Ariyanto*
| Maria Lobo | Farly Oktriany Haning | Keristina Br Ginting | Jusrry Rosalina Pahnael | Elisabet Tangkonda
© 2025 The authors. This article is published by IIETA and is licensed under the CC BY 4.0 license (http://creativecommons.org/licenses/by/4.0/).
OPEN ACCESS
African swine fever (ASF) is a highly contagious viral disease that results in very high mortality rates among pigs and causes significant economic losses worldwide. In Indonesia, ASF has spread to 32 provinces since the first outbreak in 2019, with East Nusa Tenggara being one of the most severely affected areas. This study developed a nonlinear differential equation model to analyze the dynamics of ASF transmission and evaluate the combined effectiveness of biosecurity measures and vector control strategies in controlling the disease's spread. The model calculates the basic reproductive number both without and with vectors. An integrated approach that combines biosecurity measures and optimal vector control can significantly reduce the risk of infection, depending on the effectiveness of biosecurity (p) and the effectiveness of tick vector control (q). These findings suggest that the synergistic approach of combining biosecurity and optimal vector control is highly effective in reducing the spread of ASF in East Nusa Tenggara. This provides a scientific foundation for developing adaptive disease control policies in Indonesia.
African swine fever, model mathematical, basic reproduction number, biosecurity strategy, optimal control, vector ticks
ASF is a highly virulent viral disease affecting both domestic and wild pigs, causing significant socioeconomic losses to the pig industry [1]. Globally, ASF is estimated to cause annual economic losses exceeding USD 2 billion [2]. In Indonesia, ASF was first detected in North Sumatra in 2019 and subsequently spread to 32 provinces [3]. East Nusa Tenggara Province experienced ASF outbreaks in 2020 and 2021, resulting in extremely high pig mortality, with cumulative losses reaching several hundred thousand animals [4]. ASF cases in the East Nusa Tenggara region continue to exhibit a high incidence rate up to 2025, and this situation generates substantial economic losses for smallholder farmers whose livelihoods rely heavily on pig production. To date, no vaccine or commercial antiviral drug has been proven effective against ASF [5]. Implementing strict biosecurity remains the most effective prevention strategy [6]. In East Nusa Tenggara, local farmers implement traditional and adaptive practices, such as installing mosquito nets around pens, to minimize contact with potential vectors [7]. These local initiatives highlight the vital importance of community-based biosecurity in areas with limited veterinary infrastructure. Mathematical modeling serves as a key tool for understanding disease transmission dynamics and developing effective control strategies [2].
Various mathematical models have been developed to study the dynamics of ASF transmission. Chuchard et al. [7] proposed a human-mediated transmission pathway, while Ayihou et al. [8]. Developed an eight-compartment model that yielded a fundamental reproduction number value, which indicates a high level of transmission based on analysis of Benin data. Kouidere et al. [9] further proposed an ASF transmission model that integrates vector dynamics and optimal control theory to formulate effective management intervention strategies.
The mathematical models currently developed primarily focus on either the African or European context, which means they do not accurately reflect the ecological, cultural, and agricultural conditions in Indonesia, particularly in East Nusa Tenggara. Recent research on ASF in this region can be found in the articles by Bulu et al. [10] and Pandarangga et al. [11]. Small-scale pig farming in East Nusa Tenggara is closely associated with varying levels of biosecurity practices, as well as the high mobility of pigs and people. Existing models often overlook these critical factors and seldom incorporate local biosecurity knowledge with vector control strategies. As a result, a research gap remains in understanding ASF transmission within the unique context of Indonesian veterinary environmental health.
The novelty of this study lies in the development of a nonlinear differential equation system model that explicitly represents the dynamics of ASF. This model integrates biosecurity and vector control interventions to evaluate their effectiveness. The findings are expected to provide a scientific foundation for formulating ASF control policies that are adaptable to the sociocultural conditions in East Nusa Tenggara.
2.1 Mathematical model formulation
In this study, we developed a model that consists of five compartments termed as susceptible pigs ($S_p$), exposed pigs ($E_p$), infected pigs ($I_p$), susceptible ticks ($S_v$), and infected ticks ($I_v$). This compartmental structure is an extension of the model proposed by Kouidere et al. [2]. The model examined in this study can be formulated as follows.
