Introduction

Most of the species of the Brucella bacteria are involved in causing zoonotic diseases such as bronchiolosis. It is estimated that 500 000 casualties occur each year due to brucellosis despite the fact that it is relatively controlled in the Australia and the UK. Despite there being rigorous Veterinary Hygiene Regulations in these regions, brucellosis is a chronic disease in; Caribbean, South America, Mediterranean Europe, Africa, Asia and the Middle East3. In some of these regions, it constitutes major economic challenges. In many cases, contact with the bacteria is presumed to be the reason behind Brucellosis in animals either direct or though mediators4. Indirect transmission on the other hand occur when you touch objects or surface’s that has been contaminated with the virus or droppings, or when you touch contaminated surfaces and then touch your face, nose or mouth among others. Other associated systems include respiratory or digestive systems when eating may also be affected5. While human brucellosis is acquired from contact with infected animals or their products. Abortion, epididymitis, placentitis and orchitis are the conditions most frequently associated with brucellosis. Cattle, sheep, goats, pigs, rabbits, horses, dogs and man are the usual animals that are affected by Brucellosis6. The application of domestic animal vaccination control and detection methods to avoid infection of bovine Brucella averts it7. However, together with the treatments, brucellosis is quickly halted or eliminated as has been noted above.

This paper will show how further information can be gained about the transmission of infectious disease using mathematical models. There have been different models developed over the years in an attempt to define the phenomena of zoonosis. In particular, Nyerere et al.8, the only way to prevent human from developing brucellosis is by refraining from diseases that infect domestic animals. They use a mathematic model to establish the impact of various control parameters on the transmission dynamics of brucellosis in human and animals’ population. The rational approach to successfully address the optimal control is based on environmental sanitation, personal protection by human beings, mass immunization and stepwise control of cattle and small stock population within an area9. To analyse the brucellosis infection with specific preventive measures,10 formulated a mathematical model. The impact of brucellosis on the Yellowstone bison population is under analysis in11. In view of this, there is no justification to start a vaccination campaign that includes the park as a whole. In the study12, authors focus on the time related to the treatment of cattle for brucellosis.

Fractional calculus is applied almost in all scientific fields. The first advantage that is conspicuous when dealing with FDEs is the capacity to differentiate between memory and hereditary features of diverse mathematical models. Factional order models therefore appear less ad hoc and closer to fact and data than we get from traditional integer order models13. Khan along with others. Leptospirosis epidemic model was quantitatively discussed by14. As proposed in Atangana-Baleanu (AB),15 the disease leptospirosis is modeled in terms of fractional epidemic. Taking advantage of the prospective of stochastic numerical supervised neural-network, Mukdasai and colleagues. The fractional order Leptospirosis model was developed quite recently and16 has produced numerical solutions of it. At the mere end of the previous decade, Atangana17 introduced a completely novel derivative, the so called fractal-fractional derivative. Often such an approach seems to be very effective in solving complex issues in various fields of human activities. The newly developed fractional derivative with fractal feature outdoes conventional fractional as well as other types of derivatives. A short term solution was not employed in relations to the dynamics of the infection within the plant and a new model involving a fractional derivative fractal was discussed by Farmers et al.18. The current novel corona epidemic was analyzed by the authors of cite21 using mathematical analysis applying fractal-fractional techniques. Akgul Ali et al.20 investigated study groups of people with diabetes. To investigate the dynamics of the transmission of Q fever within ticks and livestock as well as the bacterial load in the environment, fractal-fractional Atangana-Baleanu derivatives and integrals in the sense of Caputo were used by the researchers21. Based on the newly derived Newton polynomial, a numerical method was given. In22, a fractional order COVID-19 model was considered via fractal-fractional analysis. The infection transmission of Brucellosis in23 is modeled using a Caputo-Fabrizio operator model.The paper30 presents a numerical approach using Haar wavelet collocation methods to solve a fractional-order model for computer virus dynamics. The work in31 analyzes COVID-19 dynamics with a fractional-order model incorporating the Mittag-Leffler kernel. Studies32,33 explore advanced numerical methods and fractional-order models for COVID-19 dynamics. The papers34,35,36 extend these models to specific contexts, such as Pakistan, the Omicron variant, and SARS-CoV-2 variants, while37 applies similar frameworks to Anthroponotic Cutaneous Leishmania transmission.

Brucellosis, caused by the Brucella species, poses a significant threat to livestock health and productivity, leading to substantial economic losses in the cattle industry worldwide. Understanding the dynamics of this infectious disease is crucial for developing effective control strategies. Traditional models often fail to capture the complex interactions inherent in disease transmission, particularly in heterogeneous populations. In this study, we employ a fractional-order derivative approach to model brucellosis dynamics, which allows for a more nuanced representation of memory effects and time delays in infection processes. The addition of fractional derivatives improves the model forecast’s accuracy since the rate at which cattle can be infected can be affected by the external environment and disease containment measures. Moreover, we include sensitivity analysis, in which we determine the factors through which particular disease outcomes are most affected, which results benefit policymakers and veterinarians. In light of this, this study advances knowledge on patterns of brucellosis transmission and mitigation in cattle through a synthesis of mathematical perspective with biological significance. It also reveals that Caputo-Fabrizio operator possesses some advantages for describing the behavior of different infectious diseases such as brucellosis. The non-integer order calculus impedes to describe memory effects and time-dependent behaviours more effectively, inherent in disease transmission processes and disregarded in the framework of integer order models. This operator is especially useful in epidemiological modeling as it allows to exhibit and prove the fundamental mathematical properties of existence, uniqueness and stability of solutions which give the model extra durability. Additionally, numerical results reveal that the Caputo-Fabrizio derivative can be used to better identify difficult infection dynamics, thus, it can be a valuable instrument in studying the transmission of brucellosis and other diseases.

The purpose of this work is to establish a fractional order model for SIRBV associated with the cattle brucellosis outbreak. The said fractional order model impacts on memory beyond what the classical order of model does. The credibility of conclusions derived from the imitation of the process of transmission of cattle brucellosis is examined in this work. We must admit that we are dealing here with a theory unique for the solutions of the fractal fractional model. The stability of the fixed point of the proposed model is examined in the global sense with the help of fix point theory. The included fractional derivatives have been approximated numerically by using the two-step Lagrange polynomial. Eight sections make up the paper: section 1 has an introduction; some simple fractional order derivatives that are useful in solving the epidemic model are presented in section 2. In section 3, disease transmission of endemic equilibrium, disease-free equilibrium, and the positivity of the fractional order model are presented, and sensitivity analysis is shown. In section 4 the fixed point theory is applied to establish the existence and uniqueness of a system of model solutions. Section 5 focus on the stability of the FO Brucellosis in the Cattle model. In section 6 of the study, the effects of the fractional parameters have been studied using an efficient numerical methods in solving the fractional order system. The last two sections, 7 and 8 focus on the findings and conclusion.

Basic definitions

The definitions of fundamental terms that are essential to comprehending fractional operators must be reviewed before we can start developing the model. Assume that the function P(t) , is a non-differentiable function with order \(0 < \eta \le 1\).

Definition 1

23 With \(0 < \eta \le 1\) and P(t), we can proceed. A fractional derivative with a power-law kernel is defined as:

$$\begin{aligned} ^{C}D_{t}^{\eta }(P(t)) = \frac{1}{{\Gamma (1 - \eta )}}\frac{d}{{d{t{ }}}}\int \limits _0^t {{{(t - \sigma )}^{- \eta }}P(\sigma )d\sigma ,} \end{aligned}$$
(1)

Definition 2

24 For (1), the corresponding fractional integral operator is provided by.

$$\begin{aligned} ^{C}I_{t}^{\eta }(P(t)) = \frac{1}{{\Gamma (\eta )}}\int \limits _0^t {{{(t - \sigma )}^{\eta - 1}}{ { }}P(\sigma )d\sigma .} \end{aligned}$$
(2)

Definition 3

24 A fractional derivative of kernel exponential decay has the following expression

$$\begin{aligned} ^{CF}D_{t}^{\eta }(P(t)) = \frac{{M(\eta )}}{{\Gamma (m - \eta )}}\frac{d}{{d{t{ }}}}\int \limits _0^t {exp\left[ { - \frac{\eta }{{1 - \eta }}{{(t - \sigma )}}} \right] P(\sigma )d\sigma .} \end{aligned}$$
(3)

Definition 4

24 Definition of the associated fractional integral operator for (3) is given by.

$$\begin{aligned} ^{CF}I_{t}^{\eta }(P(t)) = \frac{{ (1 - \eta ){{ }}P(t)}}{{M(\eta )}} + \frac{{\eta }}{{M(\eta )}}\int \limits _0^t {{\sigma ^{\eta - 1}}P(\sigma )d\sigma .} \end{aligned}$$
(4)

Definition 5

38 The Laplace transform is defined by

$$\begin{aligned} P(s) = L\{ p(t)\} = \int \limits _0^\infty {{e^{ - st}}p(t)} dt, \end{aligned}$$
(5)

where p(t)is n-dimensional vector-valued function.

Definition 6

39 The Laplace transform of the Caputo Fabrizio fractional derivative of a function P(t) of order \(\eta > 0\) is defined as

$$\begin{aligned} {\mathscr {L}}\{ _0^{CF}D_t^\eta P(t)\} = \frac{{s {\mathscr {L}}\{ P(t)\} - P(0)}}{{s + \eta (1 - s)}}, \, s > 0. \end{aligned}$$
(6)

Lemma 1

40 Let P(t) be a real continuous and differentiable function. Then, for any \(t \ge 0\) and \(0 < \eta \le 1\), we have

$$\begin{aligned} _0^{CF}D_t^\eta ({P^2}(t)) \le 2P(t) _0^{CF}D_t^\eta P(t). \end{aligned}$$
(7)

Fractional order brucellosis in cattle model

A deterministic model provided in12 will be used to examine the effects of different treatments on the spread of bovine brucellosis. The susceptible class of cattle can be defined as Cattle that does not possess immunity against brucellosis. Cattle are said to be infected (I, ) if they have been diagnosed to have the disease and are possible to infect other cattle. Recovered (R) cattle refer to a class of cattle that tested positive with bovine brucellosis in the past but have recovered fully from the disease. Bio-secured class(B) means a class of cattle which is freedom from bovine brucellosis by implementation of bio-security measures. These practices are well controlled to minimize contact with pathogenic organisms thus enhancing efforts to produce cattle in this class with out disease. Some of the attenuated movements may include restricted access to that environment, sanitation and avoidance of contact with other animals believed to be infected. Hence, under the protective measures, the cattle in the biosecured class, are presumed to be unable to contract the disease. The vaccinated class (V), on the other hand is all cattle that took a vaccine that was availed in the study for a temporary immunity against bovine brucellosis. This immnunity period may keep the cattle unharmed but it is not life span the immunity may drop low sometimes and boosters may be required after sometime. Unlike the biosecured class, vaccinated cattle still have some potential for future susceptibility to infection, especially as the vaccine’s effectiveness decreases.

