1 Introduction

Numerous diseases can affect cattle, with Lumpy skin disease being the most significant among them. LSD is a viral infection caused by Capripoxvirus. Symptoms in animals can vary depending on the way the disease progresses. However, common signs in cattle and other ruminants include distinct skin lumps and/or nodules, fever, and a loss of appetite [1,2]. It results in reduced milk supply, weight reduction, and, in the worst cases, mortality. LSD is primarily spread through vector-borne transmission, with mosquitoes, biting flies, and ticks acting as mechanical vectors for the virus among susceptible cattle [3]. While direct contact between sick and susceptible animals, through injuries to the skin or bodily fluids can facilitate transmission, this mode is less crucial in natural environments in the absence of insect vectors. Furthermore, indirect transmission via contaminated fomites, such as feed, water, and equipment, contributes to disease spread and highlights the necessity for rigorous biosecurity measures [4]. LSD significantly affects animal health, productivity, and economies, leading to trade restrictions, reduced income, and food shortages for affected farmers and communities [2]. Animals exposed to LSD typically show symptoms after an incubation period of around two to four weeks [5]. The disease was first identified in Zambia in 1929. South Africa faced an LSD outbreak in 1949, resulting in a severe economic crisis. Since the year 2000, LSD outbreaks have affected multiple countries across the Middle East. Furthermore, the disease has been documented in various European countries, including Greece, Bulgaria, Turkey, Kosovo, and Russia. Its spread has extended into Asia, with cases reported in India, Bangladesh, Myanmar, Pakistan, Nepal and Thailand [6]. While there is no cure, effective strategies to control the spread of the disease include vaccination, vector elimination, and the implementation of quarantine or isolation of cattle showing symptoms [7–9]. In [10] and [11], it was reported that restricting cattle movement, removing those with illness, quarantining them before reintegration to the herd, and preventing herd mixing during feeding and watering are effective strategies for infection prevention. Therefore, farmers, veterinarians and other stakeholders must work together to prevent the spread of LSD and minimize the negative impacts using various strategies.

Mathematical modeling has gained prominence across diverse fields as a powerful tool for analyzing and solving complex real-world problems. It facilitates the analysis of dynamic processes and supports more reliable predictions of future outcomes. One key application is in the study of infectious disease dynamics, which is crucial for developing effective treatments, control strategies, and preventive measures. Mathematical models have played a crucial role in understanding outbreaks such as measles [12], Ebola virus disease [13,14], HIV [15], Monkeypox infection [16], particularly in the context of LSD [17,18].

To enhance the accuracy and flexibility of these models, researchers have explored various extensions of classical calculus. One such advancement is fractional calculus, which generalizes the concept of derivatives and integrals to non-integer orders. This naturally leads to the intriguing question: can the notion of derivatives be extended beyond integer orders to include, for example, half-order derivatives? The answer is affirmative, and such extensions are made possible through the framework of fractional calculus [19]. Unlike classical derivatives, fractional derivatives are defined in a non-local manner, meaning they account for the entire history of a function rather than just its instantaneous rate of change. This memory-dependent nature makes them especially powerful for modeling real-world systems where past states influence present behavior. Over the past few years, fractional differential equations have become an essential tool in various research fields, including viscoelasticity [20], electrical engineering [21], diffusion processes [22], biological systems [23], etc. By employing fractional calculus the models offer more precise representations of various complex phenomena. Various types of fractional derivatives are discussed in the literature, each with unique properties and applications. Among them, the Caputo derivative is particularly well-known and widely used. Its popularity stems from its suitability for initial value problems, as it allows for straightforward interpretation when dealing with real-world applications where beginning conditions are described in terms of integer-order derivatives. The Caputo derivative has been widely used in numerous epidemic models, demonstrating its effectiveness. Calatayud et al. [24] conducted a study focused on the interpretation of Caputo-type fractional compartmental models. Saeedian et al. [25] investigate the effects of memory on epidemic dynamics using the SIR model, demonstrating that the system’s behavior significantly depends on the strength of memory effects, which are determined by the order of the fractional derivatives. Dutta et al. [26] examined the spread of infectious diseases using the Caputo derivative, illustrating how public attitudes toward vaccination, social behaviors, and government actions influence epidemic outcomes. The results suggest that implementing control measures not only reduces disease prevalence but also mitigates the economic impact of the epidemic. A Caputo-type SIR epidemic model incorporating an awareness campaign strategy was proposed in [27]. Khan et al. [16] explored the impact of vaccination on controlling monkeypox infection using the Caputo derivative. Alaje and Olayiwola [28] investigated a fractional-order COVID-19 model with a focus on vaccine distribution.