$\begin{aligned} & \frac{d S_p}{d t}=\Lambda_p-\beta_1 \frac{S_p I_p}{N_p}-\beta_2 \frac{S_p I_v}{N_p}-\mu_p S_p \\ & \frac{d E_p}{d t}=\beta_1 \frac{S_p I_p}{N_p}+\beta_2 \frac{S_p I_v}{N_p}-\gamma E_p-\mu_p E_p \\ & \frac{d I_p}{d t}=\gamma E_p-\mu_p I_p \\ & \frac{d S_v}{d t}=\Lambda_v-\beta_3 \frac{S_v I_p}{N_v}-\mu_v S_v \\ & \frac{d I_v}{d t}=\beta_3 \frac{S_v I_p}{N_v}-\mu_v I_v\end{aligned}$ (1)
where,
Let and denote the total pig and tick population, respectively, assumed to be constant. Based on Eq. (1), the following result is obtained:
$\Lambda_p=\mu_p N_p$ and $\Lambda_v=\mu_v N_v$ (2)
From Eqs. (1) and (2), we can obtain the disease-free equilibrium, which represents a system state where there is no infection.
$E_0=\left(N_p, 0,0, N_v, 0\right)$ (3)
2.2 Basic reproduction number
The Next Generation Matrix (NGM) method was used to derive the basic reproduction number, as formulated in references by Bani-Yaghoub et al. [12] and Ndii et al. [13], where the transmission matrix ($\mathcal{F}$) and transition matrix ($\mathcal{V}$) are defined.
$\mathcal{F}=\left(\begin{array}{c}\beta_1 \frac{S_p I_p}{N_p}+\beta_2 \frac{S_p I_v}{N_p} \\ 0 \\ 0\end{array}\right)$ (4)
$\mathcal{V}=\left(\begin{array}{c}-\gamma E_p-\mu_p E_p \\ \gamma E_p-\mu_p I_p \\ \beta_3 \frac{s_v I_p}{N_v}-\mu_v I_v\end{array}\right)$ (5)
From Eqs. (4) and (5), the following result is obtained:
$F=\left(\begin{array}{ccc}0 & \beta_1 & \beta_2 \\ 0 & 0 & 0 \\ 0 & 0 & 0\end{array}\right)$ (6)
$V^{-1}=\left(\begin{array}{ccc}-\frac{1}{\gamma+\mu_p} & 0 & 0 \\ -\frac{\gamma}{\mu_p\left(\gamma+\mu_p\right)} & -\frac{1}{\mu_p} & 0 \\ -\frac{\gamma \beta_3}{\mu_p \mu_v\left(\gamma+\mu_p\right)} & -\frac{\beta_3}{\mu_p \mu_v} & -\frac{1}{\mu_v}\end{array}\right)$ (7)
The next-generation matrix is obtained from $-T V^{-1}$, as follows:
$\begin{aligned} & K= \left(\begin{array}{ccc}\frac{\gamma \beta_1}{\mu_p\left(\gamma+\mu_p\right)}+\frac{\gamma \beta_2 \beta_3}{\mu_p \mu_v\left(\gamma+\mu_p\right)} & \frac{\beta_1}{\mu_p}+\frac{\beta_2 \beta_3}{\mu_p \mu_v} & \frac{\beta_2}{\mu_v} \\ 0 & 0 & 0 \\ 0 & 0 & 0\end{array}\right)\end{aligned}$ (8)
According to Eq. (8), the basic reproduction number ($R_0$) is termed as the spectral radius of the NGM. Hence, the value of $R_0$ is given as follows:
$\mathcal{R}_0=\frac{\gamma \beta_1}{\mu_p\left(\gamma+\mu_p\right)}+\frac{\gamma \beta_2 \beta_3}{\mu_p \mu_v\left(\gamma+\mu_p\right)}$ (9)
Furthermore, the basic reproduction number, which explains the interaction between susceptible and infected pigs without the involvement of vector ticks, is given as follows:
$\mathcal{R}_{01}=\frac{\gamma \beta_1}{\mu_p\left(\gamma+\mu_p\right)}$ (10)
If $\mathcal{R}_0<1$, the disease will be eliminated from the population, whereas if $\mathcal{R}_0>1$, it is likely to spread more rapidly [14].
2.3 Model analysis
In this section, we will discuss the key properties of system (1), including the invariant region, positivity of solutions, and stability analysis of equilibrium points.