In the study of brucellosis dynamics using fractional derivatives, key assumptions are made regarding infection rates, control measures, and interaction terms. Infection rates are derived from empirical data reflecting species susceptibility and environmental factors influencing transmission27. Control measures, such as vaccination and culling strategies, are informed by historical intervention data to assess their effectiveness28. Interaction rates between species, particularly between cattle and potential rodent reservoirs, are established through ecological studies that quantify contact frequency and disease transmission likelihood29. The use of fractional derivatives allows for capturing complex dynamics and memory effects in disease transmission, enhancing model accuracy. Overall, these assumptions facilitate a robust framework for understanding and controlling brucellosis outbreaks in cattle populations. To represent the model mathematically, it is assumed that cattle are vaccinated at a rate of \(\gamma S\) and are recruited into the susceptible at a rate of \(\omega\). The animals with increased immunity return to the susceptible compartment at a rate of \(\tau V\). As a result of stringent biosecurity protocols, cattle migrate from the susceptible compartment to the recovered compartment at a rate of \(\upsilon S\). The bovine brucellosis virus spreads from infected animals to susceptible humans at a rate of \(\beta\). The sick animals get better and move to the compartment where \(\alpha I\) recovery is happening. Furthermore, a rate known as \(\gamma I\) is used to eliminate the infected people. The cattle’s recovery and subsequent return to the susceptible at a rate, \(\kappa R\), are understood to represent the rates of disease-induced and natural death, \(\mu\) and \(\psi\), respectively. The Caputo Fabrizio operator with fractional dimension \(0 < \eta \le 1\) was used to modify the brucellosis infection model with fractional differential.

$$\begin{aligned} \begin{array}{l} ^{CF}D_t^\eta S(t) = \omega - \beta S(t)I(t) - (\mu + \gamma + \upsilon )S(t) + \kappa R + \tau V,\\ ^{CF}D_t^\eta I(t) = \beta S(t)I(t) - (\mu + \psi + \alpha + \chi )I(t),\\ ^{CF}D_t^\eta R(t) = \alpha I - (\mu + \kappa )R,\\ ^{CF}D_t^\eta B(t) = \upsilon S - \mu B,\\ ^{CF}D_t^\eta V(t) = \gamma S - (\mu + \tau )V,\end{array} \end{aligned}$$
(8)

the initial conditions are

$$\begin{aligned} S(t) \ge 0,\;I(t) \ge 0,\;R(t) \ge 0,\;B(t) \ge 0,\; V(t) \ge 0. \end{aligned}$$

Positivity and boundedness

In this subsection, the positivity and boundedness of the solution for the proposed model (8) is given.

Theorem 3.1

For any given set of positive initial conditions S(0), I(0), R(0), B(0) and V(0), the variables of the model (8) remain positive and bounded.

Proof

From the system (8), we have

$$\begin{aligned} \begin{array}{lllllllllllllll}{{{\left. {^{CF}D_t^\eta S(t)} \right| }_{S = 0}} = \omega + \kappa R + \tau V \ge 0,}\\ {{{\left. {^{CF}D_t^\eta I(t)} \right| }_{I = 0}} = 0,}\\ {{{\left. {^{CF}D_t^\eta R(t)} \right| }_{R = 0}} = \alpha I \ge 0,}\\ {{{\left. {^{CF}D_t^\eta B(t)} \right| }_{B = 0}} = \upsilon S \ge 0,}\\ {{{\left. {^{CF}D_t^\eta V(t)} \right| }_{V = 0}} = \gamma S \ge 0.}\end{array} \end{aligned}$$
(9)

Therefore, the variables of the system (8) remain positive. Next, we prove the boundedness of the model (8) solutions. Consider the total number of population:

$$\begin{aligned} N(t) = S(t) + I(t) + R(t) + B(t) + V(t). \end{aligned}$$

Using the Caputo Fabrizio fractional derivative and the Caputo Fabrizio derivative linearity property, we get

$$\begin{aligned} ^{CF}D_t^\eta N(t)= & ^{CF}D_t^\eta S(t){ + ^{CF}}D_t^\eta I(t){ + ^{CF}}D_t^\eta R(t){ + ^{CF}}D_t^\eta B(t){ + ^{CF}}D_t^\eta V(t).\nonumber \\ ^{CF}D_t^\eta N(t)= & \omega - \beta S(t)I(t) - (\mu + \gamma + \upsilon )S(t) + \kappa R + \tau V\nonumber \\ & + \beta S(t)I(t) - (\mu + \psi + \alpha + \chi )I(t) + \alpha I - (\mu + \kappa )R\nonumber \\ & + \upsilon S - \mu B + \gamma S - (\mu + \tau )V\nonumber \\ ^{CF}D_t^\eta N(t)= & \omega - \mu N - (\psi + \chi )I \le \omega - \mu N,\nonumber \\ & ^{CF}D_t^\eta N(t) + \mu N \le \omega . \end{aligned}$$
(10)

We solve the Gronwall’s inequality (10) using the Laplace transform with initial conditions

$$\begin{aligned} & {\mathscr {L}}{\{ ^{CF}}D_t^\eta N\} + \mu {\mathscr {L}}\{ N\} \le {\mathscr {L}}\{ \omega \} ,\\ & \frac{{s {\mathscr {L}}\{ N\} - N(0)}}{{s + \eta (1 - s)}} + \mu {\mathscr {L}}\{ N\} \le \frac{\omega }{s},\\ & s {\mathscr {L}}\{ N\} - N(0) + (s + \eta (1 - s))\mu {\mathscr {L}}\{ N\} \le \frac{\omega }{s}(s + \eta (1 - s)),\\ & s {\mathscr {L}}\{ N\} + (s + \eta (1 - s))\mu {\mathscr {L}}\{ N\} \le \frac{\omega }{s}(s + \eta (1 - s)) + N(0),\\ & (s + (s + \eta (1 - s))\mu ){\mathscr {L}}\{ N\} \le \frac{\omega }{s}(s + \eta (1 - s)) + N(0),\\ & {\mathscr {L}}\{ N\} \le \frac{\omega }{{s(s + (s + \eta (1 - s))\mu )}}(s + \eta (1 - s)) + \frac{{N(0)}}{{(s + (s + \eta (1 - s))\mu )}}. \end{aligned}$$

After simplification and taking inverse Laplace transform we get

$$\begin{aligned} N(t) \le N(0){E_\eta }( - \mu {t^\eta }) + \int \limits _0^t {\omega {\sigma ^{\eta - 1}}{E_{\eta ,\eta }}( - \mu } {\sigma ^\eta })d\sigma . \end{aligned}$$

After performing a simple algebra, one may write

$$\begin{aligned} & N(t) \le N(0){E_\eta }( - \mu {t^\eta }) + \int \limits _0^t {\omega {\sigma ^{\eta - 1}}\sum \limits _{k = 0}^\infty {\frac{{{{( - 1)}^k}\mu {\sigma ^\eta }}}{{\Gamma (k\eta + \eta )}}} d\sigma } ,\\ & N(t) \le \frac{\omega }{\mu } + {E_\eta }( - \mu {t^\eta })\left( {N(0) - \frac{\omega }{\mu }} \right) . \end{aligned}$$

Thus if \(N(0) \le \frac{\omega }{\mu }\) then for \(t>0\),\(N(t) \le \frac{\omega }{\mu }\). This shows that the total population N(t) , that is, the subpopulations S(t) , I(t) , R(t) , B(t) and V(t), are bounded. This proves the boundedness of the solution of system (8). \(\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad \square\)

Model analysis

Equilibrium points of the proposed fractional order model (8) are obtained by solving the system

$$\begin{aligned} _0^{CF}D_t^\eta S = _0^{CF}D_t^\eta I = _0^{CF}D_t^\eta R = _0^{CF}D_t^\eta B = _0^{CF}D_t^\eta V = 0, \end{aligned}$$

and supposing that all infectious class sizes are equal to zero which means that there is no disease eradication in the studied population, thus the disease-free equilibrium state is:

$$\begin{aligned} {E_0}&= \left( {{S_0},\;{I_0},\;{R_0},\;{B_0},\;{V_0}} \right) \nonumber \\&= \left( {\frac{{\omega (\mu + \tau )}}{{\left( {\gamma \mu + \mu \tau + \mu \upsilon + \tau \upsilon + {\mu ^2}} \right) }},0,0,\frac{{\omega \upsilon (\mu + \tau )}}{{\mu \left( {\gamma \mu + \mu \tau + \mu \upsilon + \tau \upsilon + {\mu ^2}} \right) }},\frac{{\gamma \omega }}{{\left( {\gamma \mu + \mu \tau + \mu \upsilon + \tau \upsilon + {\mu ^2}} \right) }}} \right) ,\end{aligned}$$
(11)

To find the model endemic equilibrium point (8), where at least one disease class is not zero. To find the equilibrium values (\(E^*\)), the right-hand sides of the system (8) are set to zero, representing a steady-state condition. Let \({E^ * } = ({S^ * },\ {E^ * }, \ {R^ * },\ {B^ * } \ {V^ * })\) be the endemic equilibrium point of the model (8) and is given by

$$\begin{aligned} {S^ * }= & \frac{{(\alpha + \chi + \mu + \psi )}}{\beta },\\ {I^ * }= & - \left( \begin{array}{l}(\kappa + \mu )(\alpha {\mu ^2} + \chi {\mu ^2} + \gamma {\mu ^2} + {\mu ^2}\psi + {\mu ^2}\tau + {\mu ^2}\upsilon + {\mu ^3}\\ + \alpha \gamma \mu + \chi \gamma \mu - \beta \mu \omega + \alpha \mu \tau + \alpha \mu \upsilon + \chi \mu \tau + \gamma \mu \psi \\ - \beta \omega \tau + \mu \chi \upsilon + \alpha \tau \upsilon + \chi \tau \upsilon + \mu \psi \tau + \mu \psi \upsilon + \mu \tau \upsilon \\ + \psi \tau \upsilon \end{array} \right) \\ & \times \frac{1}{{\left( {\beta \mu + \beta \upsilon } \right) \left( {\chi \kappa + \alpha \mu + \chi \mu + \kappa \mu + \kappa \psi + \mu \psi + {\mu ^2}} \right) }},\\ {R^ * }= & - \left( \begin{array}{l}\alpha (\alpha {\mu ^2} + \chi {\mu ^2} + \gamma {\mu ^2} + {\mu ^2}\psi + {\mu ^2}\tau + {\mu ^2}\upsilon + {\mu ^3}\\ + \alpha \gamma \mu + \chi \gamma \mu - \beta \mu \omega + \alpha \mu \tau + \alpha \mu \upsilon + \chi \mu \tau + \gamma \mu \psi \\ - \beta \omega \tau + \mu \chi \upsilon + \alpha \tau \upsilon + \chi \tau \upsilon + \mu \psi \tau + \mu \psi \upsilon + \mu \tau \upsilon \\ + \psi \tau \upsilon \end{array} \right) \\ & \times \frac{1}{{\left( {\beta \mu + \beta \upsilon } \right) \left( {\chi \kappa + \alpha \mu + \chi \mu + \kappa \mu + \kappa \psi + \mu \psi + {\mu ^2}} \right) }},\\ {B^ * }= & \frac{{\upsilon (\alpha + \chi + \mu + \psi )}}{{\beta \mu }},\\ {V^ * }= & \frac{{\gamma (\alpha + \chi + \mu + \psi )}}{{\left( {\beta \mu + \beta \tau } \right) }}. \end{aligned}$$

Also from12 \({\mathscr {R}}_0\):

$$\begin{aligned} {\mathscr {R}}_0 = \frac{{\beta \omega \left( {\mu + \tau } \right) }}{{\left( {\mu + \psi + \alpha + \chi } \right) \left( {\mu \left( {\mu + \gamma + \tau } \right) + \upsilon \left( {\mu + \tau } \right) } \right) }}. \end{aligned}$$
(12)

In Fig.1(a),1(c) and Fig.1(f) \({\mathscr {R}}_0\) increased when \(\omega\), \(\beta\), \(\alpha\) and \(\psi\) increased. In Fig.1(d), Fig.1(f) \({\mathscr {R}}_0\) increased when \(\omega\) increased and \({\mathscr {R}}_0\) decreased when \(\omega\), \(\upsilon\) increased. In Fig.2(a) \({\mathscr {R}}_0\) decreased when \(\gamma\), \(\upsilon\) increased and Fig.2(b), 2(c) and 2(d) shows the results for \(\beta , \mu , \gamma\) and \(\psi\) respectively. In Fig.2(e) \({\mathscr {R}}_0\) decreased when \(\gamma\), \(\mu\) increased. In Fig.2(f) \({\mathscr {R}}_0\) increased when \(\tau\) increased and \({\mathscr {R}}_0\) increased when \(\gamma\) decreased.

