Human malaria remains a major killer disease worldwide, with nearly half (3.2 billion) of the world’s population at risk of malaria infection. The infectious protozoan disease is endemic in tropical and subtropical regions, with an estimated 212 million new cases and 429,000 malaria-related deaths in 2015. An in-host mathematical model of
Human malaria remains a major killer disease worldwide, with nearly half (3.2 billion) of the world’s population at risk of malaria infection [
The protozoan disease is caused by parasites of the genus
During their obligatory blood meals, infected female
Within a period of two days, the infected red blood cells rupture to release about 16 daughter merozoites [
The presence of the malaria parasites in the human body elicits response from numerous immune cells. The innate immune system and the adaptive immune system form the first and the second lines of defence, respectively [
Unlike the NK cells, the macrophages have been shown to effectively phagocytose malaria-infected red blood cells during the erythrocytic phase [
The rest of the paper is organized as follows: in Section
Several studies on mathematical modelling of in-host malaria and its dynamics within the human host have been done. Nearly all the earlier mathematical models (see, e.g., [
In the following sections, we extended the model in [
The hepatocytic-erythrocytic malaria model describes the dynamics of
The variables and parameters that describe in-host malaria dynamics are as in Tables
Symbols and definition of state variables considered in the model.
Variable | Description |
---|---|
| The population of susceptible hepatocytes at time |
| The population of infected hepatocytes at time |
| The population of susceptible red blood cells (erythrocytes) at time |
| The population of infected red blood cells at time |
| The density of macrophages in the human body at time |
| The population of sporozoites at time |
| The population of merozoites at time |
Symbols and description of parameters used in the model.
Parameter | Description |
---|---|
| The total rate of injection of sporozoites into liver due to mosquito bites |
| The death rate of sporozoites |
| Recruitment rate of susceptible hepatocytes from the bone marrow |
| Natural death rate of susceptible hepatocytes |
| The invasion rate of hepatocytes by sporozoites |
| Death rate of infected hepatocytes |
| Recruitment rate of susceptible RBCs by the bone marrow |
| The natural death rate of RBCs |
| The invasion rate of RBCs by merozoites |
| Death rates of IRBCs |
| The death rate of merozoites |
| Recruitment rate of macrophages from the bone marrow |
| The death rate of a macrophage |
| Elimination rate of IRBCs by macrophages |
| Production rate of hepatocytes due to presence of infected hepatocytes |
| Production rate of RBCs due to presence of IRBCs |
| Immunogenicity of IRBCs |
| Number of |
| Number of |
| Number of |
| The proportion of the merozoites that cause secondary infections |
| The average number of merozoites released per bursting IRBCs |
| The average number of merozoites released per bursting infected hepatocytes |
The above transmission dynamics of malaria are summarised in the compartmental diagram in Figure
Schematic diagram for hepatocytic-erythrocytic and malaria parasite dynamics. The dotted lines without arrows indicate cell-parasite interaction and the solid lines show progression from one compartment to another.
From the above description of the in-host dynamics of malaria and the representation in Figure
In this section, we study whether the formulated model (
For the in-host malaria model (
Let the parameters in model (
Considering the first equation in system (
Let
On substituting the derivatives in system (
Using integrating factor
In the second case, we consider
Using the above approach, let the total red blood cells population be
For the macrophage compartment
Finally, let
Based on this discussion, we have shown the existence of a bounded positive invariant region for our model system (
The disease-free equilibrium point,
The in-host reproduction number of model ( The term The second term Observe that the terms
Despite the inclusion of the liver stage dynamics, it is interesting to observe that the above in-host reproduction number and hence the disease progression are heavily driven by the dynamics at the erythrocytic stage.
In the sections that follow, we shall establish both the local stability and global stability of disease-free equilibrium point (
The Jacobian matrix of model system (
The remaining two eigenvalues can be obtained by reducing matrix (
The disease-free equilibrium
Biologically, Theorem
Using the results obtained in [
Let
In our case, matrix
For the matrix
The malaria-free equilibrium
The above result is quite significant in malaria control. The global stability of the disease-free status would be guaranteed if and only if the in-host basic reproduction number
When if if if
Analysis under (
The roots of the cubic equation (
From the above discussion, model (
In this section, we provide some numerical simulations to illustrate the behaviour of model system (
In epidemic modelling, sensitivity analysis is performed to investigate model parameters with significant influence on
Parameter values used in the numerical simulation and demonstration of the existence of endemic equilibrium point. See Table
Symbol | Interpretation | Value | Source |
---|---|---|---|
| Death rate of macrophages | 0.05/day | [ |
| Death rate of sporozoites | | [ |
| Elimination rate of IRBCs by macrophages | | [ |
| Recruitment rate of | | [ |
| Production rate of | | [ |
| Death rate of | 0.029 /day | [ |
| Rate of injection of sporozoites | 20 sporozoites/day | [ |
| Production rate of RBCs due to IRBCs | | [ |
| Hepatocyte invasion rate | | [ |
| Immunogenicity of IRBCs | | [ |
| Death rate of infected hepatocytes | 0.02/day | [ |
| Inhibition rate | 1 | [ |
| Recruitment rate of RBCs | | [ |
| Merozoites that cause secondary infections | 0.726 (unitless) | [ |
| Death rate of healthy RBCs | 0.0083/day | [ |
| Inhibition rate | 1 | [ |
| Invasion rate of RBCs | | [ |
| Merozoites per liver schizont | 10000/day | [ |
| Death rate of infected RBCs | 0.025/day | [ |
| Inhibition rate | 1 | [ |
