Analysis of Monkeypox viral infection with human to animal transmission via a fractional and Fractal-fractional operators with power law kernel

1.   Introduction

Monkeypox is a viral zoonotic disease caused by the monkeypox virus. This infection primarily occurs in Central and West Africa. MPX emerged as a serious orthopoxvirus for human health in 1980, when smallpox infection was eradicated. The first outbreak of this infection outside of African countries was reported in 2003 in the United States. Subsequently, many MPX-infected cases were reported in African and European countries. In May 2022, the infection was confirmed in several non-endemic areas. More than 3413 confirmed cases were reported with 1 death case to WHO from 50 countries/territories. The transmission of MPX occurs from animals to humans. Mostly, the hosts in animals include a range of rodents and non-human primates. The infection can also be spread from person to person through close contact, such as droplets released by talking, breathing, or sneezing. It can also be transmitted through sexual contact with an infectious person. Environmental transmission is also possible for MPX. In a recent investigation, the transmission of MPX from infected humans to pets are reported ^[1,2]. Different disease symptoms are observed in MPX-infected people. Some people have less severe symptoms, while others develop more serious illnesses. Typical clinical symptoms developed by infected individuals during recent outbreaks include headache, fever, back pain, myalgia, lack of energy, swollen lymph nodes, and a rash lasting 2–3 weeks. Initially, a vaccine used against the smallpox eradication program protects her MPX. Three vaccines against MPX are currently available. Two of them, namely MVA-BN and LC16, have been approved to prevent this infection ^[1,2,3].

Many epidemiological aspects of MPX infection are still under investigation. Studies are underway to find out more about the transmission and treatment of this infection. Mathematical models are one of the most effective tools for studying the dynamic aspects of infectious diseases. These models help design effective interventions and provide decision-makers and the health sector with useful information for understanding disease dynamics and control. In this regard, a number of epidemiological infection models based on various infectious diseases have been developed and studied in the recent literature. These models are usually formulated using various differential derivatives (or operators), usually partial, stochastic, ordinary or fractional in nature ^[4,5,6]. The impact of treatment and vaccination on the dynamics of MPX is analyzed through a classical integer case transmission model in ^[7]. Recently, a new mathematical model-based MPX transmission is proposed in ^[8]. The authors of ^[8] divided the population into human and non-human (animal) classes and performed a detailed theoretical and numerical analysis of the problem. A mathematical model for the transmission of HIV and MPX co-infection is analyzed in ^[9]. The authors in ^[9] analyzed the transmission in the HIV-infected population and suggested some effective control interventions for disease eradication. In ^[10], the authors recently reviewed a stochastic MPX transmission model with cross-infection. Most of the existing MPX transmission models are established through classical integer-order differential equations. However, from an epidemiological point of view, Prior experience and history of an epidemic have a substantial role in analyzing the disease dynamics in a quite realistic way. On the other hand, the models with classical derivatives are local in nature and cannot model the dynamics of phenomena between two integer values. Furthermore, these classical models are unable to possess memory effects and hereditary properties which are found in many biological diseases. Subsequently, fractional operators have been used in recent years to formulate the biological models. The well-known operators having fractional order used in the existing literature are Caputo ^[11], Caputo-Fabrizio (CF) ^[12] and the Atangana-Baleanu operators(ABC) ^[13]. So far, only a few transmission models have been established with these operators. A fractional-order MPX transmission model was recently developed in ^[14]. The authors of ^[14] formulated the model via the CF operator and estimated the parameters using real data reported in Nigeria over a specific time period. Furthermore, a fractional Caputo-type MPX transmission model is established in ^[15].

