## 2 COVID-19 Mathematical Modeling
### Lotka-Volterra Dynamic Model  
It is well-known that dynamic models play important roles in describing the interactions among specific behaviours. We propose a new dynamic model inspired from the generalized Lotka-Volterra (gLV) model [10; 11; 12; 13] for simulating predator-prey dynamics. The classic model has been suitably modified to build the healthy-infected individual population dynamics model. The gLV model is a first order nonlinear system of differential equations. In its discrete form, the gLV is represented as a  
Figure 8: Number of daily confirmed cases of COVID-19 from 22 January to 31 March and the prediction based on the statistical power series model in the following projected two weeks (until 14 April). group of first order nonlinear difference equations that relate the dissimilarity between the abundance levels of species at time \(t\) with respect to time \(t-1\). Let \(\{x_{i}(t);i=1,...,N\}\) be the relative abundance level of the \(i\)th species at time \(t\) whose intrinsic growth rate is \(g_{i}\). Moreover, let \(c_{ij}\) represent the strength of the influence of species \(i\) onto species \(j\) (a.k.a., the 'interaction coefficient'). Thus, the gLV model is given by:  
\[\frac{\mathrm{d}}{\mathrm{d}t}x_{i}(t)=g_{i}x_{i}(t)+x_{i}(t)\sum_{j=1}^{N}c_ {ij}x_{j}(t)\quad i=1,\ldots,N. \tag{3}\]  
The above framework was extended to model the effects of external perturbations (e.g., antibiotics, diets) onto the microbial community structure [12; 13]. This was obtained by adding another term to (3) which modulates the influence of each stimulating source into each member of the ecosystem. Mathematically, let \(\epsilon_{il}\) represent the'sensitivity' of the \(i\)th species in response to \(l\)th behaviour or control action with signal strength \(u_{l}\). The resulting gLV model is given by [13]:  
\[\frac{\mathrm{d}}{\mathrm{d}t}x_{i}(t)=g_{i}x_{i}(t)+x_{i}(t)\sum_{j=1}^{N}c_ {ij}x_{j}(t)+x_{i}(t)\sum_{l=1}^{L}\epsilon_{il}u_{l}(t). \tag{4}\]  
The proposed COVID-19 dynamical model adopts the formulation in Equation (4) where two species are considered; (1) \(X_{H}(t)\) which is the healthy individual population at time \(t\), (2) \(X_{I}(t)\) is the infected individual population at time \(t\). The population dynamics are dictated by the following differential equations:  
\[ \begin{array}{rcl}\frac{\mathrm{d}}{\mathrm{d}t}X_{H}(t)&=&\overbrace{ \alpha^{H}X_{H}}^{\text{ Immigration}}\ \ -\frac{\text{Infection}}{\beta\,X_{H}\,X_{I}}+\overbrace{\gamma\,X_{I}}^{\text{ Recovery}}\ \ +\overbrace{X_{H}\sum_{i=1}^{N}e_{i}^{H}U_{i}}^{\text{Control}}\\ \frac{\mathrm{d}}{\mathrm{d}t}X_{I}(t)&=&\underbrace{\alpha^{I}X_{I}}_{\text{ Immigration}}\ \ +\underbrace{\beta\,X_{H}\,X_{I}}_{\text{Infection}}-\ \underbrace{\gamma\,X_{I}}_{\text{ Recovery}}\ \ -\underbrace{X_{I}\sum_{i=1}^{N}e_{i}^{I}U_{i}}^{ \text{}}-\underbrace{\delta\,X_{I}}_{\text{Death}}\end{array}  \tag{5}\]  
The population dynamics represent five different states of interaction as follows:  
* **Immigration**: This due to the individual travels between different zones. When people move from a destination to another destination (of interest), the size of the population in the destination will increase linearly. \(\alpha^{H}\geqslant 0\) and \(\alpha^{I}\geqslant 0\) are the immigration rates of healthy individuals and infected individuals, respectively. For example, the travel bans between different destinations means that the values of immigration rates are zero. It is obvious that \(\alpha^{H}\leqslant 0\) and \(\alpha^{I}\leqslant 0\) when the individuals travel out of the destination. This implies that the size of the population will decrease. * **Infection**: This due to the direct interactions between the healthy individuals and infected individuals. \(\beta\geqslant 0\) is the infection rate. The negative sign in the healthy population term indicates that the infection will reduce the number of healthy individuals which are, in turn, added to the infected population.