Mathematical analysis of dengue virus antibody dynamics

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Dengue is a mosquito borne viral disease causing over 390 million infections worldwide per annum. Even though information on how infection is controlled and eradicated from the body is lacking, antibodies are thought to play a major role in clearing the virus. In this paper, a non-linear conceptual dynamical model with humoral immune response and absorption effect has been proposed for primary dengue infection. We have included the absorption of pathogens into uninfected cells since this effect causes the virus density in the blood to decrease. The time delay that arises in the production of antibodies was accounted and is introduced through a continuous function. The basic reproduction number R0 is computed and a detailed stability analysis is done. Three equilibrium states, namely the infection free equilibrium, no immune equilibrium and the endemic equilibrium were identified and the existence and the stability conditions of these steady states were obtained. Numerical simulations proved the results that were obtained. By establishing the characteristic equation of the model at infection free equilibrium, it was observed that the infection free equilibrium is locally asymptotically stable if R0 < 1. A threshold value for the antibody production rate was identified for which the infection gets completely cured even if R0 > 1. Stability regions are identified for infection free equilibrium state with respect to the external variables and it is observed as the virus burst rate increases, the stability regions would decrease. These results implied that for higher virus burst rates, other conditions in the body must be strong enough to eliminate the disease completely from the host. The effect of time delay of antibody production on virus dynamics is discussed. It was seen that as the time delay in production of antibodies increases, the time for viral decline also increased. Also it was observed that the virus count goes to negligible levels within 7 − 14 days after the onset of symptoms as seen in dengue infections.