- About this Journal ·
- Abstracting and Indexing ·
- Aims and Scope ·
- Article Processing Charges ·
- Articles in Press ·
- Author Guidelines ·
- Bibliographic Information ·
- Citations to this Journal ·
- Contact Information ·
- Editorial Board ·
- Editorial Workflow ·
- Free eTOC Alerts ·
- Publication Ethics ·
- Reviewers Acknowledgment ·
- Submit a Manuscript ·
- Subscription Information ·
- Table of Contents
Computational Biology Journal
Volume 2013 (2013), Article ID 303645, 13 pages
Assessing the Impact of Drug Resistance on the Transmission Dynamics of Typhoid Fever
Department of Mathematics, University of Zimbabwe, P.O. Box MP 167, Harare, Zimbabwe
Received 20 February 2013; Revised 17 May 2013; Accepted 20 May 2013
Academic Editor: Markus Rehm
Copyright © 2013 S. Mushayabasa et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
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 Salmonella enterica serovar Typhi (S. Typhi), a gram-negative bacterium [1–3]. It continues to be a global public health problem with over 21.6 million cases and at least 250,000 deaths occurring annually [4–6]. Almost 80% of the cases and deaths are in Asia; the rest occur mainly in Africa and Latin America [3, 7]. In developing countries such as India, the disease occurs with an incidence ranging from 102 to 2,219 per 100,000 of the population [3, 8]. Several studies in areas of endemicity and outbreaks have shown that about one-quarter to one-third of pediatric typhoid fever cases are under five years of age, and that between 6% and 21% are under two years of age . Varied presentations of typhoid fever are known in the pediatric age group, such as septicemia in neonates, as diarrhoea in infants, and as lower respiratory tract infections in older children [10–12]. Typical presentation in older children includes splenic abscess, liver abscess, cerebellar ataxia, meningitis, cholecystitis, chorea, palatal palsy, osteomyelitis, peritonitis, aphasia, and even psychosis [13–16]. Due to these varied and typical presentations, it is common for typhoid fever in children to be diagnosed late or even remain unrecognised. Also, no vaccine against typhoid fever is available commercially for children under two years of age . To complicate matters further, in the last two decades, multidrug-resistant (MDR) S. Typhi strains have emerged and spread worldwide, resulting in high rates of morbidity and mortality [14, 15, 17].
Typhoid drug resistance emerged first in the UK within 2 years of the successful use of chloramphenicol on typhoid treatment [18, 19]. Subsequently, isolates carrying transferable chloramphenicol resistance were described from Greece  and Israel . The emergence of drug resistance has great implications for therapy . For example, children infected with such strains are more ill at presentation and have a longer duration of illness and a significantly higher mortality rate .
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 [24–26]. Modeling the transmission dynamics of typhoid is an important and interesting topic for a lot of researchers; see [27–32] to mention but a few. Lauria et al.  developed an optimization model for reducing typhoid cases in developing countries without increasing public spending. Their work suggested that the magnitude of herd protection effects greatly influences the total number of cases avoided and the value of public treatment cost savings. More recently, Kalajdzievska and Li  developed a mathematical model for assessing the effects of carriers on the transmission dynamics of infectious diseases such as typhoid. They concluded that carriers play a significant role in the transmission of infectious diseases.
The paper is structured as follows. Section 2 is the model framework. Analytical results have been carried out in Section 3. Numerical results are presented in Section 4. A short discussion concludes the paper.