Besides the above-cited literature, the modeling approach with a novel fractal-fractional operator was adopted after the introduction of these operators by Atangana in 2017 ^[16]. These novel operators are actually the combination of fractional and fractal calculus. Recently, many researchers have applied these new concepts successfully in various scientific problems including epidemiology. For instance, a compartmental model to study the impact of commercial and rural banks via a fractional-fractal modeling approach ^[17]. They have shown that by using a model with the fractional-fractal operator one can provide a better fit to actual data. An application of fractional-fractal operators to analyze the dynamics of the novel coronavirus pandemic can be found in ^[18].

In existing literate, MPX transmission models are formulated with only animal-to-human transmission. Recently, transmission from humans to pets is being reported ^[2]. Based on the recent facts, in this paper, we develop the MPX transmission model with animal-to-human and human-to-animal transmission of infection. The proposed model is actually the extension of ^[10] by incorporating the human-to-animal transmission. In addition, unlike the existing models, we formulate the present problem using the fractional and fractal-fractional operators in order to gain more insights into the disease dynamics. The paper is classified into seven main sections with details as: Basic definition of the fractional, as well as the fractal-fractional operators are discussed in Section 2. Section 3 briefly introduces the model formulation for the integer case. The fractional MPX transmission model with some of the basic analysis is established in Section 4. Section 5 presents the numerical scheme and resulting simulation in the fractional MPX transmission model. The MPX model in fractal-fractional the case is developed in Section 6. Moreover, this section includes the existence, uniqueness, and novel numerical procedure along with simulation results and a brief discussion of the MPX model with the fractal-fractional operator. Finally, concluding remarks are drawn in Section 7.

2.   Basic definitions

This section provides basic details about fractional and fractal-fractional operators developed ^[11,16].

Definition 2.1. Consider the function , the Caputo derivative with $n-1 < \rho \leq n$ as fractional order and $n\in \mathbb{N}$ is given as:

approaches to integer derivative as $\rho\rightarrow 1$.

Definition 2.2. A point $x^*$ is said to be the equilibrium point of the following Caputo-type system defined by:

if and only if $f(t, x^*) = 0$.

Let $\texttt{X}(t)$ be continuously fractal differentiable function on $(a_1, a_2)$. The fractal and fractional parameters are respectively $\eta$ and $\rho$. We recalled some definitions as follows ^[16]:

Definition 2.3. The fractal-fractional (FF) operator of the Riemann-Liouville (RL) type based on the power-law kernel is defined as:

where, $n-1 < \rho, \eta\leq n$ $\in \mathbb{N}$ and

Definition 2.4. The corresponding FF integral operator of (2.2) is given as follows:

3.   Monkeypox model in integer case

This section briefly describes the proposed MPX transmission model. To develop the desired model, the whole human population is classified into four mutually-exclusive epidemiological classless including the susceptible human class $S_h(t)$, the infectious human class $I_h(t)$, isolate/quartine human class $Q_h(t)$ and the recovered human class $R_h(t)$. The non-human population (animals) is divided into two groups namely the susceptible animals $S_m(t)$ and the infectious animals $I_m(t)$. The considerable assumptions taken into account are:

● Unlike the existing models, we considered the transmission of infection to succeedable animals from infected humans.

● The disease-induced death rate is considered in infected and isolated compartments.

● The infectious human population can be recovered through proper treatment and natural recovery.

The susceptible human class is generated by a constant recruitment rate $\Lambda_h$ and reduced by acquiring infection after interaction with infectious humans and animals. The force of infection is $\beta_1 I_m + \beta_2 I_h$, where $\beta_1$ and $\beta_2$ are the effective contact rates (capable of transmitting infection) corresponding to infectious animals and humans respectively. Each class containing the human population is reduced due to natural death at the rate $\mu$. The Monkeypox-induced death rate in the infected and isolated human compartments are denoted by $\delta_1$ and $\delta_2$ respectively. The infectious human papulation becomes recovered and joins $R_h$ at the rated $r_1$ and $r_2$. The symbol $r_1$ shows the natural recovery rate and $r_2$ denotes the recovery rate through specific treatment. The infectious human population is isolated and joins $Q_h$ class at the transmission rate $\phi$. The recovery rate of the isolated/quartine human population is denoted by $\tau_2$ while $\tau_1$ shows the susceptibility rate of the isolated population.