Several recent studies have investigated the mathematical model of LSD to better understand its transmission dynamics and control strategies [7,17,29–32]. Elsonbaty et al. [7] explored the non-linear dynamics of a discrete fractional model of LSD. In [29] a fractional-order mathematical model of LSD has been investigated using the Caputo approach. Although the latest LSD mathematical models [17,31] investigated only the role of vaccination in LSD dynamics, these models usually do not consider the role of both quarantine and vaccination in the dynamics of the LSD virus. In [32], a mathematical model was introduced to examine the LSD transmission in dairy cows, accounting for both direct and mosquito-borne transmission. The model includes a vaccination component, and the analysis indicated that increasing the vaccination rate reduces the levels of infection. Mani et al. [33] investigated an SVEIR-type model for LSD using the Caputo-Fabrizio fractional derivative. Rathee et al. [34] developed a Caputo fractional SVEIR model to examine the dynamics of LSD transmission in cattle, employing a predictor–corrector numerical approach.

In our study, we propose an SVEIQR mathematical model for LSD. The novelty of our approach lies in integrating vaccination strategies for susceptible cattle and quarantine measures for the exposed cattle population. The quarantine strategy used in our model is based on the experimental studies [10] and [11]. The primary goal is to construct a reliable framework that facilitates a deeper understanding of disease dynamics. To enhance the accuracy and predictive capability of the model, we employ the Caputo fractional derivative. Fractional calculus possesses a unique ability to depict dynamics that are both non-local and reliant on memory. The choice of the Caputo fractional derivative is motivated by its ability to incorporate local initial conditions into the model formulation. Furthermore, the Caputo derivative of a constant equals zero, preserving consistency with classical differential equations. This methodology accounts for memory effects and long-range dependence, which are often neglected in the integer order model, allowing for a more precise representation of epidemiological dynamics. We aim to improve the precision of the predictions in the LSD model by applying the Caputo fractional derivative. This will result in more efficient control strategies and treatment interventions for LSD.

The document is outlined as follows. Sect 2 presents essential preliminaries regarding fractional derivatives. In Sect 3, focus on the development of the model including non-negativity, boundedness, existence of a unique solution, equilibria, bifurcation, stability and sensitivity. Sect 4 presents the numerical procedure for the solution of the model. Sect 5 focuses on the numerical investigation. Finally, the conclusion is given in Sect 6.

2 Preliminaries

In this section, some key formulas that will be applied in formulating the problem are presented [35].

Definition 2.1. The fractional integral of order of a function f is defined as:

(2.1)

where is Gamma function.

Definition 2.2. The Caputo fractional derivative of order of a function , defined as:

(2.2)

Definition 2.3. The Laplace transform of the Caputo derivative of order is expressed as:

(2.3)

where .

3 Lumpy skin disease model

To investigate the quarantine and vaccination roles on the LSD spread, we extend the mathematical model derived in [17], by considering the presence of a quarantine population. We propose an SVEIQR mathematical model for Lumpy skin disease. The total cattle population divided into six different classes: represents the number of susceptible cattle who are potentially prone to disease through interaction with cattle carrying the virus; those cattle that have been vaccinated are categorized within the vaccinated class , cattle that have previously been exposed to disease-causing viruses but are not yet infectious are placed in the exposed class , the cattle that have been identified and confirmed to be positive for LSD infection are represented by , the cattle in quarantine/isolation class denoted by . The cattle that have high immunity or have received effective medications that lead to recovery are represented by . The details about model parameters are given in Table 1. The flow pattern for LSD is illustrated in Fig 1, modelled by the following nonlinear coupled fractional differential equations:

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Fig 1. The flow diagram of the LSD model (3.1).

The model was inspired by the experimental studies in [10] and [11] which showed that the importance of quarantine and vaccination in mitigating the risk of LSD transmission, and the mathematical modelling studies in [29,17,31] which focused only on the vaccinated effect and did not consider the impact of the quarantine on the LSD infection.

