Boundedness and Positivity Preserving Numerical Analysis of a Fuzzy-Parameterized Delayed Model for Foot and Mouth Disease Dynamics

Foot-and-mouth disease (FMD) is a viral disease that affects cloven-hoofed animals including cattle, pigs, and sheep, hence causing export bans among others, causing high economic losses due to reduced productivity. The global effect of FMD is most felt where livestock rearing forms an important source of income. It is therefore important to understand the modes of transmission of FMD to control its spread and prevent its occurrence. This work intends to address these dynamics by including the efficacy of active migrant animals transporting the disease from one area to another in a fuzzy mathematical modeling framework. Historical models of epidemics are determinable with a set of deterministic parameters and this does not reflect on real-life scenarios as observed in FMD. Fuzzy theory is used in this model as it permits the inclusion of uncertainties in the model; this makes the model more of a reality regarding disease transmission. A time lag, in this case, denotes the incubation period and other time-related factors affecting the spread of FMD and, therefore, is added to the current model for FMD. To that purpose, the analysis of steady states and the basic reproduction number are performed and, in addition, the stability checks are conveyed in the fuzzy environment. For the numerical solution of the model, we derive the Forward Euler Method and the fuzzy delayed non-standard finite difference (FDNSFD) method. Analytical studies of the FDNSFD scheme are performed for convergence, non-negativity, boundedness, and consistency analysis of the numerical projection to guarantee that the numerical model is an accurate discretization of the continuous dynamics of FMD transmission over time. In the following simulation study, we show that the FDNSFD method preserves the characteristics of the constant model and still works if relatively large time steps are employed; this is a bonus over the normal finite difference technique. The study shows how valuable it is to adopt fuzzy theory and time delays when simulating the transmission of the epidemic, especially for such diseases as FMD where uncertainty and migration have a defining role in transmission. This approach gives more sound and flexible grounds for analyzing and controlling the outbreak of FMD in various situations.