Analysis of COVID-19 and comorbidity co-infection model with optimal control

Abstract

In this work, we develop and analyze a mathematical model for the dynamics of COVID-19 with re-infection in order to assess the impact of prior comorbidity (specifically, diabetes mellitus) on COVID-19 complications. The model is simulated using data relevant to the dynamics of the diseases in Lagos, Nigeria, making predictions for the attainment of peak periods in the presence or absence of comorbidity. The model is shown to undergo the phenomenon of backward bifurcation caused by the parameter accounting for increased susceptibility to COVID-19 infection by comorbid susceptibles as well as the rate of reinfection by those who have recovered from a previous COVID-19 infection. Simulations of the cumulative number of active cases (including those with comorbidity), at different reinfection rates, show infection peaks reducing with decreasing reinfection of those who have recovered from a previous COVID-19 infection. In addition, optimal control and cost-effectiveness analysis of the model reveal that the strategy that prevents COVID-19 infection by comorbid susceptibles is the most cost-effective of all the control strategies for the prevention of COVID-19.

Keywords: COVID‐19; comorbidity; data‐fitting; optimal control; reinfection.

FIGURE 1

Schematic diagram of the model (1) where ${\lambda }_{\mathit{CV}}=\frac{{\beta }_{\mathit{CV}}(I_{\mathit{CV}}+I_{\mathit{CVCM}})}{N_{\mathit{H}}}$

FIGURE 2

Bifurcation diagram for the model (1). Parameter values used are: ${\beta }_{\mathit{CV}}=0.148,{\psi }_1=25,{\psi }_2=40,{\chi }_{\mathit{CM}}=1.2$. All other parameters as in Table 1

FIGURE 3

Fitting the cumulative number of active cases

FIGURE 4

Projections for the cumulative number of active cases

FIGURE 5

Simulations of the Cumulative number of active cases (including those with comorbidity): effect of ${\chi }_{\mathit{CM}}$

FIGURE 6

Simulations of the Cumulative number of active cases (without comorbidity): effect of ${\chi }_{\mathit{CM}}$

FIGURE 7

Simulations of the Cumulative number of active cases (including those with comorbidity): effect of ${\psi }_1$ and ${\psi }_2$

FIGURE 8

Simulations of the Cumulative number of active cases (without comorbidity): effect of ${\psi }_1$ and ${\psi }_2$

FIGURE 9

Contour plots of ${ℛ}_{0\mathit{C}}$. Here, ${\beta }_{\mathit{CV}}=0.2001,$ ${\psi }_1={\psi }_2=0.4$. All other parameters as in Table 1. (A) A contour plot of ${ℛ}_{0\mathit{C}}$ as a function of detection rate for singly infected individuals (${\eta }_1$) and detection rate for co‐infected individuals (${\eta }_2$). (B) A contour plot of ${ℛ}_{0\mathit{C}}$ as a function of detection rate for singly infected individuals (${\eta }_1$ and the modification parameter for high susceptibility of those with comorbidity ${\chi }_{\mathit{CM}}$)

FIGURE 10

Plots of the total number of COVID‐19‐infected individuals when strategy A is implemented (u ₁ ≠ 0). Here, ${\beta }_{\mathit{CV}}=0.2001,$ ${\psi }_1={\psi }_2=0.4$. All other parameters as in Table 1

FIGURE 11

Control profile for the effect of the control u ₁ on the dynamics of the model (1). Here, ${\beta }_{\mathit{CV}}=0.2001$, ${\psi }_1={\psi }_2=0.4$. All other parameters as in Table 1

FIGURE 12

Plots of the total number of COVID‐19‐infected individuals when strategy B is implemented (u ₂ ≠ 0). Here, ${\beta }_{\mathit{CV}}=0.2001,$ ${\psi }_1={\psi }_2=0.4$. All other parameters as in Table 1

FIGURE 13

Control profile for the effect of the control u ₂ on the dynamics of the model (1). Here, ${\beta }_{\mathit{CV}}=0.2001$, ${\psi }_1={\psi }_2=0.4$. All other parameters as in Table 1

FIGURE 14

Plots of the total number of COVID‐19‐infected individuals when strategy C is implemented (u ₃ ≠ 0). Here, ${\beta }_{\mathit{CV}}=0.2001,$ ${\psi }_1={\psi }_2=0.4$. All other parameters as in Table 1

FIGURE 15

Control profile for the effect of the control u ₃ on the dynamics of the model (1). Here, ${\beta }_{\mathit{CV}}=0.2001$, ${\psi }_1={\psi }_2=0.4$. All other parameters as in Table 1