ABSTRACT
Rubella is a highly contagious and serious human disease caused by the rubella virus. It affects everyone around the world, but it is especially common in pregnant women and children. In particular, when pregnant women are infected with the rubella virus, it causes Congenital Rubella Syndrome (it transmit vertically from mother to fetus, which causes that the new born baby to inherit birth defect disease). In order to prevent this viral disease, children must receive an MMR (measles, mumps and rubella) vaccine twice. If children receive two doses of the vaccine, then they develop long life immunity (protected against rubella). Based on the biological behavior of rubella disease, the SVPEIRS (susceptible, vaccinated, protected, exposed, infected, recovered) deterministic mathematical model of rubella disease dynamics is proposed. From the perspective of the qualitative behavior of the model, it is bounded in the invariant region and all the solutions of the compartment are positive. In addition, the equilibrium points and the stability of the equilibrium points (local and global) are also analyzed. The basic reproductive number is determined using a next-generation matrix. The results of the sensitivity analysis show that rubella is spread in a community if the values of contact rate, vertical transmission (neonatal infection) rate, exposure rate and rate of waning out of the first vaccinating dose are increase by keeping other parameters constant. On the other hand, increasing the first and second vaccination rate and treatment rate can help to control rubella in the community. Numerical simulation results show that due to the lack of protection for women before pregnancy, the number of infections increases with the birth of infected children, and the two doses of vaccine play a significant role in reducing and eliminating rubella. Therefore, to eliminate rubella in the community, healthcare and policymakers must pay attention to these parameters.
ABSTRACT
In this paper we developed a stochastic model of measles transmission dynamics with double dose vaccination. The total population in this model was sub-divided in to five compartments, namely Susceptible S ( t ) , Infected I ( t ) , Vaccinated first dose V 1 ( t ) , Vaccinated second dose V 2 ( t ) and Recovered R ( t ) . First the model was developed by deterministic approach and then transformed into stochastic one, which is known to play a significant role by providing additional degree of realism compared to the deterministic approach. The analysis of the model was done in both approaches. The qualitative behavior of the model, like conditions for positivity of solutions, invariant region of the solution, the existence of equilibrium points of the model and their stability, and also sensitivity analysis of the model were analyzed. We showed that in both deterministic and stochastic cases if the basic reproduction number is less than 1 or greater than 1 the disease free equilibrium point is stable or unstable respectively, so that the disease dies out or persists within the population. Numerical simulations were carried out using MATLAB to support our analytical solutions. These simulations show that how double dose vaccination affect the dynamics of human population.
ABSTRACT
In this paper, we proposed a deterministic model of pneumonia-meningitis coinfection. We used a system of seven ordinary differential equations. Firstly, the qualitative behaviours of the model such as positivity of the solution, existence of the solution, the equilibrium points, basic reproduction number, analysis of equilibrium points, and sensitivity analysis are studied. The disease-free equilibrium is locally asymptotically stable if the basic reproduction number is kept less than unity, and conditions for global stability are established. Then, the basic model is extended to optimal control by incorporating four control interventions, such as prevention of pneumonia as well as meningitis and also treatment of pneumonia and meningitis diseases. The optimality system is obtained by using Pontryagin's maximum principle. For simulation of the optimality system, we proposed five strategies to check the effect of the controls. First, we consider prevention only for both diseases, and the result shows that applying prevention control has a great impact in bringing down the expansion of pneumonia, meningitis, and their coinfection in the specified period of time. The other strategies are prevention effort for pneumonia and treatment effort for meningitis, prevention effort for meningitis and treatment effort for pneumonia, treatment effort for both diseases, and using all interventions. We obtained that each of the listed strategies is effective in minimizing the expansion of pneumonia-only, meningitis-only, and coinfectious population in the specified period of time.
Subject(s)
Coinfection , Meningitis/physiopathology , Meningitis/therapy , Pneumonia/physiopathology , Pneumonia/therapy , Algorithms , Basic Reproduction Number , Computer Simulation , Disease-Free Survival , Humans , Meningitis/complications , Models, Biological , Models, Statistical , Pneumonia/complicationsABSTRACT
We propose and analyze a compartmental nonlinear deterministic mathematical model for the typhoid fever outbreak and optimal control strategies in a community with varying population. The model is studied qualitatively using stability theory of differential equations and the basic reproductive number that represents the epidemic indicator is obtained from the largest eigenvalue of the next-generation matrix. Both local and global asymptotic stability conditions for disease-free and endemic equilibria are determined. The model exhibits a forward transcritical bifurcation and the sensitivity analysis is performed. The optimal control problem is designed by applying Pontryagin maximum principle with three control strategies, namely, the prevention strategy through sanitation, proper hygiene, and vaccination; the treatment strategy through application of appropriate medicine; and the screening of the carriers. The cost functional accounts for the cost involved in prevention, screening, and treatment together with the total number of the infected persons averted. Numerical results for the typhoid outbreak dynamics and its optimal control revealed that a combination of prevention and treatment is the best cost-effective strategy to eradicate the disease.
Subject(s)
Disease Outbreaks/prevention & control , Models, Biological , Typhoid Fever/epidemiology , Typhoid Fever/prevention & control , Computational Biology , Computer Simulation , Cost-Benefit Analysis , Disease Outbreaks/economics , Humans , Mass Screening , Mathematical Concepts , Nonlinear Dynamics , Typhoid Fever/economicsABSTRACT
We propose and analyse a nonlinear mathematical model for the transmission dynamics of pneumonia disease in a population of varying size. The deterministic compartmental model is studied using stability theory of differential equations. The effective reproduction number is obtained and also the asymptotic stability conditions for the disease free and as well as for the endemic equilibria are established. The possibility of bifurcation of the model and the sensitivity indices of the basic reproduction number to the key parameters are determined. Using Pontryagin's maximum principle, the optimal control problem is formulated with three control strategies: namely disease prevention through education, treatment and screening. The cost-effectiveness analysis of the adopted control strategies revealed that the combination of prevention and treatment is the most cost-effective intervention strategies to combat the pneumonia pandemic. Numerical simulation is performed and pertinent results are displayed graphically.