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Table 1. Parameters description.

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(3.1a)

(3.1b)

(3.1c)

(3.1d)

(3.1e)

(3.1f)

with initial conditions

Eqs (3.1a)–(3.1f) incorporate the following biological mechanisms as depicted in Fig 1:

• The susceptible cattle are integrated into the population at a rate (see Eq (3.1a)). A constant input of this nature is often considered in LSD disease models (see e.g., [29]). We assume that the rate of new infections resulting from contacts between susceptible and infectious individuals is . The parameter denotes the infection or transmission rate. This bilinear incidence (i.e., the infection rate per infected individual and susceptible individual is constant), is commonly used in deterministic compartmental models (see [17,29,31]). Susceptible cattle are vaccinated at a rate , moving to the vaccinated compartment [17,31]. This represents the implementation of the vaccination program to protect cattle from LSD infection. Finally, we assume all populations have a natural mortality rate .
• The vaccinated cattle (see Eq (3.1b)), although protected, can still be exposed to LSD due to vaccine inefficacy or waning immunity, at a rate , where is the infection rate for vaccinated cattle (typically lower than ). Vaccinated cattle can recover at a rate [17,31].
• The exposed cattle (3.1c) progress to the infected compartment at a rate k, representing the incubation period after which the cattle become infectious [17,31]. Moreover, it has been experimentally shown in [10] and [11] that the strengthening quarantine and movement control measures can prevent direct LSD transmission. Thus, we assume that the rate of quarantine is represented by q.
• The infected cattle (see Eq (3.1d)) can recover from LSD at rate . This represents natural recovery or recovery aided by medical intervention. Infected cattle experience disease-induced mortality at a rate [17,31].
• The quarantined cattle (see Eq (3.1e)) recover at a rate , reflecting recovery during isolation [13]. All other terms have been described above.
• The recovered cattle (see Eq (3.1f)) and all populations have a natural mortality rate . All other terms have been described above.

3.1 Non-negativity and boundedness

We begin the analysis of model (3.1) by demonstrating its biological feasibility, ensuring that all population variables remain non-negative and bounded [36]. The following results confirm these characteristics.

Theorem 3.1. All solutions of the fractional-order model (3.1) that exist in are uniformly bounded and non-negative.

Proof The total population is defined as , then for each ,

Utilizing the standard comparison theorem (SCT) applicable to fractional order [37]

(3.2)

where denotes the Mittag-Leffler (ML) function. According to Lemma 5 and Corollary 6 in [38]

Therefore, all solutions of the fractional order model (3.1) that originate in are confined to the domain , where

Now, we are looking for nonnegative solutions for the proposed fractional LSD model (3.1). From Eq (3.1a), we have

where . By SCT for fractional order, using the property of ML function for any [38]

From Eq (3.1b), we have

where . Therefore,

From Eq (3.1c), we obtain

where . Therefore,

From Eq (3.1d), we obtain

where . Therefore,

From Eq (3.1e), we obtain

where . Therefore,

From Eq 3.1f), we get

where . Therefore,

Consequently, the solutions of system (3.1) are demonstrated to be non-negative.

3.2 Existence and uniqueness

In this section, we will establish the existence and uniqueness of solutions for the LSD model (3.1).

Lemma 3.2. [39] Consider the system

(3.3)

with primary condition , where , If local Lipschitz condition is fulfilled by f(t,x) for x, then there exists a solution of (3.1) on which is unique.

To establish the result, consider the region , where and . Denote and . Consider a mapping

, where

For any , we have

(3.4)

where Thus, meets the Lipschitz condition with respect to . Therefore, Lemma 3.2 confirms that there exists a unique solution of fractional-order system (3.1) with initial condition [40,41].

3.3 Basic reproduction ratio and equilibria

Proposition 1. For system (3.1), there exists a positive basic reproduction number such that

1. (i) There exists only one equilibrium point , when , and
2. (ii) There exist two equilibria, and , when .

Proof. (i) To find the positive equilibria of the system (3.1), we set the right-hand side to zero. The system demonstrates Lumpy skin disease free equilibrium (LSD-FE) when there are no infected cattle in the community, i.e. . The LSD-FE equilibrium point is given as .