Fig. 1
Fig. 1
Full size image

Parameters impact on basic reproductive number.

Fig. 2
Fig. 2
Full size image

Parameters impact on basic reproductive number.

Sensitivity analysis

For the relevant parameters, we can compute the partial derivative in order to examine the sensitivity of \({\mathscr {R}}_0\).

$$\begin{aligned} & \sum \limits _\beta ^{{{\mathscr {R}}_0}} { = \frac{\beta }{{\mathscr {R}}_0}\frac{{\partial {{\mathscr {R}}_0}}}{{\partial \beta }} = 1}> 0,\\ & \sum \limits _\omega ^{{{\mathscr {R}}_0}} { = \frac{\omega }{{\mathscr {R}}_0}\frac{{\partial {{\mathscr {R}}_0}}}{{\partial \omega }} = 1}> 0,\\ & \sum \limits _\mu ^{{{\mathscr {R}}_0}} { = \frac{\mu }{{\mathscr {R}}_0}\frac{{\partial {{\mathscr {R}}_0}}}{{\partial \mu }} = \left\{ {\frac{{ - \left( \begin{array}{l}\mu (\alpha {\mu ^2} + \chi {\mu ^2} - \gamma {\mu ^2} + \alpha {\tau ^2} + \chi {\tau ^2} + {\mu ^2}\psi - {\mu ^2}\upsilon + \psi {\tau ^2}\\ - {\tau ^2}\upsilon + \alpha \gamma \tau + \chi \gamma \tau + 2\alpha \mu \tau + 2\chi \mu \tau + \gamma \psi \tau + 2\mu \psi \tau - 2\mu \tau \upsilon )\end{array} \right) }}{{((\mu + \tau )(\alpha + \chi + \mu + \psi )(\gamma \mu + \mu \tau + \mu \upsilon + \tau \upsilon + {\mu ^2})}}} \right\} }< 0,\\ & \sum \limits _\tau ^{{{\mathscr {R}}_0}} { = \frac{\tau }{{\mathscr {R}}_0}\frac{{\partial {{\mathscr {R}}_0}}}{{\partial \tau }} = \frac{{\gamma \mu \tau }}{{(\mu + \tau )(\gamma \mu + \mu \tau + \mu \upsilon + \tau \upsilon + {\mu ^2})}}}> 0,\\ & \sum \limits _\gamma ^{{{\mathscr {R}}_0}} { = \frac{\gamma }{{\mathscr {R}}_0}\frac{{\partial {{\mathscr {R}}_0}}}{{\partial \gamma }} = \frac{{ - \gamma \mu }}{{(\upsilon (\mu + \tau ) + \mu (\gamma + \mu + \tau ))}}}< 0,\\ & \sum \limits _\upsilon ^{{{\mathscr {R}}_0}} { = \frac{\upsilon }{{\mathscr {R}}_0}\frac{{\partial {{\mathscr {R}}_0}}}{{\partial \upsilon }} = \frac{{ - \gamma (\mu + \tau )}}{{(\upsilon (\mu + \tau ) + \mu (\gamma + \mu + \tau ))}}} < 0,\\ & \sum \limits _\chi ^{{{\mathscr {R}}_0}} { = \frac{\chi }{{\mathscr {R}}_0}\frac{{\partial {{\mathscr {R}}_0}}}{{\partial \chi }} = \frac{\chi }{{(\alpha + \chi + \mu + \psi )}}}> 0,\\ & \sum \limits _\alpha ^{{{\mathscr {R}}_0}} { = \frac{\alpha }{{\mathscr {R}}_0}\frac{{\partial {{\mathscr {R}}_0}}}{{\partial \alpha }} = \frac{\alpha }{{(\alpha + \chi + \mu + \psi )}}}> 0,\\ & \sum \limits _\psi ^{{{\mathscr {R}}_0}} { = \frac{\psi }{{\mathscr {R}}_0}\frac{{\partial {{\mathscr {R}}_0}}}{{\partial \psi }} = \frac{\psi }{{(\alpha + \chi + \mu + \psi )}}} > 0, \end{aligned}$$

This investigation reveals that the sensitivity index could maintain consistent values based on certain factors, or it may not be affected by any independent factors. The significance of the parameters involved in \({\mathscr {R}}_0\) is represented in Fig.3 through the partial rank correlation coefficient (PRCC), and a detailed account of these findings is presented in Table 1.

Table 1 The \({\mathscr {R}}_0\) sensitivity indices in relation to the parameters.
Full size table
Fig. 3
Fig. 3
Full size image

Relevance of the parameters in \({\mathscr {R}}_0\) is determined by PRCC results.

The results derived from examining the PRCC data allow us to identify the crucial factors that significantly influence the potential for disease propagation, in addition to quantifying the scope and impact direction of these factors. Assessing the sensitivity of the initial infection rate (\({\mathscr {R}}_0\)) to modifications in individual parameters offers a broader understanding of the model’s effectiveness and the vital part played by diverse elements in facilitating the spread of Brucellosis in cattle populations.

Existence and uniqueness of the solution

Here, we demonstrate that the system has a unique solution. First written as follows is the system (8):

$$\begin{aligned} \begin{array}{lllllllllllllll}{ _0^{CF}D_t^{\eta }S(t) = {A_1}(t,\varpi )}\\ { _0^{CF}D_t^{\eta }I(t) = {A_2}(t,\varpi )}\\ { _0^{CF}D_t^{\eta }R(t) = {A_3}(t,\varpi )}\\ { _0^{CF}D_t^{\eta }B(t) = {A_4}(t, \varpi )}\\ { _0^{CF}D_t^{\eta }V(t) = {A_5}(t,\varpi ),}\end{array} \end{aligned}$$
(13)

where \(\varpi = S, I, R, B, V,\)

$$\begin{aligned} \begin{array}{lllllllllllllll}{{A_1}(t,\varpi ) = \omega - \beta S(t)I(t) - (\mu + \gamma + \upsilon )S(t) + \kappa R + \tau V}\\ {{A_2}(t,\varpi ) = \beta S(t)I(t) - (\mu + \psi + \alpha + \chi )I(t)}\\ {{A_3}(t,\varpi ) = \alpha I - (\mu + \kappa )R}\\ {{A_4}(t,\varpi ) = \upsilon S - \mu B}\\ {{A_5}(t,\varpi ) = \gamma S - (\mu + \tau )V.}\end{array} \end{aligned}$$
(14)

Integral form can be used to solve both sides of the aforementioned equations.

$$\begin{aligned} \left\{ {\begin{array}{lllllllllllllll}{S(t) - S(0) = \Phi \left( \eta \right) {A_1}(t,\varpi ) + \Theta (\eta )\int \limits _0^t {{A_1}(\sigma , \varpi )d\sigma ,} }\\ {I(t) - I(0) = \Phi \left( \eta \right) {A_2}(t, \varpi ) + \Theta (\eta )\int \limits _0^t {{A_2}(\sigma , \varpi )d\sigma ,} }\\ {R(t) - R(0) = \Phi \left( \eta \right) {A_3}(t, \varpi ) + \Theta (\eta )\int \limits _0^t {{A_2}(\sigma , \varpi )d\sigma ,} }\\ {B(t) - B(0) = \Phi \left( \eta \right) {A_4}(t, \varpi ) + \Theta (\eta )\int \limits _0^t {{P_4}(\sigma , \varpi )d\sigma ,} }\\ {V(t) - V(0) = \Phi \left( \eta \right) {A_5}(t, \varpi ) + \Theta (\eta )\int \limits _0^t {{A_5}(\sigma ,\varpi )d\sigma ,} }\end{array}} \right. \end{aligned}$$
(15)

where

\(\Phi \left( \eta \right) = \frac{{2(1 - \eta )}}{{(2 - \eta )M(\eta )}}\) and \(\Theta (\eta ) = \frac{{2\eta }}{{(2 - \eta )M(\eta )}}.\) We prove that \(A_i = 1, 2, 3, 4, 5\) kernels satisfy the Lipschitz condition and contraction.

Theorem 5.1

Assuming that the inequality that follows holds, the kernel \(A_1\) satisfies the contraction and the Lipschitz condition.

$$\begin{aligned} 0 \le \beta {k_1} + (\mu + \gamma + \upsilon ) < 1. \end{aligned}$$

Proof

For S and \(S_1\), we have

$$\begin{aligned} \left\| {{A_1}(t,S) - {A_1}(t,{S_1})} \right\|&= \left\| { - \beta I(t)\left( {S(t) - {S_1}(t)} \right) - (\mu + \gamma + \upsilon )\left( {S(t) - {S_1}(t)} \right) } \right\| \\&\le \left\| {\beta I(t)} \right\| \left\| {S(t) - {S_1}(t)} \right\| + (\mu + \gamma + \upsilon )\left\| {S(t) - {S_1}(t)} \right\| \\&\le \left( {\beta \left\| I \right\| + (\mu + \gamma + \upsilon )} \right) \left\| {S(t) - {S_1}(t)} \right\| \\&\le \left( {\beta {k_1} + (\mu + \gamma + \upsilon )} \right) \left\| {S(t) - {S_1}(t)} \right\| . \end{aligned}$$

Suppose that \({m_1} = \beta {k_1} + (\mu + \gamma + \upsilon ),\) where \(\left\| {{I}} \right\| \le {k_1},\) is bounded function. So

$$\begin{aligned} \left\| {{A_1}(t,S) - {A_1}(t,{S_1})} \right\| \le {m_1}\left\| {S(t) - {S_1}(t)} \right\| . \end{aligned}$$
(16)

Thus, if \(0 \le \beta {k_1} + (\mu + \gamma + \upsilon ) < 1\), then \(A_1\) is a contraction and it satisfies the Lipschitz condition. Similarly, \(A_i, i = 2, 3, 4, 5\) satisfy the Lipschitz conditions:

$$\begin{aligned} \left\{ \begin{array}{l}\left\| {{A_2}(t,I) - {A_2}(t,{I_1})} \right\| \le {m_2}\left\| {I(t) - {I_1}(t)} \right\| ,\\ \left\| {{A_3}(t,R) - {A_3}(t,{R_1})} \right\| \le {m_3}\left\| {R(t) - {R_1}(t)} \right\| ,\\ \left\| {{A_4}(t,B) - {A_4}(t,B)} \right\| \le {m_4}\left\| {B(t) - B(t)} \right\| ,\\ \left\| {{A_5}(t,V) - {A_5}(t,V)} \right\| \le {m_5}\left\| {V(t) - V(t)} \right\| ,\end{array} \right. \end{aligned}$$

where \({m_2} = \beta {k_2} + (\mu + \psi + \alpha + \chi ),\) \({m_3} = (\mu + \kappa ),\) \({m_4} = \mu\) and \({m_5} = (\mu + \tau )\) are bounded functions. Under the condition that \(0 \le m_i <1,\) \(A_i\) are contractions for \(i = 2, 3, 4, 5\).