Invariant region and positivity of solution
Attention is restricted to the feasible region $\Omega=\Omega_p \times \Omega_v \subset R_{+}^3 \times R_{+}^2$, with:
$\Omega_p=\left\{\left(S_p, E_p, I_p\right) \in R_{+}^3: N_p \leq \frac{\Lambda_p}{\mu_p}\right\}$ (11)
$\Omega_v=\left\{\left(S_v, I_v\right) \in R_{+}^2: N_v \leq \frac{\Lambda_v}{\mu_v}\right\}$ (12)
Here are the results for the feasible region. The following three theorems are derived from the reference by Dayap and Rabajante [15] and Sasongko et al. [16].
Theorem 2.3.1 The region $\Omega=\Omega_p \times \Omega_v \subset R_{+}^3 \times R_{+}^2$ constitutes a positive invariant set for system (1) under non-negative initial conditions.
Proof: The summation of the ticks and pigs population in system (1) yields the following result:
$\frac{d N_v}{d t}=\Lambda_v-\mu_v N_v$ (13)
$\frac{d N_p}{d t}=\Lambda_p-\mu_p N_p$ (14)
Solving Eqs. (13) and (14) for $N_v$ and $N_p$ yields the bounded system $N_v \leq \frac{\Lambda_v}{\mu_v}$ and $N_p \leq \frac{\Lambda_p}{\mu_p}$.
Consequently, all feasible solutions of system (1) lie within the region.
Theorem 2.3.2 Let the system (1) have non-negative initial conditions. Then, its solution set is given by $\left(S_p(t), E_p(t), I_p(t), S_v(t), I_v(t)\right)$ and solution is non-negative for all $t>0$.
Proof: To prove that system (1) has non-negative solutions for all $t>0$, we first establish the positivity of $E_p(t)$, while the positivity of the remaining state variables follows by analogous arguments.
Assume the system is subject to non-negative initial conditions. From system (1), the following inequality holds:
$\frac{d E_p}{d t} \geq-\left(\gamma+\mu_p\right) E_p$ (15)
Eq. (15) takes the following form:
$E_p(t) \geq E_p(0) e^{-\left(\gamma+\mu_p\right) t}$ (16)
Since the initial condition $E_p(0)$ is non-negative and $e^{-\left(\gamma+\mu_p\right) t}$ is also non-negative, it follows that $E_p(t)$ remains non-negative for all $t \geq 0$.
Theorems 2.3.1 and 2.3.2 establish that system (1) well-posed and biologically meaningful.
Theorem 2.3.3 The disease-free equilibrium (DEF) $E_0$ of the system (1) is locally asymptotically stable when $\mathcal{R}_0<1$; otherwise, it becomes unstable.
Proof: To find the local stability of the DFE at $E_0$, the Jacobian matrix of system (1) will be derived as follows:
$J\left(E_0\right)=\left(\begin{array}{ccccc}-\mu_p & 0 & -\beta_1 & 0 & -\beta_2 \\ 0 & -\gamma-\mu_p & \beta_1 & 0 & \beta_2 \\ 0 & \gamma & -\mu_p & 0 & 0 \\ 0 & 0 & -\beta_3 & -\mu_v & 0 \\ 0 & 0 & \beta_3 & 0 & -\mu_v\end{array}\right)$ (17)
From the Jacobian matrix $J\left(E_0\right)$ above, the following characteristic polynomial is obtained:
$p(\lambda)=\left(\mu_p+\lambda\right)\left(\mu_v+\lambda\right)\left(\lambda^3+a_1 \lambda^2+a_2 \lambda+a_3\right)=0$
with
$\begin{gathered}a_1=2 \mu_p+\mu_v+\gamma \\ a_2=\mu_v\left(\gamma+\mu_p\right)+\mu_p \mu_p+\mu_p\left(\gamma+\mu_v\right)\left(1-R_0\right)+\frac{\gamma \beta_2 \beta_3\left(\gamma+\mu_v\right)}{\mu_v\left(\gamma+\mu_p\right)} \\ a_3=\mu_p \mu_v\left(\gamma+\mu_p\right)\left(1-R_0\right)\end{gathered}$
The characteristic equation $p(\lambda)$ of $J\left(E_0\right)$ yields the eigenvalues $\lambda_1=-\mu_p$, $\lambda_2=-\mu_v$, and $\lambda^3+a_1 \lambda^2+a_2 \lambda+ a_3=0$. By the Routh-Hurwitz, all roots are negative provide that $a_1>0$, $a_3>0$ and $a_1 a_2>a_3$ whenever $R_0<1$. Hence, the equilibrium point $E_0$ is stable if $\mathcal{R}_0<1$. Conversely, if $\mathcal{R}_0>1$, the equilibrium point $E_0$ becomes unstable.
