## 5 Numeric Results  
This section presents the simulation results for an epidemiological system using a complex network approach where each node represents a population whose dynamic evolutions are defined by the MSEIR model (2) with ten nodes and all nodes have a certain degree of connectivity as shown in Figure 3; however, such connections are considered time-varying and are updated daily. Results of the scheduled vaccination strategy are obtained using MATLAB(r). An impulse controller is designed which is updated weekly based on expressions (19) and (20), starting on day 60 after the first infected individual and 365 days are simulated. The parameters for the controller are selected as  
\[R_{2d}=\left[15\;0\;0\;10\;\right],\;P_{2}=\left[1\;0\;0\;1\;\right],\;P_{12}= \left[5\;1\;\right] \tag{21}\]  
Matrix \(P_{2}\) directly affects the vaccination strategy. \(R_{2d}\) and \(P_{12}\) affect the node dynamics to be controlled by MSEIR. Simulations show that the dynamics of the MSEIR model are parameterized in the case of infection caused by the SARS COVID-19 virus. The parameters are selected as follows \(N=10,000\), \(b=0.025\), \(\delta=\frac{1}{180}\), \(\epsilon=114\) and \(\gamma=17\) with the time unit given in days, considering a life expectancy of 76 years and under the assumption that 65% of the population of nodes 1 and 2 will be vaccinated during months 2 and 12 while also considering that the mother give immunity to their offspring during the first 6 months. The initial values for the simulation of the MSEIR model are selected as \(m(0)=0.0045,\epsilon(0)=0\), \(i(0)=50\), \(N\), and \(r(0)=0.65\), \(\beta=4.5\) and \(\alpha=0.5\). In the case of this network without a controller, the following behavior is detected:  
* As a result of the disease not being contained, the behavior of Class \(M\) stays constant while class \(S\) exhibits an ascending trend during simulation time. Also, the behavior of node three is more noticeable than other nodes, which can be expected since, in addition to being the largest node, it is also where the disease begins and starts to spread to the other nodes. * Class \(E\) presents a growth of exposed individuals since there is no control of the disease. As time advances the number of exposed individuals decreases. * The model parameters used in this work determine that the number of infected individuals tends to reduce substantially if the individuals remain in class \(E\) and the number of people in class, \(I\) remain low, although this might not be the case. The quantity of people in class \(R\) is high, but it is lower compared to the class \(R\) of the number of individuals who received treatment. Figure 4 presents the simulation results of the network with a control algorithm. It is important to highlight that the dynamic population considers all the nodes. The disease originates from the node number three and spreads to the other nodes. Also, the control inputs (PIN nodes) are considered as node one and node two. Control measures start in  
Figure 3: Graphic representation of a complex network with ten nodes. node three since this node is the largest, with vaccines administered to a small portion of the population in order to decrease the infection rate. * Figure 4 shows the decline that in passive immunity (class \(M\)) in node three. This occurs because vaccination targets the susceptible population, which includes women of reproductive age, where the reduced probability of infection is considered. In contrast, at node five the number of births increases over time. If the mother is unvaccinated, she may eventually contract the disease which can temporarily deliver passive immunity to her newborn. * Class \(S\), node three shows that the susceptible population is initially the largest as the propagation begins here.