Considering the first equation from equation (15), the following recursive forms should be taken into account:

$$\begin{aligned} \left\{ {\begin{array}{l}\begin{array}{l}{\varphi _{1n}}(t) = {S_n}(t) - {S_{n - 1}}(t) = \Phi (\eta )\left( {{A_1}(t,{S_{n - 1}}) - {A_1}(t,{S_{n - 2}})} \right) \\ + \Theta (\eta )\int \limits _0^t {\left( {{A_1}(\sigma ,{S_{n - 1}}) - {A_1}(\sigma ,{S_{n - 2}})} \right) } d\sigma ,\end{array}\end{array}} \right. \end{aligned}$$

with the initial condition \({S_0}(t) = S(0)\). The norm in the equation above is defined by

$$\begin{aligned} \left\| {{\varphi _{1n}}(t)} \right\|&= \left\| {{S_n}(t) - {S_{n - 1}}(t)} \right\| \\&\le \left\| {\Phi (\eta )\left( {{A_1}(t,{S_{n - 1}}) - {A_1}(t,{S_{n - 2}})} \right) } \right\| \\&\quad + \Theta (\eta )\int \limits _0^t {\left\| {\left( {{A_1}(\sigma ,{S_{n - 1}}) - {A_1}(\sigma ,{S_{n - 2}})} \right) } \right\| } d\sigma , \end{aligned}$$

Lipschitz condition (16) gives us

$$\begin{aligned} \left\| {{\varphi _{1n}}(t)} \right\| \le \Phi (\eta ){m_1}\left\| {{\varphi _{1(n - 1)}}(t)} \right\| + \Theta (\eta ){m_1}\int \limits _0^t {\left\| {{\varphi _{1(n - 1)}}(\sigma )} \right\| } d\sigma . \end{aligned}$$
(17)

Similar to this, we get

$$\begin{aligned} \begin{array}{lllllllllllllll}{\left\| {{\varphi _{2n}}(t)} \right\| \le \Phi (\eta ){m_2}\left\| {{\varphi _{2(n - 1)}}(t)} \right\| + \Theta (\eta ){m_2}\int \limits _0^t {\left\| {{\varphi _{2(n - 1)}}(\sigma )} \right\| } d\sigma ,}\\ {\left\| {{\varphi _{3n}}(t)} \right\| \le \Phi (\eta ){m_3}\left\| {{\varphi _{3(n - 1)}}(t)} \right\| + \Theta (\eta ){m_3}\int \limits _0^t {\left\| {{\varphi _{3(n - 1)}}(\sigma )} \right\| } d\sigma ,}\\ {\left\| {{\varphi _{4n}}(t)} \right\| \le \Phi (\eta ){m_4}\left\| {{\varphi _{4(n - 1)}}(t)} \right\| + \Theta (\eta ){m_4}\int \limits _0^t {\left\| {{\varphi _{4(n - 1)}}(\sigma )} \right\| } d\sigma ,}\\ {\left\| {{\varphi _{5n}}(t)} \right\| \le \Phi (\eta ){m_5}\left\| {{\varphi _{5(n - 1)}}(t)} \right\| + \Theta (\eta ){m_5}\int \limits _0^t {\left\| {{\varphi _{5(n - 1)}}(\sigma )} \right\| } d\sigma .}\end{array} \end{aligned}$$
(18)

Hence, we can state that

$$\begin{aligned} {S_n}(t) = \sum \limits _{j = 1}^n {{\varphi _{1j}}(t),} {I_n}(t) = \sum \limits _{j = 1}^n {{\varphi _{2j}}(t),} {R}_n(t) = \sum \limits _{j = 1}^n {{\varphi _{3j}}(t),} {B}_n(t) = \sum \limits _{j = 1}^n {{\varphi _{4j}}(t),} {V_n}(t) = \sum \limits _{j = 1}^n {{\varphi _{5j}}(t).} \end{aligned}$$

The following theorem shows that there is a solution. \(\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\square\)

Theorem 5.2

If \(\xi _1\) exists such that \(\Phi (\eta ){m_i} + \Theta (\eta ){\xi _1}{m_i} < 1\), then the fractional model (8) has a system of solutions.

Proof

Given the recursive method and equations (17) and (18), we deduce that

$$\begin{aligned} \begin{array}{lllllllllllllll}{\left\| {{\varphi _{1n}}(t)} \right\| \le \left\| {{S_n}(0)} \right\| {{\left[ {\Phi (\eta ){m_1} + \Theta (\eta )\xi {m_1}} \right] }^n},}\\ {\left\| {{\varphi _{2n}}(t)} \right\| \le \left\| {{E_n}(0)} \right\| {{\left[ {\Phi (\eta ){m_2} + \Theta (\eta )\xi {m_2}} \right] }^n},}\\ {\left\| {{\varphi _{3n}}(t)} \right\| \le \left\| {{I_{rn}}(0)} \right\| {{\left[ {\Phi (\eta ){m_3} + \Theta (\eta )\xi {m_3}} \right] }^n},}\\ {\left\| {{\varphi _{4n}}(t)} \right\| \le \left\| {{I_{sn}}(0)} \right\| {{\left[ {\Phi (\eta ){m_4} + \Theta (\eta )\xi {m_4}} \right] }^n},}\\ {\left\| {{\varphi _{5n}}(t)} \right\| \le \left\| {{R_n}(0)} \right\| {{\left[ {\Phi (\eta ){m_5} + \Theta (\eta )\xi {m_5}} \right] }^n}.}\end{array} \end{aligned}$$

This suggests that the system is continuous and has a solution. Currently, we show how the model solution (15) is produced by the previously mentioned functions. In light of that

$$\begin{aligned} \begin{array}{l}{S(t) - S(0) = {S_n}(t) - {\Upsilon _{1n}}(t).} \end{array} \end{aligned}$$

So

$$\begin{aligned} \left\| {{\Upsilon _{1n}}(t)} \right\|&= \left\| \begin{array}{l}\Phi (\eta )\left( {{A_1}(t,{S_{n - 1}}) - {A_1}(t,{S_{n - 2}})} \right) \\ + \Theta (\eta )\int \limits _0^t {\left( {{A_1}(\sigma ,S) - {A_1}(\sigma ,{S_{n - 1}})} \right) d\sigma } \end{array} \right\| \\&\le \Phi (\eta ){m_1}\left\| {S - {S_{n - 1}}} \right\| + \Theta (\eta ){m_1}\left\| {S - {S_{n - 1}}} \right\| \xi . \end{aligned}$$

A repetition of the process yields

$$\begin{aligned} \left\| {{\Upsilon _{1n}}(t)} \right\| \le {\left[ {\Phi (\eta ) + \Theta (\eta )\xi } \right] ^{n + 1}}m_1^{n + 1}b. \end{aligned}$$

At \(\xi _1 ,\) we get

$$\begin{aligned} \left\| {{\Upsilon _{1n}}(t)} \right\| \le {\left[ {\Phi (\eta ) + \Theta (\eta ){\xi _1}} \right] ^{n + 1}}m_1^{n + 1}b. \end{aligned}$$

We obtain \(\left\| {{\Upsilon _{in}}(t)} \right\| \rightarrow 0,i = 2,3,4,5\) by limiting the current equation as n approaches \(\infty\). In the same way, we could show that \(\left\| {{\Upsilon _{in}}(t)} \right\| \rightarrow 0,i = 2,3,4,5.\) This completes the proof.

Assuming that there is another possible solution for the system, such as \(S_1(t),\) \(I_1(t),\) \(R_1(t),\) \(R_1(t),\) \(B_1(t),\) and \(V_1(t),\) then we have

$$\begin{aligned} S(t) - {S_1}&= \Phi (\eta )\left( {{A_1}(t,{S_{n - 1}}) - {A_1}(t,{S_{n - 2}})} \right) \\&\quad + \Theta (\eta )\int \limits _0^t {\left( {{A_1}(\sigma ,S) - {A_1}(\sigma ,{S_{n - 1}})} \right) d\sigma } .\end{aligned}$$

We make use of the equation mentioned above’s norm.

$$\begin{aligned} \left\| {S(t) - {S_1}(t)} \right\|&\le \Phi (\eta )\left\| {{A_1}(t,S) - {A_1}(t,{S_1})} \right\| \\&\quad + \Theta (\eta )\int \limits _0^t {\left\| {\left( {{A_1}(\sigma ,S) - {A_1}(\sigma ,{S_1})} \right) } \right\| d\sigma } . \end{aligned}$$

Lipschitz condition (16) implies that

$$\begin{aligned} \left\| {S(t) - {S_1}} \right\| \le \Phi (\eta ){m_1}\left\| {{A_1}(t,S) - {A_1}(t,{S_1})} \right\| + \Theta (\eta ){m_1}\xi \left\| {S(t) - {S_1}} \right\| . \end{aligned}$$

Thus

$$\begin{aligned} \left\| {S(t) - {S_1}} \right\| \left( {1 - \Phi (\eta ){m_1} - \Theta (\eta ){m_1}\xi } \right) \le 0. \end{aligned}$$
(19)

\(\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad \square\)

Theorem 5.3

If the conditions listed below are satisfied, then the model (8) solution is distinct.

$$\begin{aligned} 1 - \Phi (\eta ){m_1} - \Theta (\eta ){m_1}\xi > 0. \end{aligned}$$

Proof

Since \(|S(t) - {S_1}(t)| = 0,\) presuming condition (19) is met. Consequently, we get \(S(t) = S_1(t)\). For I,  R,  B,  and V, an analogous equality can be shown. \(\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad \square\)

Stability analysis

The stability of an epidemic model determines whether the disease will die out or persist over time. Typically, this involves analyzing the system’s equilibrium points disease-free and endemic and assessing their stability using methods such as eigenvalue analysis or Lyapunov functions. If the disease-free equilibrium is stable, the infection will eventually vanish; if the endemic equilibrium is stable, the infection will persist at a constant level within the population.

Local stability at disease free point

Theorem 6.1

At the DFE point \(E_0\), the system is locally asymptotically stable provided

$$\begin{aligned} \beta S_0 < (\mu + \psi + \alpha + \chi ). \end{aligned}$$

Proof

The jacobian matrix for the system (8) at the DFE \(E_0 (S_0, E_0, R_0, B_0, V_0)\) is given by

$$\begin{aligned} J = \begin{bmatrix} - (\mu + \gamma + \epsilon ) & -\beta S_0 & \kappa & 0 & \tau \\ 0 & \beta S_0 - (\mu + \psi + \alpha + \chi ) & 0 & 0 & 0 \\ 0 & \alpha & -(\mu + \kappa ) & 0 & 0 \\ \epsilon & 0 & 0 & -\mu & 0 \\ \gamma & 0 & 0 & 0 & -(\mu + \tau ) \end{bmatrix} \end{aligned}$$
(20)

The characteristic equation of (20) given by \(\left| {J({E_0}) - \lambda I} \right| = 0,\) The roots of the characteristic equation are \(\lambda _1 = \beta S - (\mu + \psi + \alpha + \chi )\), \(\lambda _2 = -(\mu + \kappa )\) and \(\lambda _3 = -\mu\). Here, the roots \(\lambda _2\), \(\lambda _3\) are negative and the root \(\lambda _1\) is negative if \(\beta S_0 < (\mu + \psi + \alpha + \chi )\) and the remaining eigenvalues are those of the following submatrix:

$$\begin{aligned} J^1(E_0)= & \begin{bmatrix} -(\mu + \gamma + \epsilon ) & \tau \\ \gamma & -(\mu + \tau ) \end{bmatrix} \end{aligned}$$
(21)
$$\begin{aligned} \lambda _4= & -\frac{\epsilon }{2} - \frac{\gamma }{2} - \mu - \frac{\tau }{2} - \frac{\sqrt{\epsilon ^2 + 2\epsilon \gamma - 2\epsilon \tau + \gamma ^2 + 2\gamma \tau + \tau ^2}}{2} \end{aligned}$$
(22)
$$\begin{aligned} \lambda _5= & -\frac{\epsilon }{2} - \frac{\gamma }{2} - \mu - \frac{\tau }{2} +\frac{\sqrt{\epsilon ^2 + 2\epsilon \gamma - 2\epsilon \tau + \gamma ^2 + 2\gamma \tau + \tau ^2}}{2} \end{aligned}$$
(23)

Clearly all eigenvalues are negative. Thus, at the DFE point \(E_0\), the system is locally asymptotically stable provided

$$\begin{aligned} \beta S_0 < (\mu + \psi + \alpha + \chi ). \end{aligned}$$

Global stability at endemic equilibrium point

Theorem 6.2

At the EF point \(E^*\), the system is the system is globally asymptotically stable.

