Mathematical modeling and optimal control of depression dynamics influenced by saboteurs

Abstract

Depression disorder affects millions globally, characterized by symptoms such as profound sadness, loss of interest in activities, and disruptions in eating and sleeping patterns. Understanding depression within the context of chronic pain is essential for developing effective management and intervention strategies. This study utilizes mathematical modeling to analyze depression trends using empirical data from Spain spanning from 2011 to 2022. Our depression model incorporates distinct compartments for primary and secondary depressed populations, along with a category for individuals categorized as saboteurs, who may actively influence the depression prevalence. We calculated the basic reproduction number [Formula: see text] and identified four equilibrium points and evaluated their stability. Additionally, sensitivity analysis was conducted to assess the impact of [Formula: see text] on depression prevalence. Furthermore, optimal control strategies were explored for the model. These strategies aim to improve treatment adherence, encourage doctor consultations, promote self-medication practices, and enhance recovery rates, ultimately aiming to reduce spread of depressive disorders and associated mortality. Data fitting was conducted using Python, and simulations were carried out in MATLAB to ensure rigorous validation of the model.

Fig. 1

Flow diagram for depression model.

Fig. 2

Graphs showing the model’s fit with real depression disorder cases in Spain. The yellow dots show the actual data, and the magenta line shows the model’s solution.

Fig. 3

Normalized forward sensitivity index of the basic reproduction number of the depression model. The parameter values are as in Table 2.

Fig. 4

Contour plots showing the effect of (a) and and (b) and on the basic reproduction number.

Fig. 5

Contour plots showing the effect of (a) and and (b) and on the basic reproduction number.

Fig. 6

Contour plots showing the effect of (a) and and (b) and on the basic reproduction number.

Fig. 7

Variation in total depressed individuals with respect to .

Fig. 8

Variation in Total depressed individuals with respect to (a); (b) .

Fig. 9

Variation in Recovered individuals with respect to (a); (b) .

Fig. 10

Variation in total depressed individuals with respect to (a); (b) .

Fig. 11

Variation in recovered individuals with respect to (a) ; (b) .

Fig. 12

With and without interaction of Saboteurs.

Fig. 13

Variations of secondary depressed individuals with and without control for only.

Fig. 14

Variations of secondary depressed individuals with and without optimal control for only.

Fig. 15

Variations of secondary depressed individuals with and without optimal control for only.

Fig. 16

Variations of secondary depressed individuals, with and without optimal control for and .

Fig. 17

Variations of secondary depressed individuals with and without optimal control for and .

Fig. 18

Variations of secondary depressed individuals with and without optimal control for all and .

Fig. 19

Variations of Secondary depressed individuals with and without optimal control for , and .

Fig. 20

Control profile of (a) , (b) and (c) .