In the animal population, $\Lambda_m$ shows the recruitment rate while the natural death rate denoted is $\nu$. The force of infection is $\zeta_1 I_h + \zeta_2 I_m$, where $\zeta_1$ and $\zeta_2$ are the respective transmission rates corresponding to infected humans and animals.

Keeping the above discussion in mind the proposed MPX mathematical model is given in the following system.

The initial conditions for the above system are

In order to make the onward calculation simpler, we take

Thus, the model (3.1) can be written as follows:

4.   Fractional extension of MPX model

This section presents the extension of the integer MPX model to fractional order. The well-known Caputo operator with a power law kernel is utilized to obtain the fractional model. The transmission models establish with fractional operators provides biologically more significant output than integer case problems and have a widespread application in real-life situations including epidemiology. It is due to the memory and heredity properties of fractional operators. The procedure of developing fractional-order mathematical model we proceed as follows:

In the above problem $k(t-t^{'})$ describes the time-dependent kernel. Moreover,

Replacing (4.2) in (4.1), and taking the Caputo having order $\rho-1$, it leads to

After some manipulation the fractional MPX transmission model leads to the following system

subject to ICs (3.2). In the system (4.4), ${}^CD^{\rho}_{t}$ shows the Caputo derivative with order $\rho\in(0, 1]$.

4.1.   Basic analysis of the fractional MPX model

In this section we address some of the basic and necessary mathematical analysis of fractional MPX disease model.

4.1.1.   Invariant region

The feasible domain in terms of biological point view for the MPX fractional model (4.4) is constructed as follows:

Such that and we have $\Omega = \Omega _h \times \Omega _m \subset \mathbb{R}_ + ^4 \times \mathbb{R}_ + ^2$ is positively invariant.

The following theorem prove the positivity and bounded properties of solution of Caputo MPX model.

Theorem 4.1. Let $S_h \left(0 \right) = S_{h_0 }, \; \; I_h \left(0 \right) = I_{h_0 }, \; \; Q_h \left(0 \right) = Q_{h_0 }, \; \; R_h\left(0 \right) = R_{h_0 }$, $S_m \left(0 \right) = S_{m_0 }$, and $I_m \left(0 \right) = I_{m_0}$ be the initial values of state variables. For the positive initial conditions, the region defined in the set $\Omega$ will be positively invariant for all $t > 0.$ Furthermore, $N_{h_0 } \left(t \right) \le \frac{{\Lambda _h }}{{\mu }}\, \, \, \, and\, \, N_{m_0 } \left(t \right) \le \frac{{\Lambda _m }}{{\nu }}$.

Proof. We start the proof by considering only the first equation of system (4.4). Let $S_{h_0 }, \; I_{h_0 }, \; \; Q_{h_0 }, \; R_{h_0 }, \; S_{m_0 }\; and \; \, I_{m_0 },$ be all positive, the desired result is to prove that state variables are positive. From system (4.4) we have

then

Since $\Lambda _h + \tau _1 \, Q_h \ge 0$ implies that $^C D_t^\rho S_h + \left({\lambda _h + \mu } \right)S_h \ge 0,$ by using the Laplace transform we have

By applying the inverse Laplace transform we have

Since quantities on right hand side of (4.8) are positive therefore, we have $S_{h}\ge 0$ for $t\ge 0$. Similarly, it can be shown that $I_h, Q_h, R_h, S_m, \, and\, I_m \ge 0\, \, \, \forall t \ge 0.$ Thus, the solutions in $\mathbb{R}_ + ^4 \times \mathbb{R}_ + ^2$ are positive.