Proof

Let the Lyapunov function be

$$\begin{aligned} V(S,I,R,B,V) = {(S - {S^*})^2} + {(I - {I^*})^2} + {(R - {R^*})^2} + {(B - {B^*})^2}+{(V - {V^*})^2}. \end{aligned}$$
(24)

Using the linearity property of the Caputo Fabrizio operator

$$\begin{aligned} _0^{CF}D_t^\eta V(S,I,R,B,V)= & _0^{CF}D_t^\eta {(S - {S^*})^2} + _0^{CF}D_t^\eta {(I - {I^*})^2} + _0^{CF}D_t^\eta {(R - {R^*})^2} \nonumber \\ & + _0^{CF}D_t^\eta {(B - {B^*})^2} + _0^{CF}D_t^\eta {(V - {V^*})^2} . \end{aligned}$$
(25)

Using Lemma 1, we get

$$\begin{aligned} _0^{CF}D_t^\eta V(S,I,R,V,B)\le & 2(S - {S^*}) _0^{CF}D_t^\eta S + 2(I - {I^*}) _0^{CF}D_t^\eta I + 2(R - {R^*}) _0^{CF}D_t^\eta R\\ & + 2(B - {B^*}) _0^{CF}D_t^\eta B+ 2(V - {V^*}) _0^{CF}D_t^\eta V . \end{aligned}$$

By proper choice of

$$\begin{aligned} & {\omega - \beta SI + \kappa R + \tau V = (\mu + \gamma + \upsilon ){S^*},\,\beta SI = (\mu + \psi + \alpha + \chi ){I^*}},\\ & {\alpha I = (\mu + \kappa ){R^*},\upsilon S = \mu {B^*},\gamma S = (\mu + \tau ){V^*}} \end{aligned}$$

from (8), \(_0^{CF}D_t^\eta V(S,I,R,B,V)\) becomes

$$\begin{aligned} _0^{CF}D_t^\eta V(S,E,R,V,B)&\le 2(S - {S^*})(\mu + \gamma + \upsilon )({S^*} - S) + 2(I - {I^*})(\mu + \psi + \alpha + \chi )({I^*} - I)\\&\quad + 2(R - {R^*})(\mu + \kappa )({R^*} - R) + 2(B - {B^*})\mu ({B^*} - B) + 2(V - {V^*})(\mu + \tau )({V^*} - V). \end{aligned}$$

That is,

$$\begin{aligned} _0^{CF}D_t^\eta V(S,E,R,V,B)&\le - 2{(S - {S^*})^2}(\mu + \gamma + \upsilon ) - 2{(I - {I^*})^2}(\mu + \psi + \alpha + \chi )\\&\quad - 2{(R - {R^*})^2}(\mu + \kappa ) - 2{(B - {B^*})^2}\mu - 2{(V - {V^*})^2}(\mu + \tau ) \le 0.\end{aligned}$$

Hence, at its EE point, the system is globally asymptotically stable. \(\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad \square\)

Iterative scheme for caputo fabrizio with stability results

When both sides of model (8) are subjected to the Laplace transform, the ensuing system results:

$$\begin{aligned} \begin{array}{lllllllllllllll}{{{{\mathscr {L}}}}\left\{ {^{CF}D_t^\eta S(t)} \right\} = {{{\mathscr {L}}}}\left\{ {A_1 (t, \varpi } \right\} ,}\\ {{{\mathscr {L}}}\left\{ {^{CF}D_t^\eta I(t)} \right\} = {{{\mathscr {L}}}}\left\{ {A_2(t, \varpi ) } \right\} ,}\\ {{{{\mathscr {L}}}}\left\{ {^{CF}D_t^\eta R(t)} \right\} = {{{\mathscr {L}}}}\left\{ {A_3(t, \varpi )} \right\} ,}\\ {{{{\mathscr {L}}}}\left\{ {^{CF}D_t^\eta B(t)} \right\} = {{{\mathscr {L}}}}\left\{ {A_4(t, \varpi )} \right\} ,}\\ {{{{\mathscr {L}}}}\left\{ {^{CF}D_t^\eta V(t)} \right\} = {{\mathscr {L}}}\left\{ {A_5(t, \varpi )} \right\} ,}\end{array} \end{aligned}$$
(26)

With the initial conditions, we obtain

$$\begin{aligned} \begin{array}{lllllllllllllll}{{{{\mathscr {L}}}}\left\{ {S(t)} \right\} = \frac{{{S_0}}}{s} + \left[ {\frac{{s + \eta (1 - s)}}{s}{{\mathscr {L}}}\left\{ {A_1(t, \varpi )} \right\} } \right] ,}\\ {{{{\mathscr {L}}}}\left\{ {I(t)} \right\} = \frac{{{I_0}}}{s} + \left[ {\frac{{s + \eta (1 - s)}}{s}{{{\mathscr {L}}}}\left\{ {A_2(t, \varpi )} \right\} } \right] ,}\\ \begin{array}{l}{{{\mathscr {L}}}}\left\{ {R(t)} \right\} = \frac{{{R_0}}}{s} + \left[ {\frac{{s + \eta (1 - s)}}{s}{{\mathscr {L}}}\left\{ {A_3(t, \varpi )} \right\} } \right] ,\\ {{{\mathscr {L}}}}\left\{ {B(t)} \right\} = \frac{{{B_0}}}{s} + \left[ {\frac{{s + \eta (1 - s)}}{s}{{{\mathscr {L}}}}\left\{ {A_4(t, \varpi )} \right\} } \right] ,\end{array}\\ {{{{\mathscr {L}}}}\left\{ {V(t)} \right\} = \frac{{{V_0}}}{s} + \left[ {\frac{{s + \eta (1 - s)}}{s}{{\mathscr {L}}}\left\{ {A_5(t, \varpi )} \right\} } \right] .}\end{array} \end{aligned}$$
(27)

Remember that is providing the solutions as infinite series, with S(t) , I(t), R(t) , B(t) , and V(t)

$$\begin{aligned} S(t) = \sum \limits _{v = 0}^\infty {{S_v}} , \hspace{1em}I(t) = \sum \limits _{v = 0}^\infty {{I_v}} , \hspace{1em}C(t) = \sum \limits _{v = 0}^\infty {{R_v}} , \hspace{1em}B(t) = \sum \limits _{v = 0}^\infty {{B_v}} , \hspace{1em}V(t) = \sum \limits _{v= 0}^\infty {{V_v}} , \end{aligned}$$
(28)

we resolve nonlinear terms as follows:

$$\begin{aligned} S(t)I(t) = \sum \limits _{v = 0}^\infty {{G_v}} . \end{aligned}$$
(29)

Substituting (28) and (29) into (27) results in

$$\begin{aligned} \begin{array}{lllllllllllllll}{{{{\mathscr {L}}}}\left\{ {\sum \limits _{v = 0}^\infty {{S_v}} } \right\} = \frac{{{S_0}}}{s} + \left[ {\frac{{s + \eta (1 - s)}}{s}{{{\mathscr {L}}}}\left\{ {\omega - \beta \sum \limits _{v = 0}^\infty {{G_v}} - (\mu + \gamma + \upsilon )\sum \limits _{v = 0}^\infty {{S_v}} + \kappa \sum \limits _{v = 0}^\infty {{R_v}} + \tau \sum \limits _{v= 0}^\infty {{V_v}} } \right\} } \right] ,}\\ {{{{\mathscr {L}}}}\left\{ {\sum \limits _{v = 0}^\infty {{I_v}} } \right\} = \frac{{{I_0}}}{s} + \left[ {\frac{{s + \eta (1 - s)}}{s}{{{\mathscr {L}}}}\left\{ {\beta \sum \limits _{v = 0}^\infty {{G_v}} - (\mu + \psi + \alpha + \chi )\sum \limits _{v = 0}^\infty {{I_v}} } \right\} } \right] ,}\\ {{{{\mathscr {L}}}}\left\{ {\sum \limits _{v = 0}^\infty {{R_v}} } \right\} = \frac{{{R_0}}}{s} + \left[ {\frac{{s + \eta (1 - s)}}{s}{{{\mathscr {L}}}}\left\{ {\alpha \sum \limits _{v= 0}^\infty {{I_v}} - (\mu + \kappa )\sum \limits _{v = 0}^\infty {{R_v}} } \right\} } \right] ,}\\ \begin{array}{l}{{{\mathscr {L}}}}\left\{ {\sum \limits _{v= 0}^\infty {{B_v}} } \right\} = \frac{{{B_0}}}{s} + \left[ {\frac{{s + \eta (1 - s)}}{s}{{{\mathscr {L}}}}\left\{ {\upsilon \sum \limits _{v = 0}^\infty {{S_v}} - \mu \sum \limits _{v = 0}^\infty {{B_v}} } \right\} } \right] ,\\ {{{\mathscr {L}}}}\left\{ {\sum \limits _{v= 0}^\infty {{V_v}} } \right\} = \frac{{{B_0}}}{s} + \left[ {\frac{{s + \eta (1 - s)}}{s}{{{\mathscr {L}}}}\left\{ {\gamma \sum \limits _{v = 0}^\infty {{S_v}} - (\mu + \tau )\sum \limits _{v = 0}^\infty {{V_v}} } \right\} } \right] .\end{array}\end{array} \end{aligned}$$
(30)

Theorem 6.3

Let \(({\Xi },\left| . \right| )\) be Banach space and \(\wp :\Xi \rightarrow \Xi\) be a map satisfying:

$$\begin{aligned} \left\| {{\wp _m} - {\wp _n}} \right\| \le \Psi \left\| {m - {\wp _m}} \right\| + \iota \left\| {m - n} \right\| , \end{aligned}$$

for all \(m,n \in {\Xi }\), where \(0 \le \iota\), \(0 \le \iota < 1\). In that case, \(\wp\) is Picard \(\wp\)-stable.

