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Optimal Control of Microcephaly Under Vertical Transmission of Zika

Dilara Yapışkan, Cristiana J. Silva, Delfim F. M. Torres

Abstract

The Zika virus, known for its potential to induce neurological conditions such as microcephaly when transmitted vertically from infected mothers to infants, has sparked widespread concerns globally. Motivated by this, we propose an optimal control problem for the prevention of vertical Zika transmission. The novelty of this study lies in its consideration of time-dependent control functions, namely, insecticide spraying and personal protective measures taken to safeguard pregnant women from infected mosquitoes. New results provide a way to minimize the number of infected pregnant women through the implementation of control strategies while simultaneously reducing both the associated costs of control measures and the mosquito population, resulting in a decline in microcephaly cases.

Optimal Control of Microcephaly Under Vertical Transmission of Zika

Abstract

The Zika virus, known for its potential to induce neurological conditions such as microcephaly when transmitted vertically from infected mothers to infants, has sparked widespread concerns globally. Motivated by this, we propose an optimal control problem for the prevention of vertical Zika transmission. The novelty of this study lies in its consideration of time-dependent control functions, namely, insecticide spraying and personal protective measures taken to safeguard pregnant women from infected mosquitoes. New results provide a way to minimize the number of infected pregnant women through the implementation of control strategies while simultaneously reducing both the associated costs of control measures and the mosquito population, resulting in a decline in microcephaly cases.

Paper Structure

This paper contains 5 sections, 1 theorem, 11 equations, 11 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Uncontrolled Zika Model
  3. Optimal Control Problem
  4. Numerical Simulations: Case Study in Brazil
  5. Conclusions

Key Result

Theorem 1

Let $\mathopen{}\mathclose{\left( u_{1}^{\ast },{u}_{2}^{\ast }\right) \in U$ be the optimal controls that minimize the objective functional eq5 and $\mathopen{}\mathclose{\left( S^{\ast },I^{\ast },W^{\ast },M^{\ast },A_{m}^{\ast },S_{m}^{\ast },E_{m}^{\ast },I_{m}^{\ast }\right)$ be the optimal so with transversality conditions In addition,

Figures (11)

  • Figure S1: Effect of control strategies on women population, with and without controls. (Top left): susceptible pregnant women $S$; (Top right): infected pregnant women $I$; (Bottom left): women who gave birth to babies without microcephaly $W$; (Bottom right): women who gave birth to babies with microcephaly $M$.
  • Figure S2: Effect of control strategies on mosquitoes, with and without controls. (Top left): mosquitoes in the aquatic phase $A_m$; (Top right): susceptible mosquitoes $S_m$; (Bottom left): exposed mosquitoes $E_m$; (Bottom right): infected mosquitoes $I_m$.
  • Figure S3: Comparative impact of the control strategies on women population, considering both controls and one single control $u_1$ or $u_2$. (Top left): susceptible pregnant women $S$; (Top right): infected pregnant women $I$; (Bottom left): women who gave birth to babies without microcephaly $W$; (Bottom right): women who gave birth to babies with microcephaly $M$.
  • Figure S4: Comparative impact of the control strategies on mosquitoes, considering both controls and one single control $u_1$ or $u_2$. (Top left): mosquitoes in the aquatic phase $A_m$; (Top right): susceptible mosquitoes $S_m$; (Bottom left): exposed mosquitoes $E_m$; (Bottom right): infected mosquitoes $I_m$.
  • Figure S5: Control strategies $u_1$ and $u_2$.
  • ...and 6 more figures

Theorems & Definitions (1)

  • Theorem 1