2. Model Formulation
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 , sensitive strain nonsymptomatic infectious individuals , sensitive strain symptomatic infectious individuals , drug-resistant infectious carriers , and drug-resistant symptomatic infectious individuals . Thus, the total population is given by . We assume that individuals can be infected through direct contact with an infectious individual. Further, assuming homogeneous mixing of the population, individuals are recruited at constant rate (assumed susceptible). The model takes the form: is the percapita natural mortality rate, is the rate at which individuals infected with sensitive strain recover after successful treatment, and denote the percapita disease-induced death rate for typhoid patients in class and , respectively, and denote the percapita disease-induced mortality rate for typhoid patients in class and , and respectively, , where and denote the effective contact rate for typhoid transmission (sensitive strain) by nonsymptomatic infectious individuals and symptomatic infectious individuals, respectively. Reports on cases of reinfection with typhoid for successfully treated individuals have been on an increase since the year 2000 [33, 34]. In this study we assume that individuals infected with drug-sensitive strain can be reinfected with drug-resistant strain at rate , where and denote the effective contact rate for typhoid transmission (sensitive strain) by nonsymptomatic infectious individuals and symptomatic infectious individuals, respectively. Upon infection a fraction of the infected individuals becomes chronic patients and the remainder becomes symptomatic patients. Nonsymptomatic infectious individuals becomes infectious at rate and , for sensitive and resistant strain, respectively. The model flow diagram is depicted in Figure 1.
3. Analytical Results
3.1. Basic Properties of the Model
In this section, we study the basic results of solutions for model system (1), which are essential in the proofs of stability.
Lemma 1. The equations preserve positivity of solutions.
Proof. The vector field given by the right hand side of (1) points inward on the boundary of . For example, if , then . All the other components are similar.
Lemma 2. Each positive solution is bounded in norm by .
Proof. The norm of each positive solution is , and it satisfies the inequality . Solutions to the arbitrary equation are monotone increasing and bounded by if . They are monotone decreasing and bounded above if . Since , the claim follows.
Corollary 3. The region is invariant and attracting for system (1).
Theorem 4. For every strictly positive initial value, solutions of model system (1) exist for all .
Proof. Local existence of solutions follows from standard arguments since the right-hand side of (1) is locally Lipschitz. Global existence follows from the a priori bounds.
3.2. Disease-Free Equilibrium and Its Stability
The reproductive ratio, , measures the average number of new infections generated by a singly infected typhoid individual during his/her entire infectious period when he/she is introduced into a susceptible population in the presence of the aforementioned intervention strategies; and denote the sensitive strain-induced reproductive ratio and drug-resistant-induced reproductive ratio, respectively.
Using [35, Theorem 2], the following result is established.
Theorem 5. The disease-free equilibrium of model system (1) is locally asymptotically stable (LAS) if and unstable if .
Theorem 6. The disease-free equilibrium of model system (1) is globally asymptotically stable (GAS) if and unstable if .
3.3. Endemic Equilibrium and Its Stability
System (1) has three possible endemic equilibrium points, namely (a)sensitive strain only, (b)resistant strain only, (c)coexistence of sensitive strain and resistant strain.
3.3.1. Sensitive Strain Endemic Equilibrium
In the absence of drug-resistant strain in the community, that is, , system (1) becomes For system (5), it can be shown that the region is invariant and attracting. Thus, the dynamics of typhoid sensitive strain will be considered in . System (5) has an endemic equilibrium given by which makes biological sense only whenever .
Theorem 7. If , then the endemic equilibrium point exists.
We claim the following result (see Appendix B for proof).
3.3.2. Drug-Resistant Strain Endemic Equilibrium
In the absence of drug-sensitive strain in the community, that is, , system (1) becomes For system (8), it can be shown that the region is invariant and attracting. Thus, the dynamics of typhoid sensitive strain will be considered in . System (8) has an endemic equilibrium given by Substituting and (from (10)) into , one gets Equation (11) is a quadratic equation, with one of the solutions (which defines the existence of disease-free equilibrium). The second solution is given by which defines the existence of the endemic equilibrium , whenever . We have the following Theorem.
Theorem 9. If , then the endemic equilibrium point exists.
By Theorem 9 if , the infection persists. Regarding the fashion in which the infection persists, we establish in this section that when , the infection eventually stabilizes at an equilibrium level. More specifically, we have the following result. Theorem 10 has been established using the general method of Li and Muldowney  (see Appendix C for the proof).
Theorem 10. If , then exists and is globally asymptotically stable (GAS).