The basic reproduction number, commonly denoted by , is used to assess the transmission potential of an infectious disease. It represents the average number of secondary infections caused by a single infected individual introduced into a completely susceptible population [42]. To compute , we apply the next generation matrix approach under the assumption that

(3.5)

where , denotes the influx of new infections and signifies the transfer of individuals into and out from the compartments. The Jacobian matrices of and are examined at the equilibrium point and are given by

(3.6)

Then we can calculate as:

(3.7)

So, can be obtained by using , where is spectral radius of matrix X. Therefore, we have

(3.8)

(ii) To find the other equilibrium point LSD endemic equilibrium (LSD-EE) , in addition to , let be any equilibrium point of model (3.1) satisfying the equations:

(3.9)

After solving Eq (3.9), we obtain the following expressions for and as follows:

(3.10)

where , , , and . After substituting the values of and in the first equation from Eq (3.9), we get the quadratic equation

(3.11)

where

The solutions of Eq (3.11) are given by:

It is clear that a₁>0, therefore if a₃>0 then and . Let , then from Eq (3.10) we obtain:

Therefore, the LSD endemic equilibrium (LSD-EE); exists when a₃>0 or . In the case of , the system’s behavior will be dominated by the endemic endemic equilibrium , which means that the infection persists in the population indefinitely in the long term (the disease becomes endemic). Therefore, intervention is needed to eliminate disease, for instance, increasing , and or decreasing and , which may reduce the to less than 1, which results in shifting the system toward the disease-free equilibrium. Furthermore, in Fig 2 we show how the two equilibria and change as we vary three parameters: (a) rate of infection of susceptible cattle, (b) rate of infection of vaccinated cattle, (c) vaccination rate of susceptible cattle. We can see that an increase in above or an increase in above or a decrease in below leads to the bifurcation of a state (which contains LSD infection) from the state (with no LSD infection).

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Fig 2. Bifurcation diagram for the two steady states calculated of the model (3.1); and , as the parameters are varied: (a) rate of infection of susceptible cattle (), ; (b) rate of infection of vaccinated cattle (), ; (c) vaccinated rate of of susceptible cattle (), .

Sub-plot (i) shows S vs. parameter, while Sub-plot (ii) show I vs. parameter. Here "BP1" presents the bifurcation point where the state bifurcates out of the state as we increase above , and "BP2" presents the bifurcation point where the state bifurcates out of the state as we increase above and "BP3" presents the bifurcation point where the state bifurcates out of the state as we decrease below .

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3.4 Stability analysis

In this section, we study the local and global stability of the LSD model (3.1) at its equilibria.

3.4.1 Local asymptotic stability.

Theorem 3.3. The LSD-FE is locally asymptotically stable (LAS) if and otherwise unstable.

Proof. The linearization of the model (3.1) around the LSD-FE point is given by the Jacobian matrix as:

From the above Jacobian matrix, the four eigenvalues are negative, i.e., , , , and the rest eigenvalues can be obtained by the characteristic equation below:

(3.12)

Moreover, using the Routh-Hurwitz criteria, Eq (3.12) has negative real parts when

and

Therefore, the LSD-FE is LAS when , otherwise unstable.

3.4.2 Global asymptotic stability.

In this subsection, the global stability at the equilibrium points of the system (3.1) is demonstrated using Lyapunov theory and justified by the following theorems.

Theorem 3.4. The LSD-FE is globally asymptotically stable (GAS) if and otherwise unstable.

Proof. We have the Lyapunov function

(3.13)

where and are some positive constants. Applying the Caputo derivative on , we have

(3.14)

From the system (3.1), we get

(3.15)

Let and , we get the following

(3.16)

Because all parameters are positive, if then for . We note that the solutions of system (3.1) converge to , the largest invariant subset of . From Eq (3.16) we have if and only if . The set is invariant, and for any element that belongs to satisfies then . We can see from system (3.1) that

(3.17)

as t approaches . From LaSalle’s invariance principle [43], is GAS.

Theorem 3.5. The LSD-EE is globally asymptotically stable (GAS) in , if .

