# Assessing the Impact of Drug Resistance on the Transmission Dynamics of Typhoid Fever

Authors: S. Mushayabasa; C. P. Bhunu; E. T. Ngarakana-Gwasira
Journal: Computational Biology Journal
 (2013)

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## Abstract

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.


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## Body

## 1. Introduction

Typhoid fever is caused bySalmonella 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 [9]. 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 [14]. 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 [20] and Israel [21]. The emergence of drug resistance has great implications for therapy [22]. 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 [23].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. [29] 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 [30] 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. Section2 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: susceptibleS, sensitive strain nonsymptomatic infectious individuals Ic, sensitive strain symptomatic infectious individuals I, drug-resistant infectious carriers Icr, and drug-resistant symptomatic infectious individuals Ir. Thus, the total population N is given by N=S+Ic+I+Icr+Ir. 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 Λ (assumed susceptible). The model takes the form:
(1)S′=Λ+γI-(λ+λr)S-μS,Ic′=pλS-λrIc-(μ+α+d)Ic,I′=(1-p)λS+αIc-λrI-(μ+γ+δ)I,Icr′=pλr(S+Ic)-(μ+ϕ+ω)Icr,Ir′=(1-p)λr(S+Ic)+λrI+ϕIcr-(μ+ν)Ir,μis the percapita natural mortality rate, γ is the rate at which individuals infected with sensitive strain recover after successful treatment, d and δ denote the percapita disease-induced death rate for typhoid patients in class Ic and I, respectively, ω and ν denote the percapita disease-induced mortality rate for typhoid patients in class Icr and Ir, and respectively, λ=(βIc+θI), where β and θ 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 λr=(βrIcr+θrIr), where βr and θr denote the effective contact rate for typhoid transmission (sensitive strain) by nonsymptomatic infectious individuals and symptomatic infectious individuals, respectively. Upon infection a fraction p of the infected individuals becomes chronic patients and the remainder (1-p) becomes symptomatic patients. Nonsymptomatic infectious individuals becomes infectious at rate α and ϕ, for sensitive and resistant strain, respectively. The model flow diagram is depicted in Figure 1.Figure 1
Model flow diagram.


## 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 ℝ+5∖{0}. For example, if S=0, then S′=Λ+γI≥0. All the other components are similar.Lemma 2.
Each positive solution is bounded inL1 norm by max{N(0),Λ/μ}.Proof.
TheL1 norm of each positive solution is N, and it satisfies the inequality N′≤Λ-μN. Solutions to the arbitrary equation M′=Λ-μM are monotone increasing and bounded by Λ/μ if M(0)<Λ/μ. They are monotone decreasing and bounded above if M(0)≥Λ/μ. Since N′≤M′, the claim follows.Corollary 3.
The region(2)Φ={(S,Ic,I,Icr,Ir)∈ℝ+5:N≤Λμ}
is invariant and attracting for system (1).Theorem 4.
For every strictly positive initial value, solutions of model system (1) exist for all t≥0.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

Model system (1) has an evident disease-free equilibrium (DFE) given by
(3)ℰ0=(S0,Ic0,I0,Icr0,Ir0)=(Λμ,0,0,0,0).
Following Driessche and Watmough [35], the reproductive number for system (1) is given by
(4)ℛ0=max(ℛs,ℛr)=(Λ[θ(1-p)k1+p(αθ+βk2)]μk1k2,Λ[θr(1-p)k3+p(αθr+βrk4)]μk3k4),
where k1=(d+α+μ),  k2=(γ+δ+μ),  k3=(μ+ϕ+ω), and k4=(μ+ν).The reproductive ratio,ℛ0, 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; ℛs and ℛr 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(ℰ0) of model system (1) is locally asymptotically stable (LAS) if ℛ0≤1 and unstable if ℛ0>1.Following Kamgang and Sallet, (2008) [36] we have established the results in Theorem 6. The computations for Theorem 6 are presented in Appendix A.Theorem 6.
The disease-free equilibrium(ℰ0) of model system (1) is globally asymptotically stable (GAS) if ℛ0≤1 and unstable if ℛ0>1.


