research Archive, 2025, 33(2): 600-612. doi: 10.3934/era.2025028 |
Monkeypox (MPX) is a global public health concern. This infectious disease affects people all over the world, not just those in West and Central Africa. Various approaches have been used to study epidemiology, the source of infection, and patterns of transmission of MPX. In this article, we analyze the dynamics of MPX using a fractional mathematical model with a power law kernel. The human-to-animal transmission is considered in the model formulation. The fractional model is further reformulated via a generalized fractal-fractional differential operator in the Caputo sense. The basic mathematical including the existence and uniqueness of both fractional and fractal-fractional problems are provided using fixed points theorems. A numerical scheme for the proposed model is obtained using an efficient iterative method. Moreover, detailed simulation results are shown for different fractional orders in the first stage. Finally, a number of graphical results of fractal-fractional MPX transmission models are presented showing the combined effect of fractal and fractional orders on model dynamics. The resulting simulations conclude that the new fractal-fractional operator added more biological insight into the dynamics of illness.
Digital topology with interesting applications has been a popular topic in computer science and mathematics for several decades. Many researchers such as Rosenfeld [21,22], Kong [18,17], Kopperman [19], Boxer, Herman [14], Kovalevsky [20], Bertrand and Malgouyres would like to obtain some information about digital objects using topology and algebraic topology.
The first study in this area was done by Rosenfeld [21] at the end of 1970s. He introduced the concept of continuity of a function from a digital image to another digital image. Later Boxer [1] presents a continuous function, a retraction, and a homotopy from the digital viewpoint. Boxer et al. [7] calculate the simplicial homology groups of some special digital surfaces and compute their Euler characteristics.
Ege and Karaca [9] introduce the universal coefficient theorem and the Eilenberg-Steenrod axioms for digital simplicial homology groups. They also obtain some results on the Künneth formula and the Hurewicz theorem in digital images. Ege and Karaca [10] investigate the digital simplicial cohomology groups and especially define the cup product. For other significant studies, see [13,12,16].
Karaca and Cinar [15] construct the digital singular cohomology groups of the digital images equipped with Khalimsky topology. Then they examine the Eilenberg- Steenrod axioms, the universal coefficient theorem, and the Künneth formula for a cohomology theory. They also introduce a cup product and give general properties of this new operation. Cinar and Karaca [8] calculate the digital homology groups of various digital surfaces and give some results related to Euler characteristics for some digital connected surfaces.
This paper is organized as follows: First, some information about the digital topology is given in the section of preliminaries. In the next section, we define the smash product for digital images. Then, we show that this product has some properties such as associativity, distributivity, and commutativity. Finally, we investigate a suspension and a cone for any digital image and give some examples.
Let
and
A
[x,y]Z={a∈Z | x≤a≤y,x,y∈Z}, |
where
In a digital image
A function
Definition 2.1. [2]
Suppose that
F:X×[0,m]Z→Y |
with the following conditions, then
(ⅰ) For all
(ⅱ) For all
(ⅲ) For all
A digital image
A
g∘f≃(κ,κ)1X and f∘g≃(λ,λ)1Y |
where
For the cartesian product of two digital images
Definition 2.2. [3]
A
Theorem 2.3. [5] For a continuous surjection
The wedge of two digital images
Theorem 2.4. [5] Two continuous surjections
f:(A,α)→(C,γ) and g:(B,β)→(D,δ) |
are shy maps if and only if
Sphere-like digital images is defined as follows [4]:
Sn=[−1,1]n+1Z∖{0n+1}⊂Zn+1, |
where
S0={c0=(1,0),c1=(−1,0)}, |
S1={c0=(1,0),c1=(1,1),c2=(0,1),c3=(−1,1),c4=(−1,0),c5=(−1,−1), |
c6=(0,−1),c7=(1,−1)}. |
In this section, we define the digital smash product which has some important relations with a digital homotopy theory.
Definition 3.1. Let
Before giving some properties of the digital smash product, we prove some theorems which will be used later.
Theorem 3.2.
Let
∏a∈Afa≃(κn,λn)∏a∈Aga, |
where
Proof. Let
F:(∏a∈AXa)×[0,m]Z→∏a∈AYa |
defined by
F((xa),t)=(Fa(xa,t)) |
is a digital continuous function, where
Theorem 3.3. If each
Proof. Let
(∏a∈Aga)(∏a∈Afa)=∏a∈A(ga×fa)≃(λn,κn)∏a∈A(1Xa)=1∏a∈AXa, |
(∏a∈Afa)(∏a∈Aga)=∏a∈A(fa×ga)≃(κn,λn)∏a∈A(1Ya)=1∏a∈AYa. |
So we conclude that
Theorem 3.4.
Let
p×1:(X×Z,k∗(κ×σ))→(Y×Z,k∗(λ×σ)) |
is a
Proof. Since
(p×1Z)−1(y,z)=(p−1(y),1−1Z(z))=(p−1(y),z). |
Thus, for each
(p×1Z)−1({y0,y1},{z0,z1})=(p−1({y0,y1}),1−1Z({z0,z1}))=(p−1({y0,y1}),({z0,z1})). |
Hence for each
Theorem 3.5.