3.3.3. Interior Endemic Equilibrium
When both sensitive and resistant strains exist system (1) has an endemic equilibrium given by (not computed due to its complex nature) The following symmetric conditions will be used to determine the stability: (a) (typhoid-sensitive strain dies out) and (typhoid drug-resistant strain persists),(b) (typhoid-sensitive strain persists) and (typhoid-resistant strain dies out),(c) (typhoid-sensitive strain persists) and (typhoid drug-resistant strain persists). The symmetric conditions above may be summarized diagrammatically as shown in Figure 2.
3.3.4. Impact of Drug-Resistant Strain Typhoid Dynamics
In many epidemiological models, the magnitude of the reproductive number is associated with the level of infection. The same is true for model (1). From the expressions of and we observe that and are the two parameters which are common; hence to investigate the impact of drug-resistant strain on typhoid dynamics, we will use implicit differentiation as follows: Equation (14) suggests that an increase of drug-resistant cases may increase the prevalence of typhoid in the community.
4. Numerical Results
4.1. Sensitivity Analysis
Sensitivity analysis assesses the amount and type of change inherent in the model as captured by the terms that define the reproductive number [38, 39]. If the reproductive number is very sensitive to a particular parameter, then a perturbation of the conditions that connect the dynamics to such a parameter may prove useful in identifying policies or intervention strategies that reduce epidemic prevalence. In this section we computed the partial rank correlation coefficients (PRCCs) in order to estimate the correlation between the reproductive number and the model parameters which define the reproductive number. We begin by assessing the relationship between and the model parameters which define it.
Figure 3 illustrates the PRCCs using as an output variable. Results on Figure 3 suggest that an increase in human recruitment rate, typhoid transmission for both nonsymptomatic and symptomatic individuals, and the proportion of carriers may increase the magnitude of . We also noted that an increase on treatment level has the greatest influence on reducing the magnitude of ; hence typhoid treatment should be encouraged in order to control cumulative typhoid cases.
Results on Figure 4 suggest that an increase in human recruitment and typhoid transmission by nonsymptomatic individuals has more influence on increasing the magnitude of , compared to other model parameters. Since this paper considers no treatment for drug-resistant typhoid strain, results shown here suggest the need for typhoid intervention strategies in order to reduce cases of drug-resistant strain; these may include provision of clean water and typhoid vaccination.
The effects of increasing (the proportion of individuals who become infectious but nonsymptomatic upon infection) in cumulative drug-sensitive typhoid cases and cumulative drug-resistant typhoid cases are shown on Figure 5. In Figure 5(a), we note that if increases, (despite its value) then cumulative drug-sensitive nonsymptomatic cases increase from the first month and reaches the peak on the twentieth month; then the cases will start to decline to a low level. Figure 5(b) shows similar results to Figure 5(a); the only difference is that increasing does not give a more significant change as observed on Figure 5(a). We also note that an increase on may lead to an increase on drug-resistant cases. Further analysis of Figure 5 suggests that in a community where there is drug-sensitive typhoid and drug-resistant typhoid, drug-resistant strain cases may outnumber drug-sensitive cases with time.
In Figure 6, we observe the effects of different initial conditions on future trends of typhoid epidemic. Figure 6(a) suggests 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 an average of 15–20 months for nonsymptomatic drug-resistant cases to outnumber both symptomatic and nonsymptomatic drug-sensitive cases. However, for an outbreak with slightly higher cases of drug-resistant than drug-sensitive cases, it may take less than 10 months for non-symptomatic and symptomatic drug resistant cases to outnumber all forms of drug sensitive cases (see Figure 6(b)). When a typhoid outbreak has almost the same cases of drug-sensitive and drug resistant then, it may take 10–14 months for symptomatic drug-resistant cases to outnumber all typhoid cases, and it takes an average of 15–20 months for nonsymptomatic drug-resistant cases to outnumber all drug-sensitive cases.
Simulation in Figure 7 suggests that treatment has a positive impact on reducing cumulative typhoid cases since an increase in treatment is associated with low cumulative drug-sensitive and drug-resistant cases even though the proposed model does not account for drug-resistant treatment.