| Death rate of merozoites | 48/day | [ |
| Merozoites per blood schizont | 16 | [ |
| Recruitment rate of macrophages | 30 | [ |
Sensitivity indices of
Parameter | SI | Parameter | SI |
---|---|---|---|
K | +1.0000 | | +1.0000 |
| +0.920422 | | -0.920422 |
| +0.920422 | | -0.998585 |
| +0.998585 | | -0.998585 |
| +0.998585 | | -0.920422 |
A positive sign on the SI indicates that an increase (decrease) in the value of such a parameter increases (decreases) the value of
The average number of merozoites released per bursting infected erythrocyte
The parameters
The rate of generation of macrophages from the bone marrow,
The parameters
Any therapeutic effort that clears the blood schizonts and the infectious merozoites at the blood stage would definitely guarantee immense reduction in model
Since the local sensitivity indices are relatively close, we carry out further investigation on parameter influence on disease progression by generating the partial rank correlation coefficients (PRCCs) for each parameter value in model
A global sensitivity analysis (GSA) is performed to examine the response of an epidemic model to parameter variation within a wider range of parameter space [
Tornado plots of PRCCs of parameters that influence model
Unlike the results in Table
The merozoites’ death rate
Based on these results of sensitivity analysis, we make the following remarks: (1) results of global sensitivity analysis are robust and a lot more realistic for implementation, (2) malaria control should target elimination of merozoites and infected red blood cells, (3) an effective and efficient malaria vaccine that deactivates infectious merozoites could be helpful in limiting erythrocyte invasion rate, and (4) a vaccine that is protective of susceptible erythrocytes could further ensure reduced density of second and future generation of merozoites that are responsible for disease progression.
Model system (
For
Graphs showing the simulation of in-host malaria model (
Hepatocyte dynamics
Sporozoite dynamics
Blood stage dynamics
At the blood stage, the rising density of infected erythrocytes declines in a similar fashion to that of the infective merozoites when
When
Graphs showing population dynamics of the liver hepatocytes and malaria sporozoites when
Susceptible hepatocytes
Infected hepatocytes
Sporozoites
Malaria infection dynamics are most rapid in the first 2 weeks within the host liver as illustrated in Figures
Graphs showing population dynamics of red blood cells, macrophages, and malaria merozoites when
Susceptible erythrocytes
Infected erythrocytes
Merozoites
Macrophages
An early sharp rise in the density of merozoites in the first one week of the blood stage is noted in Figure
The invasion of healthy erythrocytes prompts an immune response from host’s macrophages. These macrophages phagocyte on the generated blood schizonts. At the onset of erythrocytic infection, several macrophages are generated. The rise in the density of macrophages is proportional to that of infected erythrocytes as shown in Figure
From these discussions, we make the following observations: (1) if
Hematological parameters such as the density of healthy and infected erythrocytes in malaria hosts have considerable influence on malaria infection and possible impacts [
Graphs showing the behaviour of (a) susceptible RBCs and (b) infected RBCs. They were obtained by varying the death rate of merozoites
Effect of
Effect of
Graphs showing the behaviour of (a) susceptible RBCs and (b) infected RBCs. They were obtained by varying the merozoite invasion rate
Effect of
Effect of
Observe that increased death rate of malaria merozoites
Results in Figure
The severity of malaria infection can easily increase if the density or production of macrophages is compromised [
Graphs showing the effect of varying the rate of phagocytosis of IRBCs by macrophages,
Effect of
Effect of
Like the senescent red blood cells, aberrant infected erythrocytes formed during malaria infection are eliminated phagocytically by the host’s macrophage cells in the red pulp of the spleen [
In this paper, a mathematical model of in-host malaria infection in [
We proved that the formulated model is biologically and mathematically well posed in an invariant region
Our numerical results show that intervention during malaria infection should focus on minimizing merozoite invasion rate on healthy erythrocytes and the density of merozoites in circulation, which are responsible for secondary invasion at the blood stage. In the absence of malaria treatment, the immune cells (macrophages) are shown to be vital in eliminating infected red blood cells at the blood stage. The higher the rate of phagocytosis of infected erythrocytes by macrophages, the lower the density of infected red blood cells and hence malaria parasitemia. Patients suffering from such infections as HIV/AIDS and TB that have deleterious effect on the protective immune cells should seek immediate medical treatment when infected with malaria. Their compromised immune system exposes them to severe malaria attacks and possible untimely death.
For quick and timely reduction of parasitemia, an increased merozoite death rate using antimalarial drugs such as ACT would be necessary. This would further ensure reduced density of infected red blood cells and hence future generation merozoites. By killing a single blood schizont, we are likely to avoid the production of sixteen merozoites at maturity. Moreover, an appropriate vaccine that targets erythrocyte invasion process may equally guarantee minimal erythropoiesis. The erythrocyte invasion-avoidance vaccine would minimize the density of infected erythrocytes and hence malaria disease severity. This intervention could help terminate the erythrocytic schizont, leading to minimal parasite transmission to mosquito vector for further development and sexual reproduction.
In this study, drug resistance was not analyzed; this can be considered as a potential area for future investigation.
The authors declare that there are no conflicts of interest regarding the publication of this article.
The authors acknowledge with gratitude the support from the Institute of Mathematical Sciences, Strathmore University, and the National Research Fund (NRF), Kenya, for the production of this manuscript.