Cholera, an acute intestinal infection caused by the bacterium
The global burden of cholera remains substantial. In 2005, 131,943 cases and 2,272 deaths were reported to the WHO (World Health Organization), and recently major, sustained epidemics have been reported in many parts of Africa, such as Zimbabwe and Nigeria to mention a few [
Antimicrobial drug resistance can undermine the success of antimicrobial therapy.
A brief survey on previous studies provides the context of this paper. Various theoretical studies have been carried out on mathematical modeling of cholera transmission dynamics, focusing on a number of different issues, see [
The paper is structured as follows. The cholera transmission model is formulated in Section
Our objective is to formulate a model for cholera that includes drug resistant and drug sensitive bacteria. Thus, based on the individual’s epidemiological status the total human population
Model flow diagram.
For system (
System (
The diseasefree equilibrium
Following Kamgang and Sallet (2008) [
The system is defined on a positively invariant set
The subsystem
The matrix
There exists an upperbound matrix
We express the subsystem
which is an irreducible matrix. The upper bound for
The diseasefree equilibrium
There are three possible endemic equilibrium states for system (
if
if
if
If either case (a) or case (b) exists, then the global stability of resultant system follows from [
Before starting our main results, we give the following lemma which will be useful in the subsequent section.
Consider the following general system of ordinary differential equations with a parameter
matrix
Let
To apply this method, the following change of variables is required. Let
from (
Consider a case
Thus, the linearized system of the transformed equation (
It can be shown that the Jacobian
For the sign of
It follows from (
The endemic equilibrium
The reproduction number of an epidemic (the mean number of secondary cases infected by a single infectious case) is a key parameter for the analysis of infectious diseases because it summarizes the potential transmissibility of the disease and indicates whether an epidemic is under control. Now, we investigate the impact of increasing
Results in (
In order to support analytical findings in this study, we now simulate system (
Model parameters and their interpretations.
Parameter definition  Symbol  Units  Point estimate  Range  Source 

Recruitment rate 

People/year  100 000  —  [ 
Bacteria death rate 

/day  0.033  0.03–0.033  [ 
Bacteria shedding rate 

cells/mL day  30.0  10.0–50.0  [ 
Bacteria shedding rate 

cells/mL day  50.0  10.0–50.0  [ 
Natural mortality rate 

/year  0.0142  0.0142–0.02  [ 
Bacteria ingestion rate 

/day  0.5  0.15–1.5  [ 
Half saturation constant 

cells/mL 


[ 
Half saturation constant 

cells/mL 


[ 
Cholera related mortality rate 

/year  0.0240  0.0240–0.4622  [ 
Cholera related mortality rate 

/year  0.4622  0.0240–0.4622  [ 
Recovery rate for sensitive strain 

/year  22.6  11.3–43.2  [ 
Recovery rate for resistant strain 

/year  21.6  11.3–43.2  [ 
Proportion of infected individuals who develop drug resistance 

—  0.5  0.001–0.5  Assumed 
Numerical results in Figure
Simulations of model system (
Here (Figure
Small simulations of model system (
Simulations in Figure
Simulation of system (
Sensitivity analysis of model parameters is very important to design and control strategies as well as a direction to future research. There are many methods [
Results in Figure
Partial rank correlation coefficients showing the effects of parameter variation on,
Scatter plots for the basic reproductive number
Overall, results in Figure
Partial rank correlation coefficients showing the effects of parameter variation on
As the causative agent of cholera, the bacterium
In conclusion, our model suggests that the level of cholera infection depends highly on the rate of human recruitment and bacteria ingestion rate. We note that for bacteria ingestion rate above 0.5 cumulative cholera cases have a sharp increase within the period of 20 days, while for bacteria have ingestion less than 0.5 cumulative cholera cases a gradual increase and attain maximum after 20 days, and they also decrease steadily until stability is attained. We also observed that if a cholera outbreak have higher drug resistant cases than drug sensitive cases on the outbreak, then there will be a pronounced increase of cumulative drug resistant strain cases on a period, averagely of 10 days or less.