In the second part, we prove the bounded property of the problem solution. The total population for human is given by $N_h \left(t \right) = S_h \left(t \right) + I_h \left(t \right) + Q_h \left(t \right) + R_h \left(t \right),$ Such that

Laplace transform on both sides yields to the following

By applying inverse Laplace transform we have

By taking limit $t \to \infty$ we have implies that $N_h \left(t \right) \le \frac{{\Lambda _h }}{\mu },$ If $N_{h_0 } \left(t \right) \le \frac{{\Lambda _h }}{\mu }\, \, then\, N_h \left(t \right) \le \frac{{\Lambda _h }}{\mu }$ implies $N_{h } \left(t \right)$ is bounded. In similar way we can prove that $N_{m } \left(t \right)$ is bounded. So we conclude that the region $\Omega$ is epidemiological feasible and bounded.

4.2.   Equilibria and $\mathcal{R}_0$

For equilibrium points, we consider the following system

The solution of (4.11) at disease free state (i.e., $I_h = Q_h = R_h = I_m = 0$) gives the following disease free equilibrium ($\texttt{DFE}$).

By using the next generation method we compute $\mathcal{R}_{0}$.

Let $X = (I_h, Q_h, I_m)^t$ then have Jacobian of the linearized system at DFE is as follows

The next generation matrix is given by

Endemic equilibrium point The endemic equilibrium ($\texttt{EE}$) of the fractional MPX infection model is obtain as follows:

5.   Numerical treatment of fractional MPX model

In this section, the iterative solution of the MPX model (4.4) in fractional case is being investigated. Moreover, the resulting simulation will be analyzed in detail.

5.1.   Numerical scheme

The fractional Euler's scheme as discussed in the recent literature ^[19,20] is utilized to obtain numerical scheme of the proposed model. For this purpose, we express the fractional model (4.4) in the following problem

where, $p(t) = (S_h, I_h, Q_h, R_h, S_m, I_m)\in \mathbb{R}^6$ and $p_0$ stands for initial state vector. Further, $\mathcal{H}(t, p(t)) = (\mathcal{H}_1(t, p(t)), \mathcal{H}_2(t, p(t)), \mathcal{H}_3(t, p(t)), \mathcal{H}_4(t, p(t)),$ $\mathcal{H}_5(t, p(t)), \mathcal{H}_6(t, p(t)))$ denotes continuous real valued vector function satisfying the Lipschitz condition such that

Taking the Caputo integral on both sides of (5.1) we obtained

Further, a uniform grid on the interval $[0, \mathcal{T}]$, with $\mathfrak{h} = \frac{\mathcal{T}-0}{m}$, denotes the step size and $m\in \mathbb{N}$ is considered. Using the Euler method ^[20], the Eq (5.3) leads to the following result:

Utilizing the numerical procedure (5.4) and the expressions in (5.2), the following iterative formulae are establish for MPX transmission model (4.4):

5.2.   Simulation

In this section, we simulate the fractional case MPX infection model (4.4) using the iterative scheme derived in Eq (5.5). The time level considered in the graphical results of the model is 500 days. The model is simulated for two sets of variables and five different values of fractional order $\rho$. The values of parameters are taken from ^[10].

Case 1: In this case, we consider the parameters values as $\Lambda_h = 10, \; \beta_1 = 0.00001, \; \; \beta_2 = 0.00002, \; \mu = 0.05,$ $\delta = 0.0003, \; \tau_1 = 0.32, \; \tau_2 = 0.2;$ $\varphi = 0.5, \; r_1 = 0.041, \; r_2 = 0.043,$ $\alpha_1 = 0.004, \; \delta_2 = 0.002, \; \Lambda_m = 10,$ $\zeta_1 = 0.000027, \; \zeta_2 = 0.000017, \; \nu = 0.02$. The graphical interpretation for this case is shown in Figure 1(a)–(f). The dynamics of susceptible, infected, isolated and recovered human population are depicted in Figure 1(a)–(d) respectively. The dynamics of susceptible and infected animals population is shown in Figure 1(e), (f) respectively. These graphical interpretations demonstrated that all solution trajectories converge to the $\texttt{DFE}$ for all values of $\rho$.

