Abstract
An SEIR epidemic model with a nonconstant vaccination strategy is studied. This SEIR model has two disease transmission rates β1 and β2 which imitate the fact that, for some infectious diseases, a latent person can pass the disease into a susceptible one. Here we study the spread of some childhood infectious diseases as good examples of diseases with infectious latent. We found that our SEIR model has a unique disease free solution (DFS). A lower bound and an upper bound of the basic reproductive number, R0 are estimated. We show that, the DFS is globally asymptotically stable when and unstable if Computer simulations have been conducted to show that non trivial periodic solutions are possible. Moreover the impact of the contact rate between the latent and the susceptibles is simulated. Different periodic solutions with different periods including one, two and three years, are obtained. These results give a clearer view for the decision makers to know how and when they should take action against a possible new wave of these infectious diseases. This action is mainly, applying a suitable dose of vaccination just before a severe peak of infection occurs.
Highlights
Some infectious diseases have a latent period, that is the time from being exposed to develop diseases symptoms
We can summarise the results obtained in this stability analysis as follows: The disease free solution for our SEIR model represented by Equations (2.1)–(2.4) with a nonconstant periodic and continuous vaccination rate ρ(t) in [0, T ], is globally asymptotically stable if R0sup < 1 and not stable if R0inf > 1
Our simulations have been conducted for two different states first when R0 < 1 and the second when R0 > 1 and we found that, the vaccination of the disease has a threshold level pc depending on the values of the vaccination parameters ρ(t) and the constant convectional vaccination one p
Summary
Some infectious diseases have a latent period, that is the time from being exposed to develop diseases symptoms. Moneim [19] studied the global behavior of the dynamics of the HBV disease He considered an SEIR model with a constant vaccination rate and infectivity during the incubation period. Jan and Xiao introduce a pulse vaccination strategy to study a dynamic model of dengue disease with periodic contact rates They found that, the disease free periodic solution of the their impulsive system is globally asymptotically stable if R0 < 1 and is unstable otherwise [7]. Two studies applied several simple and continuous time linear vaccinations based control strategies for a SEIR model which takes account the total population amounts as a refrain for the illness transmission They found that under these vaccination strategies, the susceptibles, infected and infectious population tend asymptotically to zeros [9, 10]. A simulation results are conducted for our model with different types of vaccination functions and for estimated parameter values from the literature
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