Abstract
In this paper, the dynamics of SVEIR model with saturated incidence force of infection and saturated vaccination function for Streptococcus pneumonia (that is, model that monitors the temporal transmission dynamics of the disease in the presence of preventive vaccine) was formulated and analyzed. The basic reproduction number that determines disease extinction and disease survival was revealed. The existing threshold conditions of all kinds of the equilibrium points are obtained and proved to be locally asymptotic stable for disease-free equilibrium using linearization method and Lyapunov functional method for Endemic equilibrium. Qualitative Analysis of the model was obtained and the positive of solution obtained. It was revealed that the model is positively –invariant and attracting. Thus the region is positively invariant. Hence, it is sufficient to consider the dynamics of the model (1) in the given region. In this region, the model can be considered as been epidemiologically and mathematically well-posed. The governing model was normalized and also Adomian Decomposition method was used to compute an approximate solution of the non-linear system of differential equations governing the model. Maple was used in carrying out the simulations (numerical solutions) of the model. Graphical results were presented and discussed to illustrate the solution of the problem. The achieved results reveal that the disease will die out within the community if the vaccination coverage is above the critical vaccination proportion. The study indicates that we should improve the efficiency and enlarge the capacity of the treatment to control the spread of disease.
Highlights
Streptococcus pneumonia is a facultative anaerobic bacterium, gram-positive which have the shape of a lancet
Theorem 3.3: The disease free equilibrium (ξ 0 ) is locally asymptotically stable if R0 1 Proof: We prove the locally asymptotically stability of the disease free equilibrium ( E 0 ) of model (1) using linearization approach
We describe the trend of each of the classes to understand exactly what happened in Figure 9 and it reveals that, if the basic reproduction is greater than one, S. pneumonia becomes endemic in the population and persist since the exposed and infected classes were not reduced to zero, see Figure 10 and Figure 9
Summary
Streptococcus pneumonia is a facultative anaerobic bacterium, gram-positive which have the shape of a lancet. The saturated incidence rate β SI (1+αI) was introduced by [11] This reveals that if βI (which estimates the infection force at time of disease total invasion in the susceptible population) is large together with 1/(1+αI) (which estimates the reacting effect out of the behavioral change of the susceptible population at the time we have a crowding effect of the infected population), the model is certainly to be saturated. The continuous and differentiable saturated treatment function was introduced and used given as h (I) = rI/(1 +kI), where r>0, k>0, r implies rate of cure, and k estimates the treatment delay level of the infected individuals This reveals that if Iis small, the treatment function tends to rI, while it tends to r/k if I is large, [10, 45]. Other works which we looked at in this work are: [8, 15, 18, 28, 34, 25, 29, 45, 46, 47]
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