Proof: To demonstrate the GAS of , we use the same methodology as in [44]. Consider the Lyapunov function as:

(3.18)

The differentiation of with respect to t, will give:

(3.19)

By considering the following equation:

(3.20)

At the equilibrium point , we get:

(3.21)

Subtracting Eqs 3.21 from 3.20, we have:

(3.22)

Now, consider the 1st term of the right hand side of Eq (3.19) and apply Eq (3.22), we have:

(3.23)

Extending the product, (), gives:

(3.24)

Therefore,

(3.25)

Applying Cauchy–Schwarz inequality in the 2nd term of Eq (3.25), we get:

(3.26)

To simplify the LHS of Eq (3.26), we use Young’s Inequality, we get

Multiplying both sides by , we get:

(3.27)

Therefore, from the 2nd term of equation of (33):

(3.28)

So, Eq (3.25) will become:

(3.29)

Thus, from Eq (3.23) we have:

Therefore,

(3.30)

This implies that:

(3.31)

Hence, the inequality becomes:

(3.32)

where

where C₁ and C₂ are constants which depend on the bounds of and . Similar strategies may be employed to eliminate the cross terms of the remaining terms in Eq (3.19). Consequently, we have demonstrated that exclusively at the endemic equilibrium point EEP and at other positive solutions. Consequently, employing the LaSalle invariance principle, we have established that the endemic equilibrium point is globally asymptotically stable.

3.5 Sensitivity analysis

Sensitivity analysis is essential for identifying optimal strategies for reducing the transmission of infection. Computing sensitivity indices helps analyze the impact of model parameters on . This analysis identifies the most influential parameters for reducing disease transmission, providing critical insights into optimizing intervention measures to control Lumpy skin disease (LSD).

We employ the normalized forward sensitivity index as described in [45] to determine the sensitivity of model parameters. For a given parameter , the sensitivity index is defined as:

(3.33)

The sensitivity indices are presented in Table 2 and illustrated in Fig 3.

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Fig 3. Sensitivity index vs. different parameters.

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Table 2. Sensitivity index of to the model parameters.

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(3.34)

Identifying the most influential parameters is essential for effectively reducing disease transmission and prevalence. Understanding both the sign and magnitude of their sensitivity indices is key to guiding control strategies. A positive sensitivity index indicates that an increase in the corresponding parameter leads to a rise in the basic reproduction number, , whereas a negative index suggests that increasing the parameter reduces .

The results in Table 2 highlight that parameters such as the recruitment rate , transmission rate of susceptible cattle , vaccination rate , progression rate (k), and recovery rate significantly affect of the model. Notably, the transmission rate exhibits a direct proportional relationship with . The sensitivity index shows that increasing (or decreasing) the transmission rate by, say, 10%, increases (or decreases) by 8.764%. This indicates that a higher contact rate between cattle increases the likelihood of infection. As a result, these conditions contribute to establishing an endemic system with a high prevalence of Lumpy skin disease. Reducing through biosecurity measures, such as controlling insect vectors (e.g., mosquitoes and flies) and limiting cattle movement, is essential for lowering disease prevalence.

Conversely, the vaccination rate has the sensitivity index –0.4764, which means that increasing the vaccination rate 10% will decrease by 4.764%. This highlights the effectiveness of vaccination as a control strategy against Lumpy skin disease, as it reduces the susceptible population of cattle and limits disease transmission. Similarly, the quarantine rate q shows a negative sensitivity index, indicating that isolating exposed cattle reduces by limiting the spread of the virus before cattle become infectious.

Therefore, disease management strategies should focus on parameters with high sensitivity indices, as targeted improvements in these areas can lead to substantial reductions in transmission. For example, strengthening biosecurity practices on cattle farms, implementing timely and widespread vaccination campaigns, enhancing recovery rates through prompt treatment, and restricting the movement of infected animals are all critical control measures. Additionally, ensuring the well-being of livestock and maintaining cleanliness in locations where cattle gather, such as watering sites and shelters, are crucial steps in reducing the spread of Lumpy skin disease.

4 Numerical procedure

This section presents a numerical method described in [46,47] for solving the model (3.1). Consider a fractional differential equation of the form

(4.1)

Using the fundamental theorem of fractional calculus

(4.2)

at the time instance t = t_n + 1,n = 0,1,2,..., the above equation is reformulated as:

(4.3)

Inside the limited range the function is approximated using the Lagrangian interpolation method [46,47], we have

(4.4)

Eq (4.4) can be included in Eq (4.3), and by performing the same steps in [46], we obtain

(4.5)

5 Numerical simulations

This section presents the numerical solution of the fractional LSD model (3.1), obtained using the Lagrange polynomial interpolation numerical scheme for the Caputo derivative [46,47]. The simulations are conducted using the parameter values provided in Table 1 and the initial condition as .