Let
Proof. Let
(p×1Z)−1({y0,y1},{z0,z1})=(p−1({y0,y1}),1−1Z({z0,z1}))=(p−1({y0,y1}),({z0,z1})). |
Since
We are ready to present some properties of the digital smash product. The following theorem gives a relation between the digital smash product and the digital homotopy.
Theorem 3.6. Given digital images
(h∧k)∘(f∧g)=(h∘f)∧(k∘g). |
f∧g≃(k∗(κ,λ),k∗(σ,α))f′∧g′. |
Proof. The digital function
(f×g)(X∨Y)⊂A×B. |
Hence
f≃(κ,σ)f′ and g≃(λ,α)g′. |
By Theorem 3.2, we have
f×g≃(k∗(κ,λ),k∗(σ,α))f′×g′. |
Theorem 3.7.
If
Proof. Let
f∘f′≃(λ,λ)1Y and f′∘f≃(κ,κ)1X. |
Moreover, let
g∘g′≃(α,α)1B and g′∘g≃(σ,σ)1A. |
By Theorem 3.6, there exist digital functions
f∧g:X∧A→Y∧B and f′∧g′:Y∧B→X∧A |
such that
(f∧g)∘(f′∧g′)=1Y∧B, |
(f∘f′)∧(g∘g′)=1Y∧B, |
and
(f′∧g′)∘(f∧g)=1X∧A, |
(f′∘f)∧(g′∘g)=1X∧A. |
So
The following theorem shows that the digital smash product is associative.
Theorem 3.8.
Let
Proof. Consider the following diagram:
(f′∘f)∧(g′∘g)=1X∧A. |
where
f:(X∧Y)∧Z→X∧(Y∧Z) and g:X∧(Y∧Z)→(X∧Y)∧Z. |
These functions are clearly injections. By Theorem 2.3,
The next theorem gives the distributivity property for the digital smash product.
Theorem 3.9.
Let
Proof. Suppose that
f:(X∧Y)∧Z→X∧(Y∧Z) and g:X∧(Y∧Z)→(X∧Y)∧Z. |
From Theorem 2.4,
f:(X∧Z)×(Y∧Z)→(X×Z)×(Y×Z). |
Obviously
Theorem 3.10.
Let
Proof. If we suppose that
f:(X∧Z)×(Y∧Z)→(X×Z)×(Y×Z). |
The switching map
Definition 3.11. The digital suspension of a digital image
Example 1. Choose a digital image
Theorem 3.12. Let
(X×[a,b]Z)/(X×{a}∪{x0}×[a,b]∪X×{b}), |
where the cardinality of
Proof. The function
[a,b]Zθ⟶S1 |
is a digital shy map defined by
X×[a,b]Z1×θ⟶X×S1p⟶X∧S1 |
is also a digital shy map, and its effect is to identify together points of
X×{a}∪{x0}×[a,b]Z∪X×{b}. |
The digital composite function
(X×[a,b]Z)/(X×{a}∪{x0}×[a,b]Z∪X×{b})→X∧S1=sX. |
Definition 3.13. The digital cone of a digital image
Example 2. Take a digital image
Theorem 3.14. For any digital image
Proof. Since
cX=X∧I≃(2,2)X∧{0} |
is obviously a single point.
Corollary 1. For
Proof. Since
For each
This paper introduces some notions such as the smash product, the suspension, and the cone for digital images. Since they are significant topics related to homotopy, homology, and cohomology groups in algebraic topology, we believe that the results in the paper can be useful for future studies in digital topology.
We would like to express our gratitude to the anonymous referees for their helpful suggestions and corrections.
