Fig 1.
Transmission diagram for the model.
The Transmission diagram of Eq (3) is constructed to represent the key epidemiological processes and transitions considered in the study. The system of differential equations is subsequently derived based on the flow structure shown in this diagram.
Table 1.
Parameter definitions for model (3).
Fig 2.
HIV/AIDS-only transmission diagram.
Transmission diagram of HIV/AIDS-only model in system (4).
Fig 3.
COVID-19-only transmission diagram.
Transmission diagram of COVID-19-only model in system (10).
Table 2.
Descartes rules of sign for polynomial P(z).
Fig 4.
Dynamical response of the HIV-COVID co-infection model (3).
Solution was computed for the parameter values given in Table 1, with ,
,
,
,
,
,
and
.
Fig 5.
Continuation of parameter-dependent equilibria of model (3).
This experiment conducted with respect to ,
,
,
,
and
, calculated for the parameter values used in Fig 4. Panels (a)–(f) show in the vertical axis on the left (blue) the variations of the HIV-related compartments of the model. The vertical axis on the right (red) displays the parameter-dependence of the COVID compartment C. In the bifurcation pictures, stable and unstable equilibria are represented by solid and dashed lines, respectively. The bifurcation analysis reveals the presence of several branching points listed as follows: BP1 (
), BP2 (
), BP3 (
) and BP4 (
). Panel (g) shows time responses of system (3) computed at the test points P1 (
), P2 (
) and P3 (
). In these plots, the red series corresponds to C(t) while the blue series to H(t). In what follows, this color code will be employed to present time plots of the H and C compartments in the same graph.
Fig 6.
Codimension-2 of model (3).
(a) Continuation of the bifurcation points found in Fig 5(e) with respect to the parameters and
. The diagram shows the continuation of the bifurcations BP3 (
) and BP2 (
), which intersect each other at the point BP5 (
,
). From this point, two further curves of branching points emanate, labeled
and
. The resulting curves divides the parameter space into four regions as indicated in the diagram. Panel (b) shows time plots of system (3) calculated at the selected points P1 (
,
), P2 (
,
), P3 (
,
) and P4 (
,
).
Fig 7.
Optimal control simulation for Strategy A.
Simulation of (a) the effect of the control profiles u1(t), u3(t), and u4(t) on COVID-19 infected individuals (b) the effect of the control profiles u1(t), u3(t), and u4(t) on HIV infected individuals (c) the effect of the control profiles u1(t), u3(t), and u4(t) on AIDS infected individuals (d) the effect of the control profiles u1(t), u3(t), and u4(t) on individuals co-infected with both COVID-19 and HIV (e) the effect of the control profiles u1(t), u3(t), and u4(t) on individuals co-infected with both COVID-19 and AIDS (f) the control profiles u1(t), u3(t), and u4(t).
Fig 8.
Optimal control simulation for Strategy B.
Simulation of (a) the effect of the control profiles u2(t), u3(t), and u4(t) on COVID-19 infected individuals (b) the effect of the control profiles u2(t), u3(t), and u4(t) on HIV infected individuals (c) the effect of the control profiles u2(t), u3(t), and u4(t) on AIDS infected individuals (d) the effect of the control profiles u2(t), u3(t), and u4(t) on individuals co-infected with both COVID-19 and HIV (e) the effect of the control profiles u2(t), u3(t), and u4(t) on individuals co-infected with both COVID-19 and AIDS (f) the control profiles u2(t), u3(t), and u4(t).
Fig 9.
Optimal control simulation for Strategy C.
Simulation of (a) the effect of the control profiles u1(t), u2(t), u3(t), and u4(t) on COVID-19 infected individuals (b) the effect of the control profiles u1(t), u2(t), u3(t), and u4(t) on HIV infected individuals (c) the effect of the control profiles u1(t), u2(t), u3(t), and u4(t) on AIDS infected individuals (d) the effect of the control profiles u1(t), u2(t), u3(t), and u4(t) on individuals co-infected with both COVID-19 and HIV (e) the effect of the control profiles u1(t), u2(t), u3(t), and u4(t) on individuals co-infected with both COVID-19 and AIDS (f) the control profiles u1(t), u2(t), u3(t), and u4(t).