Mathematical Modeling of Foot and Mouth Disease Spread on Livestock using Saturated Incidence Rate

ABSTRACT

Foot and Mouth Disease (FMD) is an acute infectious disease that attacks livestock, thus threatening the availability of food and the husbandry industry. This paper discusses the formulation of a mathematical model for the spread of FMD in livestock with a saturated incidence rate. The research method used is quantitative mathematical modeling with simulation, with stages including problem identification, determining assumptions, model formulation, analysis and model simulation. The discussion results obtained two equilibrium points, namely the non-endemic equilibrium point and the endemic equilibrium point, and then analyzed for stability. Numerical simulation is presented using Runge-Kutta approximation with MATLAB. Furthermore, after a sensitivity analysis, the parameters that greatly influenced the spread of FMD were direct or indirect contact (which led to the entry of the FMD virus) and the supporting capacity of livestock. Then the most influential parameter in reducing the spread of FMD is the application of culling on exposed animals and infected animals. The FMD modeling is a form of mathematical application to simulate the spread of disease on livestock.

Keywords:
Foot and mouth disease; Mathematical model; Saturated incidence rate; Stability.

A. INTRODUC TION
Foot and Mouth Disease (FMD) is an acute infectious disease that attacks livestock, so it becomes a serious threat to the husbandry industry and food stability (Bahiru & Assefa, 2022. FMD is caused by type A virus from the family Picornaviridae, genus Apthovirus (Belsham, 2020). Even though the fatality is low, this disease causes a decrease in body weight and milk production of livestock, thereby lowering the selling price. Cattle, sheep, goats, deer and pigs are very susceptible to FMD, with clinical symptoms seen after an incubation period of 2 to 14 days, even longer, especially for sheep and goats (Lazarus et al., 2021).
FMD transmission occurs when there is direct or indirect contact with infected animals (Mohr et al., 2018) (Schnell et al., 2019). Livestock are susceptible to infection through aerosols, inhalation, ingestion, and through natural or artificial mating (Arzt et al., 2018). Clinical symptoms in livestock infected with FMD include pyrexia, anorexia, excessive mucus discharge from the mouth and foaming, sores such as canker sores in the oral cavity and tongue, sores on the feet and loose nails, limping, shaking, and drastic drop in milk production (Dubie & Amare, 2020). FMD has seven different variants and more than 60 viral subtypes (Mushayabasa et al., 2011). In general, there is no universal vaccine against this disease, vaccines for FMD must be compatible with the types and subtypes present in the affected area. Some of the symptoms of FMD can be seen in Figure 1. Animals that infected by FMD cannot be treated. The efforts that can be done are increasing the immunity and body resistance of animals through supportive therapy by providing vitamin and feed supplements. Then for symptoms, appropriate therapy can be carried out, such as: giving fever reducers, pain relievers, and antibiotics to prevent infection (Cabezas et al., 2018).
The efforts to eradicate FMD are focused on preventive control. To increase the immunity of animals that are susceptible to contracting FMD, it is necessary to intensify the mass vaccination program Ringa & Bauch (2014) and carry out mitigation program in areas that have not been infected these efforts can be in the form of surveillance and the formation of early awareness and implementing disease resilience (Elnekave et al., 2016). This needs to be done in order to obtain a disease spread map as a basis for determining control measures, in addition to communicating, providing information and educating the farming community. In this article, in addition to using vaccination, researchers also involve the culling of infected animals as an effort to control the spread of FMD.
Mathematical models have been widely used to describe the problem of infectious diseases spread (Fahcruddin, 2019). Research on the spread of FMD has been carried out by several researchers such as (Mushayabasa et al., 2011), (Belayneh et al., 2020), (Gashirai et al., 2020), (Chanchaidechachai et al., 2021), and (Sseguya et al., 2021). (Mushayabasa et al., 2011) have studied the spread of FMD by paying attention the bilinear incidence rate. Then, (Belayneh et al., 2020) conducted studies on the spread of FMD using the SIR model in Ethiopia. (Gashirai et al., 2020) studied the spread of FMD by focused to vaccine failures and environmental transmission. Furthermore, (Chanchaidechachai et al., 2021) conducted a study on the spread of FMD using a spatial model in Thailand. Next, (Sseguya et al., 2022) conducted research on the spread of FMD in two locations that were close to each other. The four studies used the incidence rate form in formulating mathematical models. The mathematical model of the spread of FMD on livestock in this article refers to the model used by (Mushayabasa et al., 2011). The model has been modified, by changing the form of the incidence rate which was originally a bilinear incidence rate ( ) to a saturated incidence rate 1+ . Farther, the growth of livestock is also developed in the FMD distribution model using a logistic model ℎ (1 − 1 ).
The main contribution in this article is that by applying the saturated incidence rate and logistic growth to the FMD model in livestock, the mathematical model used is more applicable, especially in limited and specific areas.