### 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,(Icr=Ir=0), system (1) becomes
(5)S′=Λ+γI-λS-μS,Ic′=pλS-(μ+α+d)Ic,I′=(1-p)λS+αIc-(μ+γ+δ)I.
For system (5), it can be shown that the region
(6)Ω={(S,Ic,I)∈ℝ+3:N≤Λμ}
is invariant and attracting. Thus, the dynamics of typhoid sensitive strain will be considered in Ω. System (5) has an endemic equilibrium given by
(7)ℰs={S*=Λμℛs,Ic*=p(ℛs-1)[(d+μ)(μ+δ+pγ)+α(δ+μ)]ℛs,I*=(pα+(1-p)(μ+α+d))(ℛs-1)[(d+μ)(μ+δ+pγ)+α(δ+μ)]ℛs,
which makes biological sense only whenever ℛs>1.Theorem 7.
Ifℛs>1, then the endemic equilibrium point ℰs exists.We claim the following result (see AppendixB for proof).Theorem 8.
The endemic equilibriumℰs is globally asymptotically stable whenever conditions outlined on (B.5) are satisfied.


#### 3.3.2. Drug-Resistant Strain Endemic Equilibrium

In the absence of drug-sensitive strain in the community, that is,(Ic=I=0), system (1) becomes
(8)S′=Λ-λS-μS,Icr′=pλrS-(μ+ϕ+ω)Icr,I′=(1-p)λrS+ϕIcr-(μ+ν)I.
For system (8), it can be shown that the region
(9)Δ={(S,Icr,Ir)∈ℝ+3:N≤Λμ}
is invariant and attracting. Thus, the dynamics of typhoid sensitive strain will be considered in Δ. System (8) has an endemic equilibrium ℰr given by
(10)ℰr={S*=Λ(μ+λr*),Icr*=pλr*Λ(μ+λr*)k3,Ir*=λr*Λ(pϕ+(1-p)k3)(μ+λr*)k3k4,withλr*=(βrIcr*+θrIr*),andk3,k4areasdefinedearlieron(4).
Substituting Icr* and Ir* (from (10)) into λr*=βrIcr+θrIr, one gets
(11)λr*(1-Λ(θr(1-p)k3+p(αθr+βrk4))(λr*+μ)k3k4)=0.
Equation (11) is a quadratic equation, with one of the solutions λr*=0 (which defines the existence of disease-free equilibrium). The second solution is given by
(12)λr*=μ(Λ(θr(1-p)k3+p(αθr+βrk4))μk3k4-1)=μ(ℛr-1),
which defines the existence of the endemic equilibrium ℰr, whenever ℛr>1. We have the following Theorem.Theorem 9.
Ifℛr>1, then the endemic equilibrium point ℰr exists.By Theorem9 if ℛr>1, the infection persists. Regarding the fashion in which the infection persists, we establish in this section that when ℛr>1, 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 [37] (see Appendix C for the proof).Theorem 10.
Ifℛr>1, then ℰr 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)
(13)ℰrs=(S*,Ic*,I*,Icr*,Ir*).
The following symmetric conditions will be used to determine the stability: (a)
ℛs<1 (typhoid-sensitive strain dies out) and ℛr>1 (typhoid drug-resistant strain persists),(b)
ℛs>1 (typhoid-sensitive strain persists) and ℛr<1 (typhoid-resistant strain dies out),(c)
ℛh>1 (typhoid-sensitive strain persists) and ℛc>1 (typhoid drug-resistant strain persists). The symmetric conditions above may be summarized diagrammatically as shown in Figure 2.Figure 2
Symmetric conditions forℛ0>1, n∈ℵ, (n>1).Using the above symmetric conditions and results deduced on Theorems6, 7, and 9, we deduce that, for ℛ0>1, the endemic equilibrium will be globally asymptotically stable.


