Mathematical analysis of the role of hospitalization/isolation in controlling the spread of Zika fever

Highlights

• A vertical and horizontal transmission model for Zika fever is constructed in the form of a system of ordinary differential equations.

• This model features the study of Zika fever by considering vertical transmission in both humans as well as vectors.

• The basic reproductive number R₀ is formulated by using next-generation matrix.

• Uncertainty and sensitivity analyses were performed.

• We proposed an optimal controlling strategy to eliminate Zika fever from the population.

Abstract

The Zika virus is transmitted to humans primarily through Aedes mosquitoes and through sexual contact. It is documented that the virus can be transmitted to newborn babies from their mothers. We consider a deterministic model for the transmission dynamics of the Zika virus infectious disease that spreads in, both humans and vectors, through horizontal and vertical transmission. The total populations of both humans and mosquitoes are assumed to be constant. Our models consist of a system of eight differential equations describing the human and vector populations during the different stages of the disease. We have included the hospitalization/isolation class in our model to see the effect of the controlling strategy. We determine the expression for the basic reproductive number R₀ in terms of horizontal as well as vertical disease transmission rates.

An in-depth stability analysis of the model is performed, and it is consequently shown, that the model has a globally asymptotically stable disease-free equilibrium when the basic reproduction number R₀ < 1. It is also shown that when R₀ > 1, there exists a unique endemic equilibrium. We showed that the endemic equilibrium point is globally asymptotically stable when it exists. We were able to prove this result in a reduced model. Furthermore, we conducted an uncertainty and sensitivity analysis to recognize the impact of crucial model parameters on R₀. The uncertainty analysis yields an estimated value of the basic reproductive number R₀ = 1.54. Assuming infection prevalence in the population under constant control, optimal control theory is used to devise an optimal hospitalization/isolation control strategy for the model. The impact of isolation on the number of infected individuals and the accumulated cost is assessed and compared with the constant control case.