B. METHODS
The type of research used is quantitative, with the following stages: 1. Problem identification is carried out to understand the problem to be formulated so that it can be notated into mathematical symbols. 2. Determine assumptions. The number of factors that influence the observed events needs to be simplified by assuming a simple relationship between variables. The assumptions here are divided into several categories, namely: a. Variable classification. Things that affect the behaviour of observations in step 1 are identified as variables, both in the form of independent variables and dependent variables. b. Determine the interrelation between the selected variables to be studied. The steps taken are to create a sub-model according to the assumptions made in the main model, then study separately on one or more independent variables.
3. The formulation of the model can be done either through functional relationships by creating compartment diagrams, mathematical equations, or with the help of a software or by analytical means. 4. Analysing the model. After the model is obtained then it is solved mathematically, in this case, the model is a system of differential equations and the analysis includes the equilibrium point, stability analysis, and sensitivity analysis. 5. Simulate model. Simulations are carried out to see whether the interpretation of the model made is rational or not, using MATLAB or MAPLE software. If the interpretation results meet the requirements and are rational, then the results can be accepted, only then can the implementation of the obtained model be carried out. The steps involved are presented in Figure 2.  The FMD epidemic model was constructed by consider the saturated infection. ( ) expressed the total population of animals, which is classified into five classes: susceptible animal ( ), vaccinated animals ( ), exposed animals ( ), and infected animals ( ), where: The assumptions used in the formation of mathematical models for spread of disease are birth rate of livestock fulfills the logistic function. Furthermore, the vaccinated livestock become susceptible again after the vaccine immunity is lost. Then, susceptible livestock become latent if there is direct or indirect contact with livestock infected with Apthovirus; according to the saturated incidence rate. Livestock that recover from infection become susceptible. Then, the following assumptions are taken in deriving the model (1): 1. Assumption 1. The population of susceptible animals increases due to growth that fulfills the logistic function ℎ ( ) (1 − ( ) 1 ), and reduced immunity in rapidly vaccinated animals ∅ > 0. 2. Assumption 2. Vulnerable livestock will become latent if there is direct or indirect contact with infected animals with a saturated incidence rate 1+ with maximum contact rate > 0 and intervention level > 0 . Then, latent animals will become infected and able to transmit the virus to susceptible animals at a rapid rate > 0. 3. Assumption 3. The efforts to control the spread of FMD are carried out by vaccinating susceptible animals at a rate 1 > 0, then extermination of infected animals and latent animals at a rate 2 > 0. 4. Assumption 4. Natural mortality of animals at a rate > 0, while the mortality due to FMD in infected animals at a rate > 0. Then, birth rate of animal more than natural mortality rate ℎ > .
The following is an illustration of the FMD transmission scheme, as shown in Figure 3. The description of the parameters used in the mathematical modelling of FMD on livestock is presented in Table 1 below. Therefore, a mathematical model of FMD transmission can be constructed based on the interaction diagram in Figure 1: with initial condition is (0) > 0, (0) > 0, (0) > 0, (0) > 0.

C. RESULT AND DISCUSSION 1. Non-Endemic Equilibrium Point
By setting the derivatives of each equation in system (1)

Basic Reproduction Number
The basic reproduction number (ℛ 0 ) is the average number of newly generated infected livestock by a single infected livestock. Using the next generation matrix [12], the non-negative matrix F, and the non-singular matrix V, from system (1), we obtained: Furthermore, the eigenvalues of V −1 are: The spectral radius of matrix V −1 is ( V −1 ) = ℛ 0 , where:

Local Stability of Equilibrium Point
We determine the Jacobian matrix and analyze its eigenvalues for all points, to investigate the local stability around each equilibrium point. The Jacobian matrix for system (1) (1) is locally asymptotically stable for 1 2 < ℛ 0 < 1, and it is unstable for ℛ 0 > 1 with = ( + 2 + )( + 2 + ) .

Sensitivity Analysis of the Reproductive Number
Sensitivity analysis is used to determine the most influential parameter in a model (Hurint et al., 2017). In this case, the sensitivity index will be determined for each parameter involved in the basic reproduction number (ℛ 0 ) from the FMD model with saturated incidence rate. The value of ℛ 0 that will be used in the sensitivity analysis of FMD model with saturated incidence rate as follows: The parameter sensitivity index is formulated as follows: where: = parameter sensitivity index = analyzed parameters.
There are 8 parameters whose sensitivity index will be calculated, namely, β, γ, u 2 , d, k 1 , u 1 , ∅, r h . The results of the sensitivity index of the FMD model parameter with the saturated incidence rate can be seen in the following equation: From the Equation 3, it can be seen that the results of the sensitivity analysis to the parameters, , , 1 are always positive, so that they are directly proportional to ℛ 0 , while the sensitivity analysis to the parameters 1 , 2 , , ∅, and ℎ can be negative, so that it is inversely proportional to the ℛ 0 . Furthermore, the most influential parameters in the spread of FMD on livestock with saturated incidence rate are and 1 . Then, the parameter that greatly influences the decrease in the spread of FMD on livestock is 2 .