Case 2: In this case, we slightly modify the parameters values. The considered parameters values are $\Lambda_h = 10, \; \beta_1 = 0.0001, \; \; \beta_2 = 0.0002, \; \mu = 0.05, \; \delta = 0.0003,$ $\tau_1 = 0.32, \; \tau_2 = 0.2; \varphi = 0.5,$ $r_1 = 0.041, \; r_2 = 0.043, \; \alpha_1 = 0.004, \; \delta_2 = 0.002,$ $\Lambda_m = 10, \; \zeta_1 = 0.00027, \; \zeta_2 = 0.00017, \; \nu = 0.02$. The graphical results, in this case, are depicted in Figure 2(a)–(f). In Figure 2(a)–(d), we have shown the dynamics of the susceptible, infected, isolated, and recovered human population respectively, while in Figure 2(e), (f) describes the dynamics of susceptible and infected animals population respectively. These simulations demonstrated that all solution trajectories converge to the $\texttt{EE}$ for all values of $\rho$.

6.   A fractal-fractional MPX model in Caputo sense

In the previous section, we extended the integer MPX transmission model to a fractional case. In this part of the manuscript, we further extend the fractional model to obtain a more generalized problem using the fractal-fractional operator in the Caputo sense. After formulating the model, we explore its basic mathematical analysis including the existence and uniqueness of the solution. Moreover, the model in the fractal-fractional case will be solved via an efficient numerical scheme to simulate the model for different values of fractional as well as fractal orders. The fractal-fractional case MPX transmission model can be established by the following system

The fractional and fractal parameters in the above model (6.1) are expressed by $\rho$ and $\eta$ respectively. We proceed with the basic analysis of the model (6.1) in the next section.

6.1.   Existence and uniqueness

The current part of the work explores the existence of the uniqueness of the proposed model solution using the well-known Picard-Lindel$\ddot{o}$f theorem coupled with the fixed point approach. To move further, the fractal-fractional system (6.1) can be expressed in the following general Cauchy problem

where, $p(t) = (S_h, I_h, Q_h, R_h, S_m, I_m)$ consists of state variables and $\mathcal{H}$ representing a continuous vector function as described in Subsection 5.1. The corresponding initial condition are stated as $p_0 = (S_h(0), I_h(0), Q_h(0), R_h(0), S_m(0), I_m(0))$.

The problem (6.2) leads to the following result after utilizing the fractional integral

After the replacement of right hand side by the Caputo derivative and then implementing the integral we obtain ^[21]:

Further, with the help of Picard-Lindel$\ddot{o}$f theorem, let we define

where,

Defining the following operator

such that

Our focus is to confirm the fact that operator defined in (6.5) maps a complete norm empty metric space into itself. Moreover, it is necessary to establish that it also possesses the contraction mapping. In the first attempt we show that

The following norm is taken into account

where,

and the norm is defined by

Further, letting $\varsigma = ty$, the above expression is converted to

Taking $\phi_1$, $\phi_2$ $\in C[I_n(t_n), A_b(t_n)]$, we obtained the following result,

Thus, we deduce that the contractive property is arrived after the fulfilment of following criteria

Hence, complete the proof.

6.2.   Numerical scheme for fractal-fractional MPX model

This section briefly presents a novel numerical scheme for the fractal-fractional MPX transmission model (6.1) in order to obtain the impact of fractional and fractal orders graphically. The procedure presented in ^[21] is utilized for this purpose. The model presented in (6.1) is firstly converted to Volterra form as the fractional integral operator is differentiable. Further, the fractal-fractional Caputo model in the form of RL operator can be expressed as:

Therefore, the problem (6.2) can be transform as

Further, the RL derivative is replaced with the Caputo derivative such that to make the use of the integer-order initial conditions. The Eq (6.13) gives the following expression:

Upon the grid points $t = t_{n+1},$ (6.14) can be written as follows:

Moreover, utilizing the Lagrangian piece-wise interpolation procedure over the finite interval $[t_v, t_{v+1}]$ for the function $\mathcal{H}(\varsigma, p(\varsigma))$ in (6.15) we have

Using the approximation made in (6.16), the (6.15) leads to the following:

Finally, the solution of (6.17) leads to the following equation:

Thus, implementing the scheme derived in (6.18) on the proposed fractal-fractal MPX transmission model (6.1), the following recursive formulae are established:

6.3.   Simulation of the fractal-fractional MPX model

The MPX transmission model (6.1) is simulated using the iterative scheme established in (6.19). The values of parameters and initial conditions are the same as considered in the fractional model. The model is simulated for three cases, considering different values of fractal order $\eta$ and fractional order $\rho$. Model simulations are analyzed numerically in Figures 3–5. In each figure, we present the impact of the fractional and fractal orders $\rho$ and $\eta$ respectively on the solution of the model.

Case 1: In this case, the fractal order $\eta$ is fixed and the fractional order $\rho$ is varied by taking four different values, i.e., 1, 0.95, 0.90, 0.80. The simulation for this set of parameters is analyzed in Figure 3(a)–(f). The dynamics of susceptible, infected, isolated, and recovered human population classes are shown in Figure 3(a)–(d) respectively. The dynamics of susceptible and infected animal population is analyzed in Figure 3(e), (f) respectively.

Case 2: This case presents the simulation of the fractal-fractional model (6.1) when the order of the Caputo fractional operator is fixed to integer case, i.e., $\rho = 1$ and considered four different values, i.e., $\eta = 1, 0.9, 0.80, 0.70$ of the order of fractal operator. The simulation obtained from this scenario is depicted in Figure 4(a)–(f). The dynamics of human population classes are shown in Figure 4(a)–(d) respectively, whereas the dynamic of susceptible and infected animal compartments are respectively analyzed in Figure 4(e), (f).

Case 3: In the third case, we simulate the fractal-fractional MPX transmission model (6.1) when both the fractional order $\rho$ and fractal order $\eta$ varied equally or differently. The simulation results produced for this scenario are analyzed in Figures 5 and 6 with subplots (a)–(f). From these figures, we observed that in all three cases the solution curves converge to steady states. However, in the third case (by taking non-integer values of both $\rho$ and $\eta$) the converges to a steady state is attained comparatively over a longer time period. Overall, we conclude that the fractal-fractional derivatives can be utilized in order to provide a better understanding of disease dynamics.

7.   Conclusions

In this manuscript, a new Monkeypox epidemic model with human-to-animal transmission is analyzed. Although few cases have been reported on such transmission. To the best of over knowledge, this is the first attempt to consider such transmission in the Monkeypox transmission model. We developed the model with integer-order derivative and then extended it to fractional order via the Caputo operator to analyze the impact of memory index on the disease incidence. The basic mathematical analysis of the fractional Monkeypox model including the existence and uniqueness of the solutions is carried out in the initial stage. The fractional model is then solved numerically and the simulation is shown for two cases choosing different values of parameters demonstrating the stability of model equilibria graphically. Moreover, the model is reformulated via the novel fractal-fractional operator in order to obtain the generalized model. The existence and uniqueness of the problem solution in this case are provided via the Picard-Lindel$\ddot{o}$f approach. An efficient numerical procedure based on the Adams-Bashforth approach is used for the solution of the fractal-fractional Monkepox model. Simulation is presented for three different cases by considering various values of fractal and fractional paraments. In all cases, we observed that the solution curves converge to the model's steady state.

Acknowledgements

The authors would like to thank the Deanship of Scientific Research at Umm Al-Qura University for supporting this work by Grant Code: 22UQU4390021DSR06.

Conflict of interest

The authors declare there is no conflict of interest.