#### 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 ℛs and ℛr we observe that Λ and μ 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:
(14)∂ℛs∂ℛr=∂ℛs/∂Λ∂ℛr/∂Λ+∂ℛs/∂μ∂ℛr/∂μ=[θ(1-p)k1+p(αθ+βk2)]k3k4[θr(1-p)k3+p(αθr+βrk4)]k1k2+(+pk3(ϕθr(μ+k4)+βrk42]k1k2)-1([(1-p)(μ+k2)θk12+(αθ+βk2)pμk2+pk1(αθ(μ+k2)+βk22)]k3k4)×([(1-p)(μ+k4)θrk32+(ϕθr+βrk4)pμk4+pk3(ϕθr(μ+k4)+βrk42)]k1k2)-1)>0.
Equation (14) suggests that an increase of drug-resistant cases may increase the prevalence of typhoid in the community.


## 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 ℝ+5∖{0}. For example, if S=0, then S′=Λ+γI≥0. All the other components are similar.Lemma 2.
Each positive solution is bounded inL1 norm by max{N(0),Λ/μ}.Proof.
TheL1 norm of each positive solution is N, and it satisfies the inequality N′≤Λ-μN. Solutions to the arbitrary equation M′=Λ-μM are monotone increasing and bounded by Λ/μ if M(0)<Λ/μ. They are monotone decreasing and bounded above if M(0)≥Λ/μ. Since N′≤M′, the claim follows.Corollary 3.
The region(2)Φ={(S,Ic,I,Icr,Ir)∈ℝ+5:N≤Λμ}
is invariant and attracting for system (1).Theorem 4.
For every strictly positive initial value, solutions of model system (1) exist for all t≥0.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

Model system (1) has an evident disease-free equilibrium (DFE) given by
(3)ℰ0=(S0,Ic0,I0,Icr0,Ir0)=(Λμ,0,0,0,0).
Following Driessche and Watmough [35], the reproductive number for system (1) is given by
(4)ℛ0=max(ℛs,ℛr)=(Λ[θ(1-p)k1+p(αθ+βk2)]μk1k2,Λ[θr(1-p)k3+p(αθr+βrk4)]μk3k4),
where k1=(d+α+μ),  k2=(γ+δ+μ),  k3=(μ+ϕ+ω), and k4=(μ+ν).The reproductive ratio,ℛ0, 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; ℛs and ℛr 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(ℰ0) of model system (1) is locally asymptotically stable (LAS) if ℛ0≤1 and unstable if ℛ0>1.Following Kamgang and Sallet, (2008) [36] we have established the results in Theorem 6. The computations for Theorem 6 are presented in Appendix A.Theorem 6.
The disease-free equilibrium(ℰ0) of model system (1) is globally asymptotically stable (GAS) if ℛ0≤1 and unstable if ℛ0>1.


## 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,(Icr=Ir=0), system (1) becomes
(5)S′=Λ+γI-λS-μS,Ic′=pλS-(μ+α+d)Ic,I′=(1-p)λS+αIc-(μ+γ+δ)I.
For system (5), it can be shown that the region
(6)Ω={(S,Ic,I)∈ℝ+3:N≤Λμ}
is invariant and attracting. Thus, the dynamics of typhoid sensitive strain will be considered in Ω. System (5) has an endemic equilibrium given by
(7)ℰs={S*=Λμℛs,Ic*=p(ℛs-1)[(d+μ)(μ+δ+pγ)+α(δ+μ)]ℛs,I*=(pα+(1-p)(μ+α+d))(ℛs-1)[(d+μ)(μ+δ+pγ)+α(δ+μ)]ℛs,
which makes biological sense only whenever ℛs>1.Theorem 7.
Ifℛs>1, then the endemic equilibrium point ℰs exists.We claim the following result (see AppendixB for proof).Theorem 8.
The endemic equilibriumℰs is globally asymptotically stable whenever conditions outlined on (B.5) are satisfied.


