This work examines the impacts of self-isolation and hospitalization on the population dynamics of the Corona Virus Disease. We developed a new nonlinear deterministic model eight classes compartment, with self-isolation and hospitalized being the most effective tools. There are (Susceptible S-C(t), Exposed E(t), Asymptomatic infected I-A(t), Symptomatic infected A(S)(t), Self-isolation T-M(t), Hospitalized T-H(t), Healed H(t), and Susceptible individuals previously infected H-C(t)). The expression of basic reproduction number R-0 comes from the next generation matrix method. With suitably constructed Lyapunov functions, the global asymptotic stability of the non-endemic equilibria sigma(0) for R-0 < 1 and that of endemic equilibria sigma* for R-0 > 1 are established. The computed value of R-0 = 3.120277403 proves the endemic level of the epidemic. The outbreak will lessen if a policy is enforced like self-isolation and hospitalization. This is related to those policies that can reduce the number of direct contacts between infected and susceptible individuals or waning immunity individuals. Various simulations are presented to appreciate self-isolation at home and hospitalized strategies if applied sensibly. By performing a global sensitivity analysis, we can obtain parameter values that affect the model through a combination of Latin Hypercube Sampling and Partial Rating Correlation Coefficients methods to determine the parameters that affect the number of reproductions and the increase in the number of COVID cases. The results obtained show that the rate of self-isolation at home and the rate of hospitalism have a negative relationship. On the other hand, infections will decrease when the two parameters increase. From the sensitivity of the results, we formulate a control model using optimal control theory by considering two control variables. The result shows that the control strategies minimize the spread of the COVID infection in the population.

The use of a vaccine and Wolbachia bacterium have been proposed as new strategies against dengue. However, the performance of Wolbachia in reducing dengue incidence may depend on the Wolbachia strains. Therefore, in this paper, the performance of two Wolbachia strains which are WMel and WAu, in combination with the vaccine, has been assessed by using an age-dependent mathematical model. An effective reproduction number has been calculated using the Extended Kalman Filter (EKF) algorithm. The results revealed that the time reproduction number varies overtime with the highest one being around 2.75. Moreover, it has also found that use of the vaccine and Wolbachia possibly leads to dengue elimination. Furthermore, vaccination on one group only reduces dengue incidence in that group but dengue infection in the other group is still high. Furthermore, the performance of the WAu strain is better than the WMel strain in reducing dengue incidence. However, both strains can still be used for dengue elimination strategies depending on the level of loss of Wolbachia infections in both strains.

This paper addresses a modified epidemic model with saturated incidence and incomplete treatment. The existence of all equilibrium points is analyzed. A reproduction number ${\mathcal{R}}_0$ is determined. Next, it is found that the non-endemic point ${\mathcal{P}}_0$ is stable in case ${\mathcal{R}}_0<1$, but unstable in case ${\mathcal{R}}_0>1$. The special conditions to analyze the local and global stability of the non-endemic and endemic points are investigated. Globally, the sensitivity analysis of the system is studied by combining the Latin Hypercube Sampling and Partial Rating Correlation Coefficients methods. By using the Pontryagins maximum principle, the optimal control problem is studied. Various numerical results are given to support our analysis.

Malaria is one of infectious diseases which become the main public health problem especially in Indonesia. Mathematically, the spread of malaria can be modeled to predict the outbreak of the disease. This research studies about mathematical model of the spread of malaria which takes into consideration the migration of human and mosquito populations. By determining basic reproduction number of the model, we analyze effects of migration parameter with respect to the reduction of malaria outbreak. Sensitivity analysis of basic reproduction number shows that mosquito migration has greater effect in reducing the outbreak of malaria compared with human migration. Basic reproduction number of the model is monotonically decreasing as mosquito migration increasing. We then confirm the analytic result by doing numerical simulation. The results show that migrations in human and mosquito populations have big influences in eliminating and eradicating the disease from the system.

We proposed and analyzed a stage-structure Rosenzweig-MacArthur model incorporating a prey refuge. It is assumed that the prey is a stage-structure population consisting of two compartments known as immature prey and mature prey. The model incorporates the functional response Holling type-II. In this work, we investigate all the biologically feasible equilibrium points, and it is shown that the system has three equilibrium points. Sufficient conditions for the local stability of the non-negative equilibrium point of the model are also derived. All points are conditionally locally asymptotically stable. By constructing Jacobian matrix and determined eigenvalues, we analyzed the local stability of the trivial equilibrium and non-predator equilibrium points. Specifically for coexistence equilibrium point, Routh-Hurwitz criterion used to analyze local stability. In addtion, we investigated the effect of immature prey refuge. Our mathematical analysis exhibits that immature prey refuge have played a crucial role in the behavioral system. When the effect of immature prey refuge (constant m) increases, it is can stabilize non-predator equilibrium point, where all the species can not exists together. And conversely, if contant m decreases, it is can stabilize coexistence equilibrium point then all the species can exists together. The work is completed with a numerical simulation to confirmed analitical results

Treatment and quarantine are efforts to cure as well as to overcome the spread of diseases including measles. The spread of measles can be expressed by mathematical modelling in the form of nonlinear dynamical systems. In this study was conducted on the spread of measles by considering the effect of treatment and quarantine on the infected individuals. By using the basic reproduction number of the model, can be analyzed the effects of treatment and quarantine to reduce the spread of measles. Basic reproduction number of models is monotonically descreasing as treatment and quarantine increasing. Numerical simulations conducted on the analysis of the results. The results showed that treatment and quarantine was given to infected individuals who were infectious has a major influence to eliminate measles from the system.