## 4 Antibiotic Fate Models  
The complexity of mathematical simulations of antibiotics has grown as a result of the multitude of inclusion of physio-biochemical, hydrological, and hydrodynamic interactions [92]. One aspect is the systematic modeling of mixing in riverine transport. The mixing in the riverine transport can be longitudinal, transverse, and vertical and is represented by one (1D), two (2D), or three (3D) dimensions depending on the nature and the purpose of the simulation required. Rivers with less storage, abstraction, and control are often studied using one-dimensional models under normal circumstances (natural flow), with the assumption that longitudinal dispersion plays a more significant role than vertical and traverse movement [2; 24]. However, in regulated rivers and stagnant water, mixing is greatly influenced by various artificial regulations and abstractions (e.g., dams, shuices, canals, and storage), hydrodynamic processes, geometry, secondary flow, and aquatic organisms [24]. In stagnant water, mixing can be influenced by longitudinal dispersion, lateral mixing (diffusion), and vertical mixing and can be represented by 2D/3D modeling. As a result, realistic mixing representation grows more complex, progressing to 1D and 3D modeling. The dynamic phenomena occurring in surface waters associated with the spread of various pollutants are often described using ordinary and partial differential equations. In a 1D to 3D process-based reactive modeling, the mass balance of a pollutant is generally expressed using the pollutant transport advection-dispersion equation (Equation (6)) [93]:  
\[\frac{\partial C}{\partial t}=-u\frac{\partial C}{\partial x}-v\frac{\partial C }{\partial y}-w\frac{\partial C}{\partial z}+\frac{\partial}{\partial x} \bigg{(}D_{x}\frac{\partial C}{\partial x}\bigg{)}+\frac{\partial}{\partial y }\bigg{(}D_{y}\frac{\partial C}{\partial y}\bigg{)}+\frac{\partial}{\partial z }\bigg{(}D_{z}\frac{\partial C}{\partial z}\bigg{)}+K_{t}C+S \tag{6}\]  
where \(u\), \(v\), and \(w\) are the velocity components in the \(x\), \(y\), and \(z\) coordinates in three dimensions; \(Dx\), \(Dy\), and \(Dz\) are components of the dispersion coefficient; \(S\) is an external source to the system due to loads, boundaries and descends, and \(t\) is the time. \(K_{t}\) is the total of all constants (Equation (7)) that account for all biotic and abiotic processes involved in the overall removal of antibiotics [94]. Very often, all the degradation processes are regarded to be first-order constants:  
\[K_{t}=K_{b}+K_{h}+K_{p}+K_{o}+K_{v}+K_{bc}+K_{s} \tag{7}\]  
where \(K_{b}\), \(K_{h}\), \(K_{p}\), \(K_{o}\), \(K_{v}\), \(K_{bc}\) and \(K_{s}\) are the lipolysis, hydrolysis, photolysis, oxidation, volatilization, bioconcentration, and sorption constants, respectively. The 1D model is better suited to study catchment-scale runoffs and assess pollutant loadings based primarily on physical transport processes. A limitation of 1D models to simulate an ecological interaction paradigm with the surrounding environment is surmounted by coupling it with fugacity models. Multimedia fate models are good at simulating the distinct behavior of antibiotics in a multimedia environment [24]. However, consideration of exposure in a multimedia environment (air, soil, water, and sediment), the fugacity concept, may increase the level of complexities in modeling. Fugacity is the tendency of a compound to prefer one phase over the other in a similar pressure and temperature. This model includes partitioning concentration among multimedia, accounting transformation, and handling variabilities. In fugacity models, a different simulation technique comprises different combination processes. The level of accounting corresponds to Level I, considering equilibrium distribution among the compartments without transformations, and to Level IV, giving advection, transformation, hysteresis (antibiotic hydrolyzed back to the parent molecule), and variabilities (without equilibrium distribution between phases) [11].