Typhoid fever continues to be a major public health problem in the developing world. Antibiotic therapy has been the main stay of treating typhoid fever for decades. The emergence of drug-resistant typhoid strain in the last two decades has been a major problem in tackling this scourge. A mathematical model for investigating the impact of drug resistance on the transmission dynamics of typhoid fever is developed. The reproductive number for the model has been computed. Numerical results in this study suggest that when a typhoid outbreak occurs with more drug-sensitive cases than drug-resistant cases, then it may take 10–15 months for symptomatic drug-resistant cases to outnumber all typhoid cases, and it may take an average of 15–20 months for nonsymptomatic drug-resistant cases to outnumber all drug-sensitive cases.
Typhoid fever is caused by
Typhoid drug resistance emerged first in the UK within 2 years of the successful use of chloramphenicol on typhoid treatment [
Motivated by the 2012 typhoid outbreak in Zimbabwe, this paper aims to assess the impact drug resistance on the transmission dynamics of typhoid fever, using a mathematical model. Mathematical models have become invaluable management tools for epidemiologists, both shedding light on the mechanisms underlying the observed dynamics as well as making quantitative predictions on the effectiveness of different control measures. The literature and development of mathematical epidemiology are well documented and can be found in [
The paper is structured as follows. Section
In this section we formulate a mathematical model for typhoid which incorporates drug resistance. The host population is divided into the following epidemiological classes or subgroups: susceptible
Model flow diagram.
In this section, we study the basic results of solutions for model system (
The equations preserve positivity of solutions.
The vector field given by the right hand side of (
Each positive solution is bounded in
The
The region
For every strictly positive initial value, solutions of model system (
Local existence of solutions follows from standard arguments since the right-hand side of (
Model system (
The reproductive ratio,
Using [
The disease-free equilibrium
Following Kamgang and Sallet, (2008) [
The disease-free equilibrium
System ( sensitive strain only, resistant strain only, coexistence of sensitive strain and resistant strain.
In the absence of drug-resistant strain in the community, that is,
If
We claim the following result (see Appendix
The endemic equilibrium
In the absence of drug-sensitive strain in the community, that is,
If
By Theorem
If
When both sensitive and resistant strains exist system (
The symmetric conditions above may be summarized diagrammatically as shown in Figure
Symmetric conditions for
Using the above symmetric conditions and results deduced on Theorems
In many epidemiological models, the magnitude of the reproductive number is associated with the level of infection. The same is true for model (
In order to illustrate the results of the foregoing analysis, we have simulated model system (
Model parameters and their interpretations.
Parameter definition | Symbol | Units | Point |
Range | Source |
---|---|---|---|---|---|
Disease-induced mortality for |
|
/year | 0.01 | 0.01–0.3 | [ |
Disease-induced mortality for |
|
/year | 0.012 | 0.01–0.3 | [ |
Disease-induced mortality for |
|
/year | 0.015 | 0.01–0.3 | [ |
Disease-induced mortality for |
|
/year | 0.02 | 0.01–0.3 | [ |
Typhoid transmission for |
|
— | 0.01 | 0.00001–0.025 | Assumed |
Typhoid transmission for |
|
— | 0.015 | 0.00001–0.025 | Assumed |
Typhoid transmission for |
|
— | 0.02 | 0.00001–0.025 | Assumed |
Typhoid transmission for |
|
— | 0.025 | 0.00001–0.025 | Assumed |
Natural mortality rate |
|
/year | 0.0142 | 0.01–0.02 | [ |
Rate of becoming symptomatic infectious for |
|
/year | 0.05 | 0.05–0.1 | [ |
Rate of becoming symptomatic infectious for |
|
/year | 0.09 | 0.05–0.1 | [ |
Proportion of individuals who join |
|
— | 0.5 | 0.0–1.0 | Assumed |
Recruitment rate |
|
People/year | 1000000 | — | [ |
Recovery rate |
|
/year | 0.150 | 0.0–0.2 | Assumed |
Sensitivity analysis assesses the amount and type of change inherent in the model as captured by the terms that define the reproductive number [
Figure
Partial rank correlation coefficients showing the effects of parameter variation on
Results on Figure
Partial rank correlation coefficients showing the effects of parameter variation on
The effects of increasing
Simulations of model system (
In Figure
Simulations of model system (
Simulation in Figure
Simulations of model system (
Typhoid fever continues to be an important cause of illness and death, particularly among children and adolescents in developing countries, where sanitary conditions remain poor. Drug resistance is becoming a major problem and treatment is becoming increasingly difficult, leading to patients taking longer to recover, suffering more complications and continuing to spread the disease to their family and to their community [
The model developed in this paper has limitation(s), which should be acknowledged. We assumed that the disease is transmitted through human contact only although the disease can be acquired through consumption, mainly of water, but sometimes of food, that has been contaminated by sewage containing the excrement of people suffering from the disease. Furthermore, recruited individuals are assumed to be susceptible which might not be case in some communities.
Following Kamgang and Sallet, (2008) [ the system is defined on a positively invariant set the subsystem the matrix there exists an upper-bound matrix
are satisfied. If conditions (
We express the subsystem
Then the condition
In order to investigate the global stability of the endemic equilibrium
Using the geometrical approach of Li and Muldowney in [
Let there exists a compact absorbing set equation (
The equilibrium
A point
Suppose that assumptions (
The following
Assume that
Since
Then,
Let
The authors are very grateful to the anonymous referee and the handling editor for their valuable comments and suggestions.