Numerical Simulation
This section presents a simulation of the FMD model on livestock using saturated incidence rate. The simulations carried out are classified into two conditions, namely conditions without spread of FMD and conditions when FMD spread occurs. The simulations were carried out using MATLAB software by entering initial values for each population. The condition without FMD occurs when there are no FMD spreading animals, in other words there is no FMD spread on livestock (ℛ 0 < 1). The simulation was carried out for 7 days. The parameter values used in the simulation are presented in the Table 2. Based on the parameter values in Table 2, it is obtained that ℛ 0 = 0.0037 < 1 and 0 = (225, 6, 0, 0). The following is a numerical simulation result when there is no FMD spread, as shown in Figure 4. From Figure 4 it can be seen that all populations produce graphs that tend to converge to the non-endemic equilibrium point of FMD model. Furthermore, the population of susceptible and vaccinated animals tends to increase because there is no spread of FMD on animals. Then, the population of exposed animals and infected animals in conditions free from the spread of FMD will tend to decrease until finally exhausted. The spread of FMD occurs when there are animals that are infected and then infect FMD to other animals (ℛ 0 > 1) . Based on the parameter values in Table 2, it is obtained that ℛ 0 = 2.508 > 1 and * = (44,0,235,38). The following is the result of a numerical simulation on the spread of FMD with a saturated incidence rate, as shown in Figure 5. The non-endemic point P * is asymptotically stable. Figure 5 it can be seen that all populations produce graphs that tend to converge to the endemic equilibrium point of FMD model. Furthermore, the population of exposed animals and infected animals tends to increase before finally being in a constant condition. The results of the analysis sensitivity simulation when FMD spread occurs are presented in Table 3.  Table 3, several conclusions were obtained: a. The sensitivity index of parameter is 1, it means if the rate of direct contact with infected animals increases by 10%, then the value of ℛ 0 will increase by 10%. b. The sensitivity index of parameter is 0.861, it means if the incubation rate on animals increases by 10%, then the value of ℛ 0 will increase by 8.6%. c. The sensitivity index of parameter 2 is -0.0184, it means if the culling rate on animals increases by 10%, then the value of ℛ 0 will decrease by 0.18%. d. The sensitivity index of parameter is -0.000049, it means if the mortality rate due to FMD on animals increases by 10%, then the value of ℛ 0 will decrease by 0.00049%. e. The sensitivity index of parameter 1 is 1, it means if the total supporting capacity of the animal population increases by 10%, then the value of ℛ 0 will increase by 10%. f. The sensitivity index of parameter 1 is -0.00072, it means if the vaccination rate of animals increases by 10%, then the value of ℛ 0 will decrease by 0.0072%. g. The sensitivity index of parameter ∅ is -0.0000000019, it means if the waning rate of animals increases by 10%, then the value of ℛ 0 will decrease by 0.000000019%. h. The sensitivity index of parameter ℎ is 0.96, it means if the birth rate of animals increases by 10%, then the value of ℛ 0 will increase by 9.6%.
Based on the explanation above, there are 8 parameters that affect the value of ℛ 0 . Furthermore, the most influential parameters in the spread of FMD on animals with a saturated incidence rate are and 1 with sensitivity index of 1. Then the parameter that is very influential in reducing the spread of FMD on animals is 2 with a sensitivity index of -0.0184. The impact of vaccination and animals culling on the spread of FMD is presented in Figure 6. Figure 6. The dynamics of all population for ℛ 0 > 1. The non-endemic point P * is asymptotically stable Figure 6 shows the estimated population of animals infected with FMD for Ro > 1 with different control applications. For cases u1 = 0.16 and u2 = 0.2, the infected animal population converges to 38, then for cases u1 = 0.30 and u2 = 0.35, the infected animal population is 37. Furthermore, for u1 = 0.5 and u2 = 0.6, it is obtained that infected by 36. Thus, from the simulation, it can be concluded that efforts to reduce FMD infected animals can be carried out maximally if large numbers of livestock are vaccinated and large numbers of infected animals are exterminated.

D. CONCLUSION AND SUGGESTIONS
In this paper discussed the model for foot-and-mouth disease (FMD) considering the saturated incidence rate. By determining the number base re-production (ℛ 0 ), the existence and the stability of equilibrium point of FMD model can be analyzed. Furthermore, the number of exposed animals and the number of infected animals in conditions without the spread of FMD, will tend to decrease until they are finally exhausted. It is different when there is a spread of FMD, the number of exposed animals and the number of infected animals tends to increase before finally being in a constant condition. Furthermore, the impact of vaccination and culling on livestock can reduce the number of animals infected by FMD. The PMK model in this article uses a system of differential equations. However, the cost analysis in implementing vaccination and culling on livestock has not been discussed in this article. So that further research is needed for the application of optimal control involving the cost function. In addition, it is necessary to review the development of PMK distribution models such as more complex discrete models.