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 . A mathematical model for investigating the impact of drug-resistant strain on the dynamics of typhoid is developed and analyzed. Comprehensive and robust mathematical techniques have been used to analyze the model steady states. It has been established that the model has a disease-free equilibrium which is globally asymptotically stable when the associated reproductive number is less than unity. Sensitivity analysis of the reproductive number has been carried out. Results from the sensitivity analysis of the reproductive number suggest that an increase in human recruitment rate, typhoid transmission by nonsymptomatic and symptomatic individuals, and the proportion of individuals who become nonsymptomatic upon infection have the greatest influence on increasing the magnitude of the associated reproductive number. We also note that treatment has the greatest influence on reducing cumulative drug-sensitive typhoid cases. Analytical and numerical results have suggested that an increase of drug-resistant typhoid cases may outnumber typhoid drug-sensitive cases with time, and this will lead to high prevalence of typhoid in the community. With the aid of parameter values in Table 1, we deduce that the drug-resistant strain-induced reproductive number is greater than the drug-sensitive strain-induced reproductive number, and this is due to the fact that our study assumes that there is no treatment for drug-resistant typhoid patients.
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.
A. Global Stability of the Disease-Free Equilibrium
Following Kamgang and Sallet, (2008) , we write system (1) in the form: on the positively invariant set . Here and . Here denotes (its components) the number of uninfected individuals and denotes the number of infected components, and . We have to prove that the following conditions: ) the system is defined on a positively invariant set of the nonnegative orthant; the system is dissipative on ,() the subsystem is globally asymptotically stable at the equilibrium on the canonical projection of on ,() the matrix is Metzler (A Metzler matrix is a matrix with off-diagonal entries nonnegative ) and irreducible for any given ,() there exists an upper-bound matrix for with the property that either or if , (i.e., ), then, for any such that , (i.e., the points where the maximum is realized are contained in the disease-free submanifold),() are satisfied. If conditions () are satisfied, then is globally asymptotically stable for .
We express the subsystem : System (A.2) is a linear system which is globally asymptotically stable at the origin. This equilibrium corresponds to , satisfying conditions and . Hence, matrix is given by Theorem 5 implies that , , , and ; hence matrix is a Metzler matrix for any , and this satisfies condition . With the notations of hypothesis (), the diagonal block matrix is bounded by the matrix which is a maximum. This maximum is realized at point of such that . This implies that these equations belong to the manifold with equations . Now the diagonal block matrix is bounded by the matrix: which is a maximum. This maximum is realized at point of such that . This implies that these equations belong to the manifold with equations . Thereby the whole hypothesis is fulfilled.
Then the condition is equivalent to . The conditions and can be expressed by and the maximum of the two quantities on the left hand side is . Thus, sytem (1) is globally asymptotically stable if and unstable if ; we summarize the result in Theorem 6.
B. Global Stability of the Endemic Equilibrium
In order to investigate the global stability of the endemic equilibrium , we adopt the approach by Korobeinikov (2006) . Let be a positive and monotonic function, and define the following continuous function in (for more details, see Korobeinikov, 2006 ): If is monotonic with respect to its variables, then the endemic state is the only extremum and the global minimum of this function. Indeed if grow monotonically, then the function has only one stationary point. Further, since are nonnegative, then the point is a minimum; that is, , and hence is a Lyapunov function, and its time derivative is given by Since , the function is concave with respect to , and . Also the monotonicity of with respect to ensures that holds for all , , . We summarize the result in Theorem 8.
C. Global Stability of the Endemic Equilibrium
Using the geometrical approach of Li and Muldowney in , we obtain simple sufficient conditions which show that is globally asymptotically stable. At first, we give a brief outline of this geometrical approach.