[1] | World Health Organization, Monkeypox, 2022. Available from: https://www.who.int/news-room/fact-sheets/detail/monkeypox. |
[2] | World Health Organization, Monkeypox, 2022. Available from: https://www.who.int/news-room/questions-and-answers/item/monkeypox. |
[3] | Centers for Disease Control and Prevention. Available from: https://www.cdc.gov/poxvirus/monkeypox/. |
[4] |
S. Ullah, M. A. Khan, M. Farooq, T. Gul, Modeling and analysis of tuberculosis (tb) in khyber pakhtunkhwa, pakistan, Math. Comput. Simul., 165 (2019), 181–199. https://doi.org/10.1016/j.matcom.2019.03.012 doi: 10.1016/j.matcom.2019.03.012
![]() |
[5] |
A. Khan, R. Ikram, A. Din, U. W. Humphries, A. Akgul, Stochastic covid-19 seiq epidemic model with time-delay, Results Phys., 30 (2021), 104775. https://doi.org/10.1016/j.rinp.2021.104775 doi: 10.1016/j.rinp.2021.104775
![]() |
[6] |
X. Liu, S. Ullah, A. Alshehri, M. Altanji, Mathematical assessment of the dynamics of novel coronavirus infection with treatment: A fractional study, Chaos, Solitons Fractals, 153 (2021), 111534. https://doi.org/10.1016/j.chaos.2021.111534 doi: 10.1016/j.chaos.2021.111534
![]() |
[7] |
S. Usman, I. I. Adamu, Modeling the transmission dynamics of the monkeypox virus infection with treatment and vaccination interventions, J. Appl. Math. Phys., 5 (2017), 81078. https://doi.org/10.4236/jamp.2017.512191 doi: 10.4236/jamp.2017.512191
![]() |
[8] |
O. J. Peter, S. Kumar, N. Kumari, F. A. Oguntolu, K. Oshinubi, R. Musa, Transmission dynamics of monkeypox virus: a mathematical modelling approach, Model. Earth Syst. Environ., 8 (2022), 3423–3434. https://doi.org/10.1007/s40808-021-01313-2 doi: 10.1007/s40808-021-01313-2
![]() |
[9] |
C. P. Bhunu, S. Mushayabasa, J. Hyman, Modelling hiv/aids and monkeypox co-infection, Appl. Math. Comput., 218 (2012) 9504–9518. https://doi.org/10.1016/j.amc.2012.03.042 doi: 10.1016/j.amc.2012.03.042
![]() |
[10] |
A. Khan, Y. Sabbar, A. Din, Stochastic modeling of the monkeypox 2022 epidemic with cross-infection hypothesis in a highly disturbed environment, Math. Biosci. Eng., 19 (2022), 13560–13581. https://doi.org/10.3934/mbe.2022633 doi: 10.3934/mbe.2022633
![]() |
[11] | 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. |
[12] |
M. Caputo, M. Fabrizio, A new definition of fractional derivative without singular kernel, Progr. Fract. Differ. Appl., 1 (2015), 1–13. http://dx.doi.org/10.12785/pfda/010201 doi: 10.12785/pfda/010201
![]() |
[13] |
A. Atangana, D. Baleanu, New fractional derivatives with nonlocal and non-singular kernel: theory and application to heat transfer model, Therm. Sci., 20 (2016), 763–769. https://doi.org/10.2298/TSCI160111018A doi: 10.2298/TSCI160111018A
![]() |
[14] |
O. J. Peter, F. A. Oguntolu, M. M. Ojo, A. O. Oyeniyi, R. Jan, I. Khan, Fractional order mathematical model of monkeypox transmission dynamics, Phys. Scr., 97 (2022), 084005. https://doi.org/10.1088/1402-4896/ac7ebc doi: 10.1088/1402-4896/ac7ebc
![]() |
[15] |
A. El-Mesady, A. Elsonbaty, W. Adel, On nonlinear dynamics of a fractional order monkeypox virus model, Chaos, Solitons Fractals, 164 (2022), 112716. https://doi.org/10.1016/j.chaos.2022.112716 doi: 10.1016/j.chaos.2022.112716
![]() |
[16] |
A. Atangana, Fractal-fractional differentiation and integration: Connecting fractal calculus and fractional calculus to predict complex system, Chaos, Solitons Fractals, 102 (2017), 396–406. https://doi.org/10.1016/j.chaos.2017.04.027 doi: 10.1016/j.chaos.2017.04.027
![]() |
[17] |
W. Wang, M. Khan, Analysis and numerical simulation of fractional model of bank data with fractal-fractional Atangana-Baleanu derivative, J. Comput. Appl. Math., 369 (2019), 112646. https://doi.org/10.1016/j.cam.2019.112646 doi: 10.1016/j.cam.2019.112646
![]() |
[18] |
X. P. Li, S. Ullah, H. Zahir, A. Alshehri, M. B. Riaz, B. A. Alwan, Modeling the dynamics of coronavirus with super-spreader class: A fractal-fractional approach, Results Phys., 34 (2022), 105179. https://doi.org/10.1016/j.rinp.2022.105179 doi: 10.1016/j.rinp.2022.105179
![]() |
[19] |
S. Qureshi, E. Bonyah, A. A. Shaikh, Classical and contemporary fractional operators for modeling diarrhea transmission dynamics under real statistical data, Phys. A, 535 (2019), 122496. https://doi.org/10.1016/j.physa.2019.122496 doi: 10.1016/j.physa.2019.122496
![]() |
[20] | C. Li, F. Zeng, Numerical Methods for Fractional Calculus, Chapman and Hall/CRC, 2015. |
[21] |
A. Atangana, S. Qureshi, Modeling attractors of chaotic dynamical systems with fractal–fractional operators, Chaos, Solitons Fractals, 123 (2019), 320–337. https://doi.org/10.1016/j.chaos.2019.04.020 doi: 10.1016/j.chaos.2019.04.020
![]() |
1. | Byungsoo Moon, Orbital stability of periodic peakons for the generalized modified Camassa-Holm equation, 2021, 14, 1937-1632, 4409, 10.3934/dcdss.2021123 |