3.1 The constructed estimation of $\mathcal{R}_0$ according to the interaction between susceptible and infected pigs without vector ticks involved
First, we construct the estimation of $\mathcal{R}_{01}$ from Eq. (10). This estimation at $E_0$ is based on the assumption that the number of infected pigs, $I_p$, grows exponentially at the same rate over a short period of time [17].
$E_p(t)=E_p(0) e^{r t}$ (18)
$I_p(t)=I_p(0) e^{r t}$ (19)
with $E_p(0)$ and $I_p(0)$ denoting the initial numbers of exposed and infectious pigs, respectively, and let $r$ represent the take-off rate of the early epidemic growth. Next, by substituting (18) and (19) into (1) and assuming $E_p \approx S_p$ and $I_p \approx S_p$ at the early stage of the epidemic, we obtain:
$\mathcal{R}_{01 { est }}=\left(1+\frac{r}{\mu_p}\right)\left(1+\frac{r}{\gamma+\mu_p}\right)$ (20)
with
$\beta_1=\frac{\left(r+\gamma+\mu_p\right)\left(r+\mu_p\right)}{\gamma}$ (21)
The values of the model parameters and their corresponding sources are presented in Table 1.
The data of ASF cases in East Nusa Tenggara Province, Indonesia, are displayed in Figure 1.
Figure 1 displays the monthly incidence of ASF in East Nusa Tenggara, Indonesia, which reached its highest peak in January 2021. This will be used as the basis for estimating the $\mathcal{R}_0$ value.
Table 1. Parameter value and references
|
Parameter |
Value |
Source |
|
$\mu_p$ |
0.16667 (month)-1 |
[18] |
|
$\gamma$ |
2.73 (month)-1 |
[19] |
|
$\mu_v$ |
0.017 (month)-1 |
[20] |
Figure 1. ASF cases incidence in East Nusa Tenggara from 2020 to 2025
Figure 1 presents the monthly ASF incidence in East Nusa Tenggara, Indonesia, with the highest peak observed in January 2021. This dataset forms the basis for estimating the early epidemic growth rate and the basic reproduction number $\mathcal{R}_0$.
Figure 2 displays the fitted model curve together with the observed ASF incidence. The close visual agreement between the model output and the reported data supports the reliability of the estimated take-off rate and the basic reproduction number $\mathcal{R}_0$. The analysis yields a take-off rate $r=0.2(95 \% C I: 0.1854-0.2566)$ with a coefficient of determination $R^2=0.7763$, indicating a strong correspondence between the model and the observed data, as well as stable estimate with minimal uncertainty. Using the parameter values provided in Table 1, together with $r=0.2$ and Eqs. (20) and (21), the basic reproduction number is estimated as $\mathcal{R}_{01}=2.35$ for $\beta_1=0.4$. The transmission potential of ASF pigs is higher than through ticks, as the virus spreads more readily via direct contact with bodily fluids [20]. Given the limited field data on ASF-infected ticks, this study assumes that $\beta_2$ is 32% of $\beta_1$, and $\beta_3$ is 22% of $\beta_2$, yielding $\beta_2$ = 0.128, and $\beta_3$ = 0.0282. Thus, based on Eq. (9) and the parameter values in Table 1, including $\beta_2$ and $\beta_3$, the basic reproduction number involving transmission through infected ticks vectors is obtained $\mathcal{R}_0=3.5$. Several $\mathcal{R}_0$ values from different countries are presented for comparison with the $\mathcal{R}_0$ value in East Nusa Tenggara.
Figure 2. The fitting between observational data and fitted model
The reproduction number of ASF transmission in East Nusa Tenggara is relatively high compared to cases in other countries, as shown in Table 2.