Let be a for in an open set . Consider the differential equation: Denote by the solution to (C.1) such that . We make the following two assumptions:() there exists a compact absorbing set , () equation (C.1) has a unique equilibrium in . The equilibrium is said to be globally stable in if it is locally stable, and all trajectories in converge to . For , by a Bendixson criterion we mean a condition satisfied by which precludes the existence of nonconstant periodic solution of (C.1). The classical Bendixson’s condition div for is robust under local perturbations of . For higher-dimensional systems, the robust properties are discussed in [37, 42, 43].
A point is wandering for (C.1) a neighborhood of and such that is empty for all . Thus, for example, all equilibria and limit points are nonwandering. The following global-stability principle is established in Li and Muldowney  for autonomous systems in any finite dimension.
Theorem 11. Suppose that assumptions () and () hold. Assume that (C.1) satisfies a Bendixson criterion that is robust under local perturbations of at all nonequilibrium nonwandering points for (C.1). Then is globally stable in .
The following Bendixson criterion is given in  and shown to have the robustness required by Theorem 11. Let be an matrix-valued function, that is, for . Assume that exists and is continuous for , the absorbing set. A quantity is defined as where and the matrix is obtained by replacing each entry its directional derivative in the direction , . The matrix is the second second additive compound matrix of the Jacobian matrix . It is an matrix. The following global-stability result is due to Li and Muldowney (see , Theorem 8) Let denote a vector norm in and the corresponding matrix norm it induces. The Lozinskii measure on matrices with respect to is defined by (see ) for an matrix . For properties and calculations of Lozinski i will refer the reader to . It is shown in  that if is simply connected, the condition rules out the presence of any orbit that gives rise to a simple closed rectifiable curve that is invariant for (C.1), such as periodic orbits, homoclinic orbits, and heteroclinic cycles. Moreover, it is robust under local perturbations of near any non-equilibrium point that is non-wandering. In particularly, the following global stability result is proved in Li and Muldowney .
Theorem 12. Assume that is simply connected and that the assumptions () and () hold. Then the unique equilibrium of (C.1) is globally stable in if .
Then, , and the matrix can be written in block form: where
Let be a vector in . We choose a vector norm in as for any vector . Let denote the Lozinskii measure with respect to this norm. We can then obtain with Here and are matrix norms with respect to the vector norm, and denotes the Lozinskii measure with respect to the norm. More specifically Therefore Recall that Thus, becomes Further recall that , so that Therefore, Thus, there exists such that when , . As a result, which implies , thus completing the proof.
The authors are very grateful to the anonymous referee and the handling editor for their valuable comments and suggestions.
- T. M. Al-Sanouri, B. Paglietti, A. Haddadin et al., “Emergence of plasmid-mediated multidrug resistance in epidemic and non-epidemic strains of Salmonella enterica serotype Typhi from Jordan,” Journal of Infection in Developing Countries, vol. 2, no. 4, pp. 295–301, 2008.
- S. Kanungo, S. Dutta, and D. Sur, “Epidemiology of typhoid and paratyphoid fever in India,” Journal of Infection in Developing Countries, vol. 2, no. 6, pp. 454–460, 2008.
- S. A. Zaki and S. Karande, “Multidrug-resistant typhoid fever: a review,” Journal of Infection in Developing Countries, vol. 5, no. 5, pp. 324–337, 2011.
- J. A. Crump, S. P. Luby, and E. D. Mintz, “The global burden of typhoid fever,” Bulletin of the World Health Organization, vol. 82, no. 5, pp. 346–353, 2004.
- A. Kothari, A. Pruthi, and T. D. Chugh, “The burden of enteric fever,” Journal of Infection in Developing Countries, vol. 2, no. 4, pp. 253–259, 2008.
- P. Srikantiah, F. Y. Girgis, S. P. Luby et al., “Population-based surveillance of typhoid fever in Egypt,” American Journal of Tropical Medicine and Hygiene, vol. 74, no. 1, pp. 114–119, 2006.
- K. Nagshetty, S. T. Channappa, and S. M. Gaddad, “Antimicrobial susceptibility of Salmonella typhi in India,” Journal of Infection in Developing Countries, vol. 4, no. 2, pp. 070–073, 2010.