With the transmission rate set to and other parameters as in Table 1, we get , indicating that the transmission of LSD will eventually decline. This outcome is confirmed through the numerical simulation of the fractional LSD model (3.1), as illustrated in Fig 4a–4f. The simulation shows that the susceptible cattle population declines for different values of the fractional order . This highlights the notable influence of on the proportion of susceptible individuals. The vaccinated cattle population initially rises due to vaccination of susceptible cattle, then gradually stabilize, with memory effects capturing the impact of historical vaccination efforts. The exposed cattle population spikes early as susceptible and vaccinated cattle contract the virus. For , where no memory effect is considered, the peak is more prominent, followed by a rapid decline. Introducing the memory effect with lowers the peak and a more gradual decline over time, which is consistent with observations from real-world epidemics. The solution curve shows that fractional order values influence the proportion of exposed cattle. Infected cattle populations exhibit a similar pattern, where the maximum levels of infection decrease as the fractional order value decreases, reflecting the memory effects on behavior. With a stronger memory effect, as observed for , the infection peak diminishes and the decline becomes more gradual. This behavior closely aligns with real-world scenarios, where infections often persist longer because of extended recovery times and the influence of past disease states on current dynamics. A similar pattern was observed in the cattle population in quarantine. The quarantined population reaches its peak and declines faster, reflecting a system with no memory influence. With the memory effect, the peak is lower and takes a longer time to stabilize, reflecting more realistic epidemic dynamics. The fractional order derivative accounts for the cumulative history of exposed cattle entering quarantine, resulting in a slower buildup and a prolonged presence of cattle in quarantine. The recovered cattle population exhibits a steady increase over time, with slight variations in trajectories depending on fractional orders, ultimately converging to the integer order solution. The solution curve takes longer to reach equilibrium for smaller values of fractional orders. However, the higher values of lead to quicker convergence to the equilibrium state. Simulations of the fractional LSD model (3.1) reveal that the solution curves in all population classes converge more rapidly to an equilibrium state with higher fractional orders, whereas convergence is slower with lower fractional orders. The incorporation of memory effects via fractional derivatives offers a more accurate representation of disease dynamics within cattle populations.

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Fig 4. Simulation of the fractional system (3.1) with varying fractional orders, we choose and other parameters as in Table 1, we get .

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To explore epidemic scenarios, we consider the transmission rate . In this case, the , which is greater than one and the results are presented in Fig 5a–5f. Visual dynamics shows that the solution curve converges to the EE for all values of fractional order. The higher fractional orders increase the peak in infected cattle, whereas the peak diminishes as the fractional order decreases. Similar to the case where , the results for indicate that the solution curve for all population groups converges to the equilibrium state more quickly at higher fractional orders compared to lower ones.

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Fig 5. Simulation of the fractional system (3.1) with varying fractional orders, we choose and other parameters as in Table 1, we get .

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Figs 6 and 7 illustrate the effectiveness of a vaccination program to vaccinate cattle against LSD. The corresponding results are summarized in Tables 3 and 4, for fractional orders and 0.8. Implementing the vaccination strategy leads to a noticeable reduction in both the exposed and infected populations, highlighting the program’s efficacy. These results underscore the importance of ensuring the availability and effective distribution of vaccines to control infection rates among cattle. The findings suggest that the proposed model is a robust framework for managing and mitigating Lumpy skin disease.

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Fig 6. Simulation of the fractional system (3.1) with and without vaccination.

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Fig 7. Effect of on exposed and infected cattle for different fractional orders.

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Table 3. Impact of vaccination on exposed cattle population at EE point for different fractional orders.

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Table 4. Impact of vaccination on infected cattle at EE point for different fractional orders.

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The findings in Fig 8 demonstrate that reducing the contact rate through targeted strategies can significantly curb epidemic spread. Effective measures to achieve this include implementing targeted prevention initiatives and conducting educational campaigns that emphasize the importance of avoiding contact with insect vectors such as mosquitoes and flies.

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Fig 8. Effect of on exposed and infected cattle for different fractional orders.

