## 2 Materials and Methods
### System Dynamics Model
#### 2.4.1 Model of Water Quality  
The DO sag curve or equation, introduced by [PERSON] and [PERSON] in 1925, presented a basis for developing the water quality model (DO model), and the equation determines the relationship between DO and BOD over time based on a first-order differentiation [38]. This DO modeling by Streeter Phelps equation is the most popular method to determine the water quality of a river, where the variation of only two parameters (DO and BOD) are considered and our model is constructed based on it [49; 50]. However, to determine the variation of these two parameters, several factors such as temperature, velocity, water depth, river coefficients (deoxygenation and reoxygenation), oxygen solubility time, DO-saturation and riverbed coefficient, are also considered [50]. Figure 5 shows the stock-flow diagram of water quality, whose idea was derived from the Streeter-Phelps model, a pioneer in river water quality modeling [51]. In Figure 5, DO is the state variable, which is the amount of gaseous oxygen (O\({}_{2}\)) present in the river water. The DO value changes over time(t) based on two rate variables: reoxygenation and deoxygenation, where reoxygenation represents addition of molecular oxygen and deoxygenation represents its removal. Change in DO can be mathematically expressed as given below:  
\[\mathrm{dDO/dt=Reoxygenation(t)-Deoxygenation(t),} \tag{2}\]  
where reoxygenation is increased when water gets agitated, resulting in more atmospheric oxygen mixing with river water. It is estimated by the multiplication of reoxygenation coefficient and the difference between DO and DO saturation. DO saturation is the greatest concentration of DO present in river water under natural pressure and temperature conditions. The reoxygenation coefficient is the multiplier that measures the property of reoxygenation, and it varies from one water body to another based on velocity and water depth. We calculated Kr, according to the Streeter-Phelps' reaeration rate equation for this river system [52]. Figure 4: Causal loop diagram to evaluate the impacts on inclusive wealth (IW) index. Note: \(+\) and \(-\) indicate the causal links (arrows), referring to how the change in the independent variable influences the dependent variable. IW formation depends on balancing and reinforcing loops of the water quality model. Improved water quality enhances IW, which attracts immigration, leading to overpopulation, increased BOD concentration through greater BOD inflows, which decreases water quality, and in turn, reduces IW. Figure 5: System dynamic stock-flow diagram representing the simulated system, including sub-systems of water quality, natural capital, produced capital, and human capital. Deoxygenation is the removal of oxygen through pollution decay caused by bacteria. Bacteria consume DO to decompose the organic pollutant deposited in the river system. The deoxygenation depends on DO, O2 solubility time, BOD, and pollution decay coefficient (Kd). O2 solubility time corresponds to when oxygen is absorbed in river water by direct diffusion and surface-water agitation, and it is estimated to take six years for O2 to diffuse from the surface to a depth of 6 m in water which is physically undisturbed [53]. Pollution decay coefficient (Kd) is a factor that determines the property of deoxygenation, and it varies from one water body to another based on water velocity, water depth, and riverbed activity coefficient. We calculated Kd according to the Streeter-Phelps' deoxygenation rate equation for Turag-Biriganga river system [54]. A riverbed is the ground or channel bottom, which a river flows over, and it is the physical confine of the normal water flow [55]. Typically, riverbeds are composed of gravel and sand beds. The riverbed of this river system is a sandbed, which is composed of sand and smaller-sized particles. The function and materials of the riverbed significantly affect the river water quality [56]. Therefore, the riverbed activity coefficient affects the decomposition of pollutants.