- M. N. Chowta and N. K. Chowta, “Study of clinical profile and antibiotic response in typhoid fever,” Indian Journal of Medical Microbiology, vol. 23, no. 2, pp. 125–127, 2005.
- F. Y. C. Lin, V. A. Ho, P. V. Bay et al., “The epidemiology of typhoid fever in the Dong Thap Province, Mekong Delta region of Vietnam,” American Journal of Tropical Medicine and Hygiene, vol. 62, no. 5, pp. 644–648, 2000.
- S. Mohanty, R. Gaind, R. Sehgal, H. Chellani, and M. Deb, “Neonatal sepsis due to Salmonella Typhi and Paratyphi A,” Journal of Infection in Developing Countries, vol. 3, no. 8, pp. 633–638, 2009.
- S. C. Karande, M. S. Desai, and M. K. Jain, “Typhoid fever in a 7 month old infant,” Journal of Postgraduate Medicine, vol. 41, no. 4, pp. 108–109, 1995.
- P. S. Olutola and J. B. Familusi, “Salmonella typhi pneumonia without gastrointestinal manifestations,” Diagnostic Imaging in Clinical Medicine, vol. 54, no. 5, pp. 263–267, 1985.
- V. Jindal and V. P. Singh, “Impending rupture of splenic abscess in enteric fever,” Indian Pediatrics, vol. 45, no. 10, p. 864, 2008.
- T. Cleary, “Salmonella,” in Feigin and Cherrys Textbook of Pediatric Infectious Diseases, R. D. Feigin, J. D. Cherry, G. J. Demmler, and S. L. Kaplan SL, Eds., pp. 1473–1487, Saunders, Philadelphia, Pa, USA, 5th edition, 2005.
- I. A. Memon, A. G. Billoo, and H. I. Memon, “Cefixime: an oral option for the treatment of multidrug-resistant enteric fever in children,” Southern Medical Journal, vol. 90, no. 12, pp. 1204–1207, 1997.
- S. A. Zaki and P. Shanbag, “Clinical manifestations of dengue and leptospirosis in children in Mumbai: an observational study,” Infection, vol. 38, no. 4, pp. 285–291, 2010.
- R. Kumar, N. Gupta, and S. Shalini, “Multidrug-resistant typhoid fever,” Indian Journal of Pediatrics, vol. 74, no. 1, pp. 39–42, 2007.
- S. H. Mirza, N. J. Beeching, and C. A. Hart, “Multi-drug resistant typhoid: a global problem,” Journal of Medical Microbiology, vol. 44, no. 5, pp. 317–319, 1996.
- J. Colquhoun and R. S. Weetch, “Resistance to chloramphenicol developing during treatment of 324 typhoid fever,” The Lancet, vol. 256, no. 6639, pp. 621–623, 1950.
- P. Kontomichalou, “Studies on resistance transfer factors,” Pathologia et Microbiologia, vol. 30, no. 1, pp. 71–93, 1967.
- D. Sompolinsky, M. Ben-Yakov, M. Aboud, and I. Boldur, “Transferable resistance factors with mutator effect in Salmonella typhi,” Mutation Research, vol. 4, no. 2, pp. 119–127, 1967.
- B. Rowe, L. R. Ward, and E. J. Threlfall, “Spread of multiresistant Salmonella typhi,” Lancet, vol. 336, no. 8722, p. 1065, 1990.
- Z. A. Bhutta, S. H. Naqvi, R. A. Razzaq, and B. J. Farooqui, “Multidrug-resistant typhoid in children: presentation and clinical features,” Reviews of Infectious Diseases, vol. 13, no. 5, pp. 832–836, 1991.
- R. M. Anderson and R. M. May, Infectious Diseases of Humans, Dynamics and Control, Oxford University Press, New York, NY, USA, 1991.
- N. Bailey, The Mathematical Theory of Infectious Diseases, Charles Griffin, 1975.
- F. Brauer and C. Castillo-Chavez, Mathematical Models in Population Biology and Epidemiology, vol. 40 of Texts in Applied Mathematics, Springer, New York, NY, USA, 2nd edition, 2012.