Theorem 6.4

Given the definition of a self map as follows, let \(\wp\)

$$\begin{aligned} \begin{array}{lllllllllllllll}{\wp \left[ {{S_v}(t)} \right] = {S_{v + 1}}(t) = {S_\delta }(0) + {{{{\mathscr {L}}}}^{ - 1}}\left[ {\frac{{s + \eta (1 - s)}}{s}{{{\mathscr {L}}}}\left\{ {\omega - \beta {S_v}{I_v} - (\mu + \gamma + \upsilon ){S_v} + \kappa {R_v} + \tau {V_v}} \right\} } \right] ,}\\ {\wp \left[ {{I_v}(t)} \right] = {I_{v + 1}}(t) = {I_\delta }(0) + {{{{\mathscr {L}}}}^{ - 1}}\left[ {\frac{{s + \eta (1 - s)}}{s}{{{\mathscr {L}}}}\left\{ {\beta {S_v}{I_v} - (\mu + \psi + \alpha + \chi ){I_v}} \right\} } \right] ,}\\ \begin{array}{l}\wp \left[ {{R_v}(t)} \right] = {R_{v + 1}}(t) = {R_\delta }(0) + {{{{\mathscr {L}}}}^{ - 1}}\left[ {\frac{{s + \eta (1 - s)}}{s}{{{\mathscr {L}}}}\left\{ {\alpha {I_v} - (\mu + \kappa ){R_v}} \right\} } \right] ,\\ {\wp } \left[ {{B_v}(t)} \right] = {B_{v + 1}}(t) = {B_\delta }(0) + {{{{\mathscr {L}}}}^{ - 1}}\left[ {\frac{{s + \eta (1 - s)}}{s}{{{\mathscr {L}}}}\left\{ {\upsilon {S_v} - \mu {B_v}} \right\} } \right] ,\end{array}\\ {\wp \left[ {{V_v}(t)} \right] = {V_{v + 1}}(t) = {V_\delta }(0) + {{{{\mathscr {L}}}}^{ - 1}}\left[ {\frac{{s + \eta (1 - s)}}{s}{{{\mathscr {L}}}}\left\{ {\gamma {S_v} - (\mu + \tau ){V_v}} \right\} } \right] ,}\end{array} \end{aligned}$$
(31)

Then, the iteration is \(\wp\)-stable in \({L^1}(m,n)\) if the following statements are achieved:

$$\begin{aligned} \begin{array}{lllllllllllllll}{\left( {1 - \beta ({N_1} + {N_2}){f_1}(\vartheta ) - (\mu + \gamma + \upsilon ){f_2}(\vartheta ) + \kappa {f_3}(\vartheta ) + \tau {f_4}(\vartheta )} \right)< 1,}\\ {\left( {1 + \beta ({N_1} + {N_2}){f_1}(\vartheta ) - (\mu + \psi + \alpha + \chi ){f_5}(\vartheta )} \right)< 1,}\\ {\left( {1 + \alpha {f_6}(\vartheta ) - (\mu + \kappa ){f_7}(\vartheta )} \right)< 1,}\\ \begin{array}{l}\left( {1 + \upsilon {f_8}(\vartheta ) - \mu {f_9}(\vartheta )} \right)< 1,\\ \left( {1 + \gamma {f_{10}}(\vartheta ) - (\mu + \tau ){f_{11}}{{(\vartheta )}}} \right) < 1.\end{array}\end{array} \end{aligned}$$
(32)

Proof

Using the following evaluation for \((v,u) \in M \times M\), we demonstrated the fixed point of wp:

$$\begin{aligned} \begin{array}{l}\wp \left[ {{S_v}(t)} \right] - \wp \left[ {{S_u}(t)} \right] = {S_v}(t) - {S_u}(t) + {{{{\mathscr {L}}}}^{ - 1}}\left[ {\frac{{s + \eta (1 - s)}}{s}{{{\mathscr {L}}}}\left\{ {\omega - \beta {S_v}{I_v} - (\mu + \gamma + \upsilon ){S_v} + \kappa {R_v} + \tau {V_v}} \right\} } \right] \\ \hspace{10em} - {{{{\mathscr {L}}}}^{ - 1}}\left[ {\frac{{s + \eta (1 - s)}}{s}{{{\mathscr {L}}}}\left\{ {\omega - \beta {S_u}{I_u} - (\mu + \gamma + \upsilon ){S_u} + \kappa {R_u} + \tau {V_u}} \right\} } \right] ,\\ {\wp }\left[ {{I_v}(t)} \right] - \wp \left[ {{I_u}(t)} \right] = {I_v}(t) - {I_u}(t) + {{{{\mathscr {L}}}}^{ - 1}}\left[ {\frac{{s + \eta (1 - s)}}{s}L\left\{ {\beta {S_v}{I_v} - (\mu + \psi + \alpha + \chi ){I_v}} \right\} } \right] \\ \hspace{10em} - {{{{\mathscr {L}}}}^{ - 1}}\left[ {\frac{{s + \eta (1 - s)}}{s}{{{\mathscr {L}}}}\left\{ {\beta {S_u}{I_u} - (\mu + \psi + \alpha + \chi ){I_u}} \right\} } \right] ,\\ \wp \left[ {{R_v}(t)} \right] - K\left[ {{R_u}(t)} \right] = {R_v}(t) - {R_u}(t) + {{{{\mathscr {L}}}}^{ - 1}}\left[ {\frac{{s + \eta (1 - s)}}{s}{{{\mathscr {L}}}}\left\{ {\alpha {I_v} - (\mu + \kappa ){R_v}} \right\} } \right] \\ \hspace{10em} - {{{{\mathscr {L}}}}^{ - 1}}\left[ {\frac{{s + \eta (1 - s)}}{s}{{{\mathscr {L}}}}\left\{ {\alpha {I_u} - (\mu + \kappa ){R_u}} \right\} } \right] ,\\ {\wp }\left[ {{B_v}(t)} \right] - \wp \left[ {{B_u}(t)} \right] = {B_v}(t) - {B_u}(t) + {{{{\mathscr {L}}}}^{ - 1}}\left[ {\frac{{s + \eta (1 - s)}}{s}{{{\mathscr {L}}}}\left\{ {\upsilon {S_v} - \mu {B_v}} \right\} } \right] \\ \hspace{10em} - {{{{\mathscr {L}}}}^{ - 1}}\left[ {\frac{{s + \eta (1 - s)}}{s}{{{\mathscr {L}}}}\left\{ {\upsilon {S_u} - \mu {B_u}} \right\} } \right] ,\\ {\wp }\left[ {{V_v}(t)} \right] - \wp \left[ {{V_u}(t)} \right] = {V_v}(t) - {V_u}(t) + {{{{\mathscr {L}}}}^{ - 1}}\left[ {\frac{{s + \eta (1 - s)}}{s}{{{\mathscr {L}}}}\left\{ {\gamma {S_v} - (\mu + \tau ){V_v}} \right\} } \right] \\ \hspace{10em}- {{{{\mathscr {L}}}}^{ - 1}}\left[ {\frac{{s + \eta (1 - s)}}{s}{{{\mathscr {L}}}}\left\{ {\gamma {S_v} - (\mu + \tau ){V_v}} \right\} } \right] .\end{array} \end{aligned}$$
(33)

By calculating the norm of both sides of the first equation of (33) we obtain

$$\begin{aligned} \begin{array}{lllllllllllllll}{\left\| {\wp \left( {{S_v}(t)} \right) - \wp \left( {{S_u}(t)} \right) } \right\| = \left\| {{S_v}(t) - {S_u}(t) + {{{{\mathscr {L}}}}^{ - 1}}\left[ {\frac{{s + \eta (1 - s)}}{s}{{{\mathscr {L}}}}\left\{ {\omega - \beta {S_v}{I_v} - (\mu + \gamma + \upsilon ){S_v} + \kappa {R_v} + \tau {V_v}} \right\} } \right] } \right. }\\ {\left. { - {{{{\mathscr {L}}}}^{ - 1}}\left[ {\frac{{s + \eta (1 - s)}}{s}{{{\mathscr {L}}}}\left\{ {\omega - \beta {S_u}{I_u} - (\mu + \gamma + \upsilon ){S_u} + \kappa {R_u} + \tau {V_u}} \right\} } \right] } \right\| ,}\end{array} \end{aligned}$$
(34)

using triangular inequality and simplifying (34), we get

$$\begin{aligned} \begin{array}{lllllllllllllll}{\left\| {\wp \left( {{S_v}(t)} \right) - \wp \left( {{S_u}(t)} \right) } \right\| \le \left\| {{S_v}(t) - {S_u}(t)} \right\| + {{{{\mathscr {L}}}}^{ - 1}}\left[ {\frac{{s + \eta (1 - s)}}{s}{{{\mathscr {L}}}}\left[ {\left\| { - \beta {S_u}({I_v} - {I_u})} \right\| + } \right. } \right. \left\| { - \beta {I_u}({S_v} - {S_u})} \right\| }\\ { + \left\| { - (\mu + \gamma + \upsilon )({S_v} - {S_u})} \right\| + \left\| {\kappa ({R_v} - {R_u})} \right\| + \left. {\left. {\left\| {\tau ({V_v} - {V_u})} \right\| } \right] } \right] .} \end{array} \end{aligned}$$
(35)

We allowed for the relative influence of both solutions:

$$\begin{aligned} \left\| {{S_v}(t) - {S_u}(t)} \right\| \cong \left\| {{I_v}(t) - {I_u}(t)} \right\| \cong \left\| {{R_v}(t) - {R_u}(t)} \right\| \cong \left\| {{B_v}(t) - {B_u}(t)} \right\| \cong \left\| {{V_v}(t) - {V_u}(t)} \right\| . \end{aligned}$$
(36)

If we replace this in (36), we obtain the relation shown below:

$$\begin{aligned} \begin{array}{lllllllllllllll}{\left\| {\wp K\left( {{S_v}(t)} \right) - \wp \left( {{S_u}(t)} \right) } \right\| \le \left\| {{S_v}(t) - {S_u}(t)} \right\| + {{{{\mathscr {L}}}}^{ - 1}}\left[ {\frac{{p + \eta (1 - p)}}{p}{{{\mathscr {L}}}}\left[ {\left\| { - \beta {S_u}({I_v} - {I_u})} \right\| + } \right. } \right. \left\| { - \beta {I_u}({S_v} - {S_u})} \right\| }\\ { + \left\| { - (\mu + \gamma + \upsilon )({S_v} - {S_u})} \right\| + \left\| {\kappa ({S_v} - {S_u})} \right\| + \left. {\left. {\left\| {\tau ({S_v} - {S_u})} \right\| } \right] } \right] .}\\ \end{array} \end{aligned}$$
(37)

Also the convergent sequence \(S_v\), \(I_v\), \(R_v\), \(B_v\) and \(V_v\) are bounded. From there, we can derive five distinct positive constants for each t such that: \(M_1\), \(M_2\), \(M_3\), \(M_4\), and \(M_5\)

$$\begin{aligned} \left\| {{S_u}} \right\|< {M_1},\left\| {{I_u}} \right\|< {M_2},\left\| {{R_u}} \right\|< {M_3},\left\| {{B_u}} \right\|< {M_4},\left\| {{V_u}} \right\| < {M_5},\,(v,u) \in M \times M. \end{aligned}$$
(38)

