Optimal control analysis of hepatocytic-erythrocytic dynamics of Plasmodium falciparum malaria

Abstract

This paper presents an in-host malaria model subject to anti-malarial drug treatment and malaria vaccine antigens combinations. Pontryagin's Maximum Principle is applied to establish optimal control strategies against infected erythrocytes, infected hepatocytes and malaria parasites. Results from numerical simulation reveal that a combination of pre-erythrocytic vaccine antigen, blood schizontocide and gametocytocide drugs would offer the best strategy to eradicate clinical P. falciparum malaria. Sensitivity analysis, further reveal that the efficacy of blood schizontocides and blood stage vaccines are crucial in the control of clinical malaria infection. Futhermore, we found that an effective blood schizontocide should be used alongside efficacious blood stage vaccine for rapid eradication of infective malaria parasites. The authors hope that the results of this study will help accelerate malaria elimination efforts by combining malaria vaccines and anti-malarial drugs against the deadly P. falciparum malaria.

Keywords: Blood schizontocide; Gametocytocide; Malaria vaccines; Optimal control; P. falciparum malaria; Pontryagin's Maximum Principle.

Fig. 1

Graph showing tornado plots of partial rank correlation coefficients (PRCCs) of the parameters that influence the effective reproduction number R_eff. Parameter values are shown in Table 2.

Fig. 2

Simulations of system (1), showing the impact of a combination of blood schizontocide u₃ and a gametocitocide u₄ only during clinical P. falciparum malaria infection. Used parameter values are shown in Table 2.

Fig. 3

Profiles of blood schizontocide u₃ and gametocytocide u₄. Here, u₁ = 0, u₂ = 0.

Fig. 4

Simulations of system (1), showing the impact of a combination of pre-erythrocytic vaccine antigens u₁ with blood stage vaccine antigens u₂ only. Used parameter values are shown in Table 2.

Fig. 5

Profiles of pre-erythrocytic vaccine antigen u₁ and blood stage vaccine antigens u₂. Here, u₃ = 0, u₄ = 0.

Fig. 6

Simulations of system (1), showing the impact of a combining pre-erythrocytic malaria vaccine u₁ and blood schizontocides u₃ only in the control of within-human P. falciparum infection. Used parameter values are shown in Table 2.

Fig. 7

Profiles of pre-erythrocytic vaccine antigen u₁ and blood schizontocide drug u₃. Here, u₂ = 0, u₄ = 0.

Fig. 8

Simulations of system (1), showing the impact of a combination of pre-erythrocytic vaccine antigens u₁, blood stage vaccine antigens u₂ and blood schizontocide u₃ only. Used parameter values are shown in Table 2.

Fig. 9

Profiles of pre-erythrocytic vaccine antigen u₁, blood stage vaccine antigen u₂ and blood schizontocide u₃. Here, u₄ = 0.

Fig. 10

Simulations of system (1), showing the impact of a combination of pre-erythrocytic malaria vaccine u₁, blood schizontocide u₃ and gametocitocidal drug u₄. Used parameter values are shown in Table 2.

Fig. 11

Profiles of pre-erythrocytic vaccine antigen u₁, blood schizontocide u₃ and gametocytocide u₄. Here u₂ = 0.

Fig. 12

Simulations of system (1), showing the impact of combining antigens of pre-erythrocytic malaria vaccine antigen u₁ and blood stage vaccine antigen u₂ together with the administration of combined blood schizontocide u₃ and gametocitocidal drug u₄. Used parameter values are shown in Table 2.

Fig. 13

Plots showing the control profiles of pre-erythrocytic vaccine antigen u₁, blood stage vaccine antigen u₂, blood schizontocide u₃ and gametocytocide u₄.