- J. Gonzalez-Guzman, “An epidemiological model for direct and indirect transmission of typhoid fever,” Mathematical Biosciences, vol. 96, no. 1, pp. 33–46, 1989.
- B. Cvjetanovic, B. Grab, and H. Dixon, “Computerized epidemiological model of typhoid fever with age structure and its use in the planning and evaluation of antityphoid immunization and sanitation programmes,” Mathematical Modelling, vol. 7, no. 5–8, pp. 719–744, 1986.
- D. T. Lauria, B. Maskery, C. Poulos, and D. Whittington, “An optimization model for reducing typhoid cases in developing countries without increasing public spending,” Vaccine, vol. 27, no. 10, pp. 1609–1621, 2009.
- D. Kalajdzievska and M. Y. Li, “Modeling the effects of carriers on transmission dynamics of infectious diseases,” Mathematical Biosciences and Engineering, vol. 8, no. 3, pp. 711–722, 2011.
- S. Mushayabasa, “Impact of vaccines on controlling typhoid fever in Kassena-Nankana district of upper east region of Ghana: insights from a mathematical model,” Journal of Modern Mathematics and Statistics, vol. 5, no. 2, pp. 54–59, 2011.
- S. Mushayabasa, C. P. Bhunu, and E. T. Ngarakana-Gwasira, “Mathematical analysis of a typhoid model with carriers, direct and indirect disease transmission,” International Journal of Mathematical Sciences and Engineering Applications, vol. 7, no. 1, pp. 79–90, 2013.
- S. Zaki, “Re-infection of typhoid fever and typhoid vaccine (comment on “An imported enteric fever caused by a quinolone-resistant Salmonella typhi”),” Annals of Saudi Medicine, vol. 31, no. 2, pp. 203–204, 2011.
- A. Griffin, D. Baraho-Hassan, and S. J. McSorley, “Successful treatment of bacterial infection hinders development of acquired immunity,” Journal of Immunology, vol. 183, no. 2, pp. 1263–1270, 2009.
- P. V. Driessche and J. Watmough, “Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission,” Mathematical Biosciences, vol. 180, pp. 29–48, 2002.
- J. C. Kamgang and G. Sallet, “Computation of threshold conditions for epidemiological models and global stability of the disease-free equilibrium (DFE),” Mathematical Biosciences, vol. 213, no. 1, pp. 1–12, 2008.
- M. Y. Li and J. S. Muldowney, “A geometric approach to global-stability problems,” SIAM Journal on Mathematical Analysis, vol. 27, no. 4, pp. 1070–1083, 1996.
- S. Mushayabasa, C. P. Bhunu, and R. J. Smith, “Assessing the impact of educational campaigns on controlling HCV among women in prison settings,” Communications in Nonlinear Science and Numerical Simulation, vol. 17, no. 4, pp. 1714–1724, 2012.
- J. C. Helton, “Uncertainty and sensitivity analysis techniques for use in performance assessment for radioactive waste disposal,” Reliability Engineering and System Safety, vol. 42, no. 2-3, pp. 327–367, 1993.
- A. Berman and R. J. Plemmons, “Nonnegative matrice in the mathematical sciences,” SIAM Review, vol. 35, pp. 43–79, 1993.
- A. Korobeinikov, “Lyapunov functions and global stability for SIR and SIRS epidemiological models with non-linear transmission,” Bulletin of Mathematical Biology, vol. 68, no. 3, pp. 615–626, 2006.
- M. Y. Li, “Dulac criteria for autonomous systems having an invariant affine manifold,” Journal of Mathematical Analysis and Applications, vol. 199, no. 2, pp. 374–390, 1996.
- M. Y. Li and J. S. Muldowney, “On R. A. Smith's autonomous convergence theorem,” The Rocky Mountain Journal of Mathematics, vol. 25, no. 1, pp. 365–379, 1995.
- W. A. Coppel, Stability and Asymptotic Behavior of Differential Equations, Health, Boston, Mass, USA, 1965.