Further, considering equations (37) and (38),we get

$$\begin{aligned} \begin{array}{lllllllllllllll}{\left\| {\wp \left( {{S_v}(t)} \right) - \wp \left( {{S_u}(t)} \right) } \right\| \le \left( {1 - \beta ({N_1} + {N_2}){f_1}(\vartheta ) - (\mu + \gamma + \upsilon ){f_2}(\vartheta )} \right. }\\ \hspace{11em} {\left. { + \kappa {f_3}(\vartheta ) + \tau {f_4}(\vartheta )} \right) \left\| {({S_v} - {S_u})} \right\| ,}\end{array} \end{aligned}$$
(39)

where \(f_1\), \(f_2\), \(f_3\) and \(f_4\) are functions of \({{{{\mathscr {L}}}}^{ - 1}}\left\{ {{{{\mathscr {L}}}}\frac{{s + \eta (1 - s)}}{s}} \right\}\). Similarly, we are able to obtain

$$\begin{aligned} \begin{array}{lllllllllllllll}{\left\| {\wp \left( {{I_v}(t)} \right) - \wp \left( {{I_u}(t)} \right) } \right\| \le \left( {1 + \beta ({N_1} + {N_2}){f_1}(\vartheta ) - (\mu + \psi + \alpha + \chi ){f_5}(\vartheta )} \right) \left\| {({I_v} - {I_u})} \right\| ,}\\ {\left\| {\wp \left( {{R_v}(t)} \right) - \wp \left( {{R_u}(t)} \right) } \right\| \le \left( {1 + \alpha {f_6}(\vartheta ) - (\mu + \kappa ){f_7}(\vartheta )} \right) \left\| {({R_v} - {R_u})} \right\| ,}\\ {\left\| {\wp \left( {{B_v}(t)} \right) - \wp \left( {{B_u}(t)} \right) } \right\| \le \left( {1 + \upsilon {f_8}(\vartheta ) - \mu {f_9}(\vartheta )} \right) \left\| {({B_v} - {B_u})} \right\| ,}\\ {\left\| {\wp \left( {{V_v}(t)} \right) - \wp \left( {{V_u}(t)} \right) } \right\| \le \left( {1 + \gamma {f_{10}}(\vartheta ) - (\mu + \tau ){f_9}{{(\vartheta )}_{11}}} \right) \left\| {({V_v} - {V_u})} \right\| ,}\end{array} \end{aligned}$$
(40)

where

$$\begin{aligned} \begin{array}{lllllllllllllll}{\left( {1 - \beta ({N_1} + {N_2}){f_1}(\vartheta ) - (\mu + \gamma + \upsilon ){f_2}(\vartheta ) + \kappa {f_3}(\vartheta ) + \tau {f_4}(\vartheta )} \right)< 1,}\\ {\left( {1 + \beta ({N_1} + {N_2}){f_1}(\vartheta ) - (\mu + \psi + \alpha + \chi ){f_5}(\vartheta )} \right)< 1,}\\ {\left( {1 + \alpha {f_6}(\vartheta ) - (\mu + \kappa ){f_7}(\vartheta )} \right)< 1,}\\ \begin{array}{l}\left( {1 + \upsilon {f_8}(\vartheta ) - \mu {f_9}(\vartheta )} \right)< 1,\\ \left( {1 + \gamma {f_{10}}(\vartheta ) - (\mu + \tau ){f_9}{{(\vartheta )}_{11}}} \right) < 1.\end{array}\end{array} \end{aligned}$$

Therefore, K has a fixed point. Considering equations (39) and (40), we assume:

$$\begin{aligned} & \Im = (0, 0, 0, 0, 0),\\ & \Im = \left\{ {\begin{array}{lllllllllllllll}{\left( {1 - \beta ({N_1} + {N_2}){f_1}(\vartheta ) - (\mu + \gamma + \upsilon ){f_2}(\vartheta ) + \kappa {f_3}(\vartheta ) + \tau {f_4}(\vartheta )} \right)< 1,}\\ {\left( {1 + \beta ({N_1} + {N_2}){f_1}(\vartheta ) - (\mu + \psi + \alpha + \chi ){f_5}(\vartheta )} \right)< 1,}\\ \begin{array}{l}\left( {1 + \alpha {f_6}(\vartheta ) - (\mu + \kappa ){f_7}(\vartheta )} \right)< 1,\\ \left( {1 + \upsilon {f_8}(\vartheta ) - \mu {f_9}(\vartheta )} \right)< 1,\end{array}\\ {\left( {1 + \gamma {f_{10}}(\vartheta ) - (\mu + \tau ){f_9}{{(\vartheta )}_{11}}} \right) < 1.}\end{array}} \right. \end{aligned}$$

This completes the proof. \(\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\square\)

Numerical scheme

The system (8) can be solved numerically using the method we present in this section, which is based on a Newton polynomial26. To numerically simulate the suggested mathematical model explaining the transmission of Brucellosis in cattle, we employ the fractal fractional operator with exponential decay kernel.

$$\begin{aligned} \begin{array}{l}S({t_{r + 1}}) = S({t_r}) + \frac{{(1 - \eta )}}{{M(\eta )}}\left[ \begin{array}{l}t_r^{1 - \zeta }{A_1}({t_r},{S_r},{I_r},{R_r},{B_r},{V_r})\\ - t_{r - 1}^{1 - \zeta }{A_1}({t_{r - 1}},{S_{r - 1}},{I_{r - 1}},{R_{r - 1}},{B_{r - 1}},{V_{r - 1}})\end{array} \right] \\ \hspace{5em}+ \frac{\eta }{{M(\eta )}}\int \limits _{{t_r}}^{{t_{r + 1}}} {{A_1}(t,\varpi )} {\sigma ^{\zeta - 1}}d\sigma ,\\ I({t_{r + 1}}) = I({t_r}) + \frac{{(1 - \eta )}}{{M(\eta )}}\left[ \begin{array}{l}t_r^{1 - \zeta }{A_2}({t_r},{S_r},{I_r},{R_r},{B_r},V)\\ - t_{r - 1}^{1 - \zeta }{A_2}({t_{r - 1}},{S_{r - 1}},{I_{r - 1}},{R_{r - 1}},{B_{r - 1}},{V_{r - 1}})\end{array} \right] \\ \hspace{5em} + \frac{\eta }{{M(\eta )}}\int \limits _0^t {{A_2}(t,\varpi )} {\sigma ^{\zeta - 1}}d\sigma ,\\ R({t_{r + 1}}) = R({t_r}) + \frac{{(1 - \eta )}}{{M(\eta )}}\left[ \begin{array}{l}t_r^{1 - \zeta }{A_3}({t_r},{S_r},{I_r},{R_r},{B_r},V)\\ - t_{r - 1}^{1 - \zeta }{A_3}({t_{r - 1}},{S_{r - 1}},{I_{r - 1}},{R_{r - 1}},{B_{r - 1}},{V_{r - 1}})\end{array} \right] \\ \hspace{5em} + \frac{\eta }{{M(\eta )}}\int \limits _0^t {{A_3}(t, \varpi )} {\sigma ^{\zeta - 1}}d\sigma ,\\ B({t_{r + 1}}) = B({t_r}) + \frac{{(1 - \eta )}}{{M(\eta )}}\left[ \begin{array}{l}t_r^{1 - \zeta }{A_4}({t_r},{S_r},{I_r},{R_r},{B_r},V)\\ - t_{r - 1}^{1 - \zeta }{A_4}({t_{r - 1}},{S_{r - 1}},{I_{r - 1}},{R_{r - 1}},{B_{r - 1}},{V_{r - 1}})\end{array} \right] \\ \hspace{5em} + \frac{\eta }{{M(\eta )}}\int \limits _0^t {{A_4}(t,\varpi )} {\sigma ^{\zeta - 1}}d\sigma ,\\ V({t_{r + 1}}) = V({t_r}) + \frac{{(1 - \eta )}}{{M(\eta )}}\left[ \begin{array}{l}t_r^{1 - \zeta }{A_5}({t_r},{S_r},{I_r},{R_r},{B_r},V)\\ - t_{r - 1}^{1 - \zeta }{A_5}({t_{r - 1}},{S_{r - 1}},{I_{r - 1}},{R_{r - 1}},{B_{r - 1}},{V_{r - 1}})\end{array} \right] \\ \hspace{5em} + \frac{\eta }{{M(\eta )}}\int \limits _0^t {{A_5}(t,\varpi )} {\sigma ^{\zeta - 1}}d\sigma ,\end{array} \end{aligned}$$
(41)

For this model, we could have the following scheme

$$\begin{aligned} {S^{r + 1}}= & {S^r} + \frac{{(1 - \eta )}}{{M(\eta )}}\left[ \begin{array}{l}t_r^{1 - \zeta }{A_1}({t_r},{S_r},{I_r},{R_r},{B_r},{V_r})\\ - t_{r - 1}^{1 - \zeta }{A_1}({t_{r - 1}},{S_{r - 1}},{I_{r - 1}},{R_{r - 1}},{B_{r - 1}},{V_{r - 1}})\end{array} \right] \\ & + \frac{\eta }{{M(\eta )}}\left\{ \begin{array}{l}\frac{{23}}{{12}}t_r^{1 - \zeta }{A_1}({t_r},{S_r},{I_r},{R_r},{B_r},{V_r})\Delta t\\ - \frac{4}{3}t_{r - 1}^{1 - \zeta }{A_1}({t_{r - 1}},{S_{r - 1}},{I_{r - 1}},{R_{r - 1}},{B_{r - 1}},{V_{r - 1}})\Delta t\\ + \frac{5}{{12}}t_{r - 2}^{1 - \zeta }{A_1}({t_{r - 2}},{S_{r - 2}},{I_{r - 2}},{R_{r - 2}},{B_{r - 2}},{V_{r - 2}})\Delta t\end{array} \right\} \\ {I^{r + 1}}= & {I^r} + \frac{{(1 - \eta )}}{{M(\eta )}}\left[ \begin{array}{l}t_r^{1 - \zeta }{A_2}({t_r},{S_r},{I_r},{R_r},{B_r},{V_r})\\ - t_{r - 1}^{1 - \zeta }{A_2}({t_{r - 1}},{S_{r - 1}},{I_{r - 1}},{R_{r - 1}},{B_{r - 1}},{V_{r - 1}})\end{array} \right] \\ & + \frac{\eta }{{M(\eta )}}\left\{ \begin{array}{l}\frac{{23}}{{12}}t_r^{1 - \zeta }{A_2}({t_r},{S_r},{I_r},{R_r},{B_r},{V_r})\Delta t\\ - \frac{4}{3}t_{r - 1}^{1 - \zeta }{A_2}({t_{r - 1}},{S_{r - 1}},{I_{r - 1}},{R_{r - 1}},{B_{r - 1}},{V_{r - 1}})\Delta t\\ + \frac{5}{{12}}t_{r - 2}^{1 - \zeta }{A_2}({t_{r - 2}},{S_{r - 2}},{I_{r - 2}},{R_{r - 2}},{B_{r - 2}},{V_{r - 2}})\Delta t\end{array} \right\} \\ {R^{r + 1}}= & {R^r} + \frac{{(1 - \eta )}}{{M(\eta )}}\left[ \begin{array}{l}t_r^{1 - \zeta }{A_3}({t_r},{S_r},{I_r},{R_r},{B_r},{V_r})\\ - t_{r - 1}^{1 - \zeta }{A_3}({t_{r - 1}},{S_{r - 1}},{I_{r - 1}},{R_{r - 1}},{B_{r - 1}},{V_{r - 1}})\end{array} \right] \\ & + \frac{\eta }{{M(\eta )}}\left\{ \begin{array}{l}\frac{{23}}{{12}}t_r^{1 - \zeta }{A_3}({t_r},{S_r},{I_r},{R_r},{B_r},{V_r})\Delta t\\ - \frac{4}{3}t_{r - 1}^{1 - \zeta }{A_3}({t_{r - 1}},{S_{r - 1}},{I_{r - 1}},{R_{r - 1}},{B_{r - 1}},{V_{r - 1}})\Delta t\\ + \frac{5}{{12}}t_{r - 2}^{1 - \zeta }{A_3}({t_{r - 2}},{S_{r - 2}},{I_{r - 2}},{R_{r - 2}},{B_{r - 2}},{V_{r - 2}})\Delta t\end{array} \right\} \\ {B^{r + 1}}= & {B^r} + \frac{{(1 - \eta )}}{{M(\eta )}}\left[ \begin{array}{l}t_r^{1 - \zeta }{A_4}({t_r},{S_r},{I_r},{R_r},{B_r},{V_r})\\ - t_{r - 1}^{1 - \zeta }{A_4}({t_{r - 1}},{S_{r - 1}},{I_{r - 1}},{R_{r - 1}},{B_{r - 1}},{V_{r - 1}})\end{array} \right] \\ & + \frac{\eta }{{M(\eta )}}\left\{ \begin{array}{l}\frac{{23}}{{12}}t_r^{1 - \zeta }{A_4}({t_r},{S_r},{I_r},{R_r},{B_r},{V_r})\Delta t\\ - \frac{4}{3}t_{r - 1}^{1 - \zeta }{A_4}({t_{r - 1}},{S_{r - 1}},{I_{r - 1}},{R_{r - 1}},{B_{r - 1}},{V_{r - 1}})\Delta t\\ + \frac{5}{{12}}t_{r - 2}^{1 - \zeta }{A_4}({t_{r - 2}},{S_{r - 2}},{I_{r - 2}},{R_{r - 2}},{B_{r - 2}},{V_{r - 2}})\Delta t\end{array} \right\} \\ {V^{r + 1}}= & {V^r} + \frac{{(1 - \eta )}}{{M(\eta )}}\left[ \begin{array}{l}t_r^{1 - \zeta }{A_5}({t_r},{S_r},{I_r},{R_r},{B_r},{V_r})\\ - t_{r - 1}^{1 - \zeta }{A_5}({t_{r - 1}},{S_{r - 1}},{I_{r - 1}},{R_{r - 1}},{B_{r - 1}},{V_{r - 1}})\end{array} \right] \\ & + \frac{\eta }{{M(\eta )}}\left\{ \begin{array}{l}\frac{{23}}{{12}}t_r^{1 - \zeta }{A_5}({t_r},{S_r},{I_r},{R_r},{B_r},{V_r})\Delta t\\ - \frac{4}{3}t_{r - 1}^{1 - \zeta }{A_5}({t_{r - 1}},{S_{r - 1}},{I_{r - 1}},{R_{r - 1}},{B_{r - 1}},{V_{r - 1}})\Delta t\\ + \frac{5}{{12}}t_{r - 2}^{1 - \zeta }{A_5}({t_{r - 2}},{S_{r - 2}},{I_{r - 2}},{R_{r - 2}},{B_{r - 2}},{V_{r - 2}})\Delta t\end{array} \right\} . \end{aligned}$$