Table 2. Values of $\mathcal{R}_0$ several countries
|
Country |
$\mathcal{R}_0$ Value |
Source |
|
Benin |
2.78 |
[8] |
|
Uganda |
3.24 |
[21] |
|
Czech Republic |
1.95 |
[22] |
|
Belgium |
1.65 |
[23] |
3.2 A mathematical model with biosecurity measures and contact control between susceptible pigs and ASF-infected ticks
The model incorporating biosecurity measures and contact control between susceptible pigs and ASF-infected ticks is formulated as follows:
$\begin{aligned} & \frac{d S_p}{d t}=\Lambda_p-(1-p) \beta_1 \frac{S_p I_p}{N_p}-\beta_2(1-q) \frac{S_p I_v}{N_p}-\mu_p S_p \\ & \frac{d E_p}{d t}=(1-p) \beta_1 \frac{S_p I_p}{N_p}+(1-q) \beta_2 \frac{S_p I_v}{N_p}-\gamma E_p-\mu_p E_p \\ & \frac{d I_p}{d t}=\gamma E_p-\mu_p I_p \\ & \frac{d S_v}{d t}=\Lambda_v-\beta_3 \frac{S_v I_p}{N_v}-\mu_v S_v \\ & \frac{d I_v}{d t}=\beta_3 \frac{S_v I_p}{N_v}-\mu_v\end{aligned}$ (22)
where, p denotes the effectiveness of biosecurity measures in reducing transmission between pigs, and q denotes the effectiveness of controlling ASF-infected ticks. By applying the NGM method as in the previous concept, the basic reproduction number derived from Eq. (22) is obtained as follows:
$\mathcal{R}_0=\frac{(1-p) \gamma \beta_1}{\mu_p\left(\gamma+\mu_p\right)}+\frac{(1-q) \gamma \beta_2 \beta_3}{\mu_p \mu_v\left(\gamma+\mu_p\right)}$ (23)
3.3 Numerical simulation
We performed numerical simulation to evaluate the time to disease extinction within the system and to identify the conditions that facilitate the emergence of an endemic state in the population. The simulation results are presented in Figures 3 and 4.
Figure 3. Numerical simulation of disease extinction in the population
Figure 3(a) shows a monotonic, near-exponential decline approximately $3.4 \times 10^5$ at the beginning of the observation period to about $0.2 \times 10^5$ by month 12, indicating a consistent depletion of the susceptible population throughout the study.
The $E_p$ compartment exhibits a pronounced transient dynamic, marked by a sharp decline during the first 0–2 months from approximately $2.0 \times 10^5$ to nearly zero, after which it remains at a very low level for the remainder of the time horizon. This pattern indicates that the exposure phase is short-lived before the system stabilizes near zero.
Figure 3(c) illustrates a wave-like infection dynamic, in which $I_p$ increases rapidly from approximately $1.6 \times 10^5$ to a peak of $3.0 \times 10^5$ arround month 1, followed by a gradual decline to about $0.8 \times 10^5$ by month 12. This pattern reflects an initial amplification phase of infection, followed by a subsequent dissipation phase as the system approaches a lower level of infectivity.
Figure 3(d) shows that the $S_v$ compartment exhibits an approximately linear upward trend, increasing from about $0.8 \times 10^6$ to around $2.7 \times 10^6$ over the 12-month horizon. This pattern indicates a steady expansion of the susceptible vector population throughout the study period.
Figure 3(e) shows a gradual decline from approximately $1.2 \times 10^6$ to about $1.11 \times 10^6$ by month 12, with a slightly steeper gradient toward the end of the time horizon. This pattern reflects a slow but continuous dissipation of the infected vector population throughout the study period.
Figure 4. The numerical solution is examined under conditions in which the disease persists within the population
Figure 4(a) shows that the susceptible pig population $S_p$ declines rapidly to near zero within approximately 5–6 months. This pattern suggests a high effective transmission pressure, leading to a rapid depletion of susceptibles through progression to infection-related compartments (or removal). Thereafter, the system approaches an endemic steady state with a very small susceptible pool.
Figure 4(b) illustrates the temporal evolution of the exposed pig population, $E_p$, over 12 months. The population experiences a sharp decline during the initial phase, with the most pronounced reduction occurring between months 2 and 3. By month 5–6, $E_p$ approaches zero and remains negligible, indicating a rapid depletion of the exposed class and the system’s convergence toward an almost disease-free state.