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To prevent the spread of the LSD epidemic among cattle, we examined the effects of different levels of quarantine on LSD dynamics. The visual representation of these dynamics is shown in Fig 9, and the detailed results are presented in Table 5. The findings indicate that disease transmission decreases as the percentage of livestock in quarantine increases. The analysis shows that quarantine can control or eliminate the disease if appropriate quarantine measures are implemented at varying levels, supported by public education.

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Fig 9. Effect of q on infected cattle populations for different fractional orders.

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Table 5. Impact of quarantine on infected cattle at EE point for different fractional orders.

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Fig 10 presents the transient dynamics as we change some parameters. In sub-plots (a) and (b), we can see that as q increases, the exposed, susceptible, vaccinated and infected cattle decrease, while the recovered population increases. Sub-plots (c) and (d) show that the increase in vaccination rate led to a decrease in the susceptible, exposed and infected populations. The last two sub-plots (e) and (d) demonstrate that increasing the infection rate of susceptible cattle increases the exposed, quarantined, vaccinated and recovered cattle populations.

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Fig 10. Change in population dynamics of model (3.1), as we vary the following parameters: (a) and (b) quarantine rate q of exposed cattle; (c) and (d) vaccination rate of susceptible cattle; (e) and (f) infection rate of susceptible cattle.

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The phase planes shown in Fig 11 illustrate the combined effects of several key parameters on the behavior of the basic reproduction number . As observed in Fig 11b, when and vary, the behavior of changes with smaller values corresponding to lower and higher . Similarly, we can see that the other parameters influence disease dynamics.

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Fig 11. Variations in the basic reproduction number with key parameters of the model (3.1).

https://doi.org/10.1371/journal.pone.0327673.g011

6 Conclusion

This study thoroughly investigates the Lumpy skin disease model, incorporating vaccination and quarantine strategies. The Caputo fractional derivative is used to extend the integer-order model to its fractional form. Using an appropriate qualitative method, the existence and uniqueness of solutions for the dynamic model were established. The validity of the model is confirmed by qualitative analysis, such as existence and uniqueness, positivity and boundedness of the solution. The equilibrium points of the model were determined, and stability analysis was conducted for both the disease-free state and the endemic state . The basic reproduction number was calculated, and the effects of key parameters were analyzed using normalized forward sensitivity indices. A Lagrange polynomial interpolation-based numerical scheme was employed to solve the fractional model, with graphical results presented for varying model parameters and fractional order . It is observed that the total number of infections during the disease is influenced by the fractional-order, which represents the memory property. The solution curve takes longer to reach the equilibrium point for lower fractional order values. The impact of vaccination compared to no vaccination was investigated by simulation. The results show that vaccination plays an important role in disease control. Increasing vaccination coverage among cattle reduces infection rates. This finding is consistent with what was achieved in the research [32]. Furthermore, we examined the effect of quarantine measures on the infected cattle population. The results indicate that quarantining exposed cattle is highly effective in curbing the spread of the disease. Compared to the previous studies [7] and [17,18] our model provides a more comprehensive framework by integrating both vaccination and quarantine, with the fractional-order approach enhancing prediction accuracy. Our findings highlight the effectiveness of the strategies adopted in improving the accuracy of the prediction, offering more reliable information on the dynamics of the LSD disease. Furthermore, the results indicate that apply vaccination before infection is more effective than quarantine after infection in reducing the prevalence of LSD diseases, but applying both strategies will better control the disease’s spread. To prevent LSD outbreaks, we recommend the following actionable strategies based on our results: ensuring widespread and timely vaccination of susceptible cattle, early detection and isolation, and implementing biosecurity measures to lower disease transmission (for more details, see, e.g., [10] and [11]). These strategies, grounded in model sensitivity analysis and simulations, provide reliable information to stakeholders, including farmers, veterinarians, and policymakers, to effectively manage LSD. Some limitations of this work are assuming a homogeneous cattle population (no spatial spread), not considering explicitly vector dynamics (e.g., flies/mosquitoes), and ignoring the impact of time delays between the infected population and susceptible/vaccinated population. In addition, we have neglected to evaluate the cost-effectiveness of vaccination compared to quarantine. Some future work can explore the model’s dynamics using fractional operators with nonlocal and nonsingular kernels.