Simulation and discussion

The initial values and parameters for system (8) are listed in Table 2 and were taken from12. Using data from Table 2 corresponding to various fractal-fractional orders, we simulate the solution behavior of model (8) for S, I, R, B, and V.

Table 2 Parameters values of brucellosis model.
Full size table

The figures illustrate the dynamics of a brucellosis model under various fractional orders and control strategies. For the susceptible population (\(S(t)\)), Figs.4(a-c) show an initial decline followed by stabilization, with higher fractional orders (\(\zeta = 1.0\)) leading to faster dynamics and lower susceptible numbers, while lower orders (\(\zeta = 0.6\)) result in slower changes and higher stability. Similarly, the infected population (\(I(t)\)), shown in Figures 5(a-c), declines over time, with sharper decreases observed for higher fractional orders, indicating effective mitigation. The recovered population (\(R(t)\)) in Figs.6(a-c) grows steadily, with faster recovery dynamics linked to higher fractional orders. Figs.7(a-c) highlight the dynamics of the Brucella-contaminated environment (\(B(t)\)), where contamination levels increase more rapidly at higher fractional orders, stabilizing at lower levels as fractional orders decrease. Vaccination dynamics (\(V(t)\)) in Figs.8(a-c) follow a similar trend, with higher orders producing faster vaccination coverage growth. Figures 9(a-f) explore the effects of combined control measures. Vaccination and culling (9a-b) significantly reduce infections (\(I(t)\)) and stabilize \(S(t)\). Biosecurity and culling (9c-d) effectively enhance \(S(t)\) recovery while sharply reducing \(I(t)\). The combination of vaccination and biosecurity (9e-f) moderately increases \(S(t)\) and reduces \(I(t)\) over time. These results collectively demonstrate that fractional-order dynamics are highly sensitive to parameter variations, with higher orders capturing faster transmission and intervention effects. The measures included vaccination, elimination of affected animals, and maintaining security against diseases which help in disease fight in the fight.

The figures offer a comprehensive view of the brucellosis dynamics in cattle and their biological implications. In Figs.4(a-c), the susceptible population (\(S(t)\)) initially decreases as the disease spreads among uninfected cattle. Over time, this population stabilizes due to the interplay between natural recruitment, disease transmission, and interventions. Higher fractional orders (\(\zeta = 1.0\)) correspond to faster disease spread and quicker adaptation to interventions, while lower orders (\(\zeta = 0.6\)) reflect a slower, more gradual response, potentially mirroring biological processes with delayed effects or less efficient control measures. The infected population (\(I(t)\)), shown in Figs.5(a–c), decreases over time as infected cattle either recover or are culled. This trend is biologically significant because it reflects effective herd immunity development and the impact of interventions like vaccination. Higher fractional orders capture more rapid clearance of infection, aligning with scenarios where vaccination and biosecurity measures are efficiently implemented. Figs.6(a-c) depict the recovered population (\(R(t)\)), which grows as infected cattle overcome the disease and gain immunity. The faster rise in \(R(t)\) for higher fractional orders indicates a stronger immune response and successful recovery mechanisms. Conversely, lower fractional orders reflect slower recovery, suggesting weaker immune responses or delayed intervention effects. Figs.7(a-c) illustrate the dynamics of the Brucella-contaminated environment (\(B(t)\)), where contamination increases due to the pathogen’s persistence in the environment. Higher fractional orders show rapid contamination build-up, indicating conditions with higher interaction rates or insufficient control measures. Lower fractional orders demonstrate slower contamination growth, potentially linked to effective biosecurity practices that reduce pathogen spread. Figs.8(a-c) show vaccination dynamics (\(V(t)\)), with higher fractional orders leading to rapid increases in vaccinated cattle. This reflects efficient vaccination programs and enhanced herd immunity, reducing the susceptible population’s size and breaking transmission chains. Lower fractional orders suggest slower vaccine uptake or less effective coverage. In Figs.9(a-f), combined control measures are analyzed. Vaccination and culling of seropositive cattle (9a-b) biologically reduce disease prevalence by removing infection sources and boosting immunity. Biosecurity and culling (9c-d) enhance the protection of the susceptible population by limiting pathogen exposure and directly removing infected individuals. The combination of vaccination and biosecurity (9e-f) demonstrates a balanced approach, where enhanced immunity and reduced environmental contamination synergistically lower infection rates. Biologically, these results emphasize the importance of fractional-order modeling in capturing the memory effects and delayed responses inherent in real-world disease dynamics. They underscore the critical role of vaccination, culling, and biosecurity in controlling brucellosis and highlight the necessity for tailored strategies that consider the specific biological and environmental factors influencing disease spread.

Changes in key parameters such as infection rates, control measures, and interaction rates significantly influence brucellosis dynamics. Higher transmission rates can lead to increased disease prevalence, necessitating stronger control strategies like vaccination and culling. Effective vaccination coverage can substantially reduce infection rates and enhance herd immunity. Additionally, managing interactions between cattle and potential reservoirs (e.g., rodents) can mitigate transmission risks. The proposed fractional-order model was validated using parameter values from studies on brucellosis dynamics in cattle populations, such as recruitment, infection, and vaccination rates12. The model’s predictions closely align with observed trends, including the decline in infected cattle populations (\(I(t)\)) after implementing vaccination and biosecurity interventions. The use of the Caputo-Fabrizio operator enhances the model’s ability to incorporate memory effects, capturing delayed responses to control measures, as seen in empirical studies on disease management. Furthermore, the model accurately reflects Brucella persistence in the environment (\(B(t)\)), consistent with documented survival patterns of the pathogen in livestock settings. These findings highlight the model’s effectiveness as a predictive tool for managing brucellosis and guiding policy decisions to mitigate its impact on cattle populations. The numerical results highlight that vaccination, as a single control measure, is highly effective in reducing the infected population, with the memory effects captured by the operator emphasizing the gradual accumulation of immunity across the herd. When paired with culling of seropositive cattle, the memory effects show enhanced long-term reductions in infection prevalence due to the combined impact of removing infection sources and preventing new cases. The Caputo-Fabrizio operator also demonstrates the compounded benefits of a three-pronged approach–vaccination, culling, and biosecurity protocols’by capturing how delayed and sustained interventions collectively stabilize susceptible populations and significantly reduce environmental contamination (\(B(t)\)) over time. This multifaceted strategy achieves the most significant long-term reduction in brucellosis prevalence, highlighting the operator’s utility in evaluating the delayed impact of combined control measures.The model’s findings on brucellosis dynamics in cattle have important real-world implications for disease management. Effective control measures can lead to significant economic benefits, such as increased livestock productivity and reduced veterinary costs. The insights emphasize the need for robust surveillance to prevent zoonotic transmission to humans, particularly through unpasteurized dairy products. Additionally, the findings can inform public policy and targeted interventions, optimizing resource allocation. Lastly, they support educational campaigns to raise awareness about brucellosis prevention among farmers and consumers, enhancing health outcomes for both livestock and humans. The model’s limitations include assuming uniform population behavior, which overlooks factors like age, immunity, and socio-environmental differences. It also does not account for pathogen resistance, behavioral changes, or environmental influences that could affect transmission dynamics. Additionally, reliance on existing data may limit the model’s accuracy, especially in capturing localized variations and real-world challenges in intervention effectiveness.

Fig. 4
Fig. 4
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The dynamics of S with different fractal fractional order. (a) At \(\zeta = 1\). (b) At \(\zeta = 0.8\). (c) At \(\zeta = 0.6\).

Fig. 5
Fig. 5
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The dynamics of I with different fractal fractional order. (a) At \(\zeta = 1\). (b) At \(\zeta = 0.8\). (c) At \(\zeta = 0.6\).

Fig. 6
Fig. 6
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The dynamics of R with different fractal fractional order. (a) At \(\zeta = 1\). (b) At \(\zeta = 0.8\). (c) At \(\zeta = 0.6\).

Fig. 7
Fig. 7
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The dynamics of B with different fractal fractional order. (a) At \(\zeta = 1\). (b) At \(\zeta = 0.8\). (c) At \(\zeta = 0.6\).

Fig. 8
Fig. 8
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The dynamics of V with different fractal fractional order. (a) At \(\zeta = 1\). (b) At \(\zeta = 0.8\). (c) At \(\zeta = 0.6\).

Fig. 9
Fig. 9
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Simulation of S(t) and I(t) with various combination of control interventions.

Conclusion

This paper presents a detailed analysis of the Brucellosis model using the Caputo-Fabrizio fractional order derivative. By applying the fixed-point theorem, we demonstrate that the solutions to the developed model exist, are unique, and remain stable, providing further evidence that the non-integer order model is more robust than the conventional integer-order model. Simulations were conducted using the iterative Laplace transform technique, with parameters derived from existing literature. The results revealed that the disease dynamics are highly sensitive to the fractional order of the derivative, underscoring the importance of using the Caputo-Fabrizio operator in predicting the spread of brucellosis. The findings establish the effectiveness and accuracy of the Caputo-Fabrizio fractional derivative in modeling Brucellosis and suggest its potential applicability to other infectious diseases.

However, this study has certain limitations that should be noted. Many of the parameters used in the model were estimated based on existing literature, which may result in discrepancies when compared to real-world data from different regions or populations. Additionally, the model does not account for population heterogeneity, meaning factors such as age, immunity levels, and socio-economic status are not differentiated, which could influence the spread of the disease. Future research could focus on developing more sophisticated models, such as those incorporating different population structures and spatial distributions, as well as time variations in the model’s parameters. It would also be valuable to apply a similar model to other infectious diseases with comparable transmission dynamics. Further empirical research is necessary to compare the model’s findings with actual data to assess its effectiveness for public health strategies. Expanding the research to include other fractional operators and comparing their efficiency could enhance our understanding of the most appropriate mathematical tools for modeling disease transmission.