Figure 4(c) illustrates the unimodal temporal of $I_p$. The infected pig population increases rapidly during the initial phase, reaching a peak of approximately $4.1 \times 10^5$ in month 2, and subsequently declines gradually and near-monotonically to about $1.0 \times 10^5$ by month 12. This trajectory suggests a transient early outbreak followed by relaxation toward a quasi-endemic regime.
Figure 4(d) indicates that the susceptible vector population $S_v$ declines sharply during the initial phase, decreasing from approximately $7.5 \times 10^5$ to a minimum of $(2.4-2.6) \times 10^5$ around months 4–5, before rising steadily to about $4.4 \times 10^5$ by month 12. This trajectory reflects an early depletion of susceptible vectors followed by progressive recovery toward a quasi-equilibrium state.
Figure 4(e) indicates that $I_v$ increases monotonically, rising rapidly during the initial phase from approximately $1.2 \times 10^6$, and then shifting to a slower growth rate that approaches saturation, reaching $(2.4-2.5) \times 10^6$ by month 12. This trajectory suggests a sustained accumulation of infected vector as the system moves toward a quasi-endemic regime.
In this section, the effects of parameters $p$ and $q$ on the dynamics of $E_p$ are examined via numerical simulations, by analyzing the temporal response of $E_p$ across a range of $(p, q)$ values and identifying the combinations that most effectively reduce exposure.
Figure 5 presents the simulation results for the reduction in the exposed compartment $E_p$ as the parameters $p$ and $q$ increase. The percentage reduction in $E_p$ is used as an indicator of the effectiveness of biosecurity ($p$) and contact control between susceptible pigs and ASF-infected ticks ($q$). Specifically, when $p$ = 50% and $q$ = 50%, $E_p$ decreases by 17.48%. When $p$ = 65% and $q$ = 65%, the reduction in $E_p$ increases to 22.73%, then to 25.18% for $p$ = 72% and $q$ = 72%, and reaches 29.72% for $p$ = 85% and $q$ = 85%. These findings indicate that strengthening biosecurity and contact control measures consistently reduces the proportion of pigs in the exposed compartment.
Figure 5. Simulation of changes in $E_p$ resulting from variation in the parameters $p$ and $q$
Next, the decline in the basic reproduction number $\mathcal{R}_0$ is analyzed in response to intervention strategies represented by variations in parameters $p$ and $q$. This analysis examines how changes in $p$ and $q$ influence $\mathcal{R}_0$ to identify the most effective combination of interventions for reducing transmission potential.
Figure 6 presents the simulated reduction in the basic reproduction number $\mathcal{R}_0$ as the parameters $p$ and $q$ increase. Quantitatively, when $p$ = 50% and $q$ = 50%, $\mathcal{R}_0$ = 1.775; at $p$ = 65% and $q$ = 65%, $\mathcal{R}_0$ decreases to 1.2425; and for $p$ = 72% and $q$ = 72%, $\mathcal{R}_0$ is further reduced to 0.994. At a higher level of intervention, namely $p$ = 85% and $q$ = 85%, $\mathcal{R}_0$ declines further to 0.5325. Overall, these results indicate that enhanced biosecurity and contact control are effective in reducing $\mathcal{R}_0$ to below one, thereby suggesting the potential for outbreak control from an epidemiological perspective.
Figure 6. Simulation of changes in $\mathcal{R}_0$ in response to variation in parameters $p$ and $q$
3.4 Sensitivity analysis
Local sensitivity analysis is applied when a small number of parameters are uncertain, with an approach based on the concept of partial derivatives [24-26]. The normalized sensitivity index formula used in this study is as follows:
$C_p=\frac{\partial \mathcal{R}_0}{\partial p} \frac{p}{\mathcal{R}_0}$
where, $p$ denotes a parameter.
The normalized sensitivity indices of the parameters $\gamma$, $\beta_1$, and $\beta_2$ with respect to $\mathcal{R}_0$ are presented in Figure 7, respectively.
Figure 7. Local sensitivity analysis: Normalized sensitivity indices
Figure 7(a) shows that the normalized sensitivity with respect to $\gamma$ decreases monotonically, indicating that larger $\gamma$ values consistently diminish its contribution to variations in $\mathcal{R}_0$. Figure 7(b) indicates that the sensitivity to $\beta_1$ increases nonlinearly, implying that direct transmission becomes progressively more influential in shaping $\mathcal{R}_0$ as $\beta_1$ increases. Meanwhile, Figure 7(c) exhibits a more gradual increase in sensitivity to $\beta_2$, with a gentler slope than that of $\beta_1$, suggesting that vector-mediated transmission contributes positively to $\mathcal{R}_0$ but with a comparatively moderate effect.
The normalized sensitivity indices for $\gamma$, $\beta_1$, and $\beta_2$, along with their corresponding interpretations, are summarized in Table 3.
Table 3. The results of the sensitivity index calculation
|
Parameter |
Sensitivity Index |
Interpretation |
|
$C_{\beta_1}=0.65$ |
The highest value |
A 1% increase in $\beta_1$ results in a 0.65% increase $\mathcal{R}_0$ |
|
$C_{\beta_2}=0.35$ |
Moderate |
A 1% increase in $\beta_2$ results in a 0.35% increase $\mathcal{R}_0$ |
|
$C_\gamma=0.06$ |
Low |
A 1% increase in $\gamma$ results in a 0.06% increase $\mathcal{R}_0$ |
This study analyzes local sensitivity by focusing on the parameters $\gamma$, $\beta_1$, and $\beta_2$, yielding the following results. The sensitivity analysis presented in Table 3 identifies $\beta_1$, and $\beta_2$ as the most influential parameters. Consequently, control strategies should prioritize reducing these parameters by enhancing biosecurity measures and minimizing contact between pigs and vector ticks.
This study is the first to examine ASF transmission in East Nusa Tenggara and, more broadly, the first in Indonesia to analyze its dynamics using a mathematical modeling approach. The results indicate that $\mathcal{R}_{01}=2.35$ in the absence of tick vector involvement, whereas with vector involvement, $\mathcal{R}_{0}=3.5$. Since the value of $\mathcal{R}_{0}$ remains well above unity, this indicates that ASF transmission in East Nusa Tenggara continues, and the disease persists within the pig population. This result is consistent with the findings of previous studies [8-27]. Effective control of ASF requires the implementation of strategic measures. These measures include improved biosecurity, disease surveillance, the establishment of ASF-free compartments, ongoing vaccine research and evaluation, and the strengthening of veterinary services. We applied biosecurity strategies and controlled contact between susceptible pigs and ASF-infected tick vectors to assess the reduction in the exposed compartment and the basic reproduction number, with the following results. At a biosecurity and vector control coverage of $p=q=50 \%$, the transmission effectiveness in the exposed compartment decreased by 17.48% with a reproduction number ($\mathcal{R}_{0}$) of 1.775. Increasing coverage to $p=q=65 \%$, reduced it to 22.73% ($\mathcal{R}_{0}$ = 1.2425), while at $p=q=72 \%$, the reduction reached 25.18% ($\mathcal{R}_{0}$ = 0.994), indicating a state near the control threshold. At higher coverage $p=q=85 \%$, a 29.72% reduction with $\mathcal{R}_{0}$ = 0.5325 confirmed that the epidemic can be effectively controlled. Based on mathematical modeling analysis, this study demonstrates that an integrated strategy of biosecurity and vector control is effective in reducing the spread of ASF in East Nusa Tenggara, Indonesia. Field evidence also indicates that the local practice of installing nets in pig pens is effective in controlling ASF. These findings are consistent with those of Boklund et al. [28] and Olesen et al. [29], who demonstrated that installing nets in pig pens is effective in controlling tick vectors and reducing the risk of ASF transmission.
The results show that ASF in East Nusa Tenggara is still spreading at a high level, with $\mathcal{R}_{01}=2.35$ and $\mathcal{R}_0=3.5$, indicating continued infection in the pig population. Numerical simulation indicates that a combination of enhanced biosecurity and vector control is effective in suppressing transmission. These findings emphasize the need for strengthened biosecurity policies, community-based vector control, and ongoing surveillance. This model has limitations due to its assumption of homogeneous mixing and simplification of host-vector interactions, necessitating field validation and the development of further stochastic models.
Our research funding comes from the Mathematics Study Program, Faculty of Science and Engineering, Nusa Cendana University (Grant No.: 247/UN15.22/PL/2025) dated on 26th March 2025.
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