### 3.3.2. Drug-Resistant Strain Endemic Equilibrium

In the absence of drug-sensitive strain in the community, that is,(Ic=I=0), system (1) becomes
(8)S′=Λ-λS-μS,Icr′=pλrS-(μ+ϕ+ω)Icr,I′=(1-p)λrS+ϕIcr-(μ+ν)I.
For system (8), it can be shown that the region
(9)Δ={(S,Icr,Ir)∈ℝ+3:N≤Λμ}
is invariant and attracting. Thus, the dynamics of typhoid sensitive strain will be considered in Δ. System (8) has an endemic equilibrium ℰr given by
(10)ℰr={S*=Λ(μ+λr*),Icr*=pλr*Λ(μ+λr*)k3,Ir*=λr*Λ(pϕ+(1-p)k3)(μ+λr*)k3k4,withλr*=(βrIcr*+θrIr*),andk3,k4areasdefinedearlieron(4).
Substituting Icr* and Ir* (from (10)) into λr*=βrIcr+θrIr, one gets
(11)λr*(1-Λ(θr(1-p)k3+p(αθr+βrk4))(λr*+μ)k3k4)=0.
Equation (11) is a quadratic equation, with one of the solutions λr*=0 (which defines the existence of disease-free equilibrium). The second solution is given by
(12)λr*=μ(Λ(θr(1-p)k3+p(αθr+βrk4))μk3k4-1)=μ(ℛr-1),
which defines the existence of the endemic equilibrium ℰr, whenever ℛr>1. We have the following Theorem.Theorem 9.
Ifℛr>1, then the endemic equilibrium point ℰr exists.By Theorem9 if ℛr>1, the infection persists. Regarding the fashion in which the infection persists, we establish in this section that when ℛr>1, 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 [37] (see Appendix C for the proof).Theorem 10.
Ifℛr>1, then ℰr 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)
(13)ℰrs=(S*,Ic*,I*,Icr*,Ir*).
The following symmetric conditions will be used to determine the stability: (a)
ℛs<1 (typhoid-sensitive strain dies out) and ℛr>1 (typhoid drug-resistant strain persists),(b)
ℛs>1 (typhoid-sensitive strain persists) and ℛr<1 (typhoid-resistant strain dies out),(c)
ℛh>1 (typhoid-sensitive strain persists) and ℛc>1 (typhoid drug-resistant strain persists). The symmetric conditions above may be summarized diagrammatically as shown in Figure 2.Figure 2
Symmetric conditions forℛ0>1, n∈ℵ, (n>1).Using the above symmetric conditions and results deduced on Theorems6, 7, and 9, we deduce that, for ℛ0>1, the endemic equilibrium will be globally asymptotically stable.


### 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 ℛs and ℛr we observe that Λ and μ 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:
(14)∂ℛs∂ℛr=∂ℛs/∂Λ∂ℛr/∂Λ+∂ℛs/∂μ∂ℛr/∂μ=[θ(1-p)k1+p(αθ+βk2)]k3k4[θr(1-p)k3+p(αθr+βrk4)]k1k2+(+pk3(ϕθr(μ+k4)+βrk42]k1k2)-1([(1-p)(μ+k2)θk12+(αθ+βk2)pμk2+pk1(αθ(μ+k2)+βk22)]k3k4)×([(1-p)(μ+k4)θrk32+(ϕθr+βrk4)pμk4+pk3(ϕθr(μ+k4)+βrk42)]k1k2)-1)>0.
Equation (14) suggests that an increase of drug-resistant cases may increase the prevalence of typhoid in the community.


## 3.3.1. Sensitive Strain Endemic Equilibrium

In the absence of drug-resistant strain in the community, that is,(Icr=Ir=0), system (1) becomes
(5)S′=Λ+γI-λS-μS,Ic′=pλS-(μ+α+d)Ic,I′=(1-p)λS+αIc-(μ+γ+δ)I.
For system (5), it can be shown that the region
(6)Ω={(S,Ic,I)∈ℝ+3:N≤Λμ}
is invariant and attracting. Thus, the dynamics of typhoid sensitive strain will be considered in Ω. System (5) has an endemic equilibrium given by
(7)ℰs={S*=Λμℛs,Ic*=p(ℛs-1)[(d+μ)(μ+δ+pγ)+α(δ+μ)]ℛs,I*=(pα+(1-p)(μ+α+d))(ℛs-1)[(d+μ)(μ+δ+pγ)+α(δ+μ)]ℛs,
which makes biological sense only whenever ℛs>1.Theorem 7.
Ifℛs>1, then the endemic equilibrium point ℰs exists.We claim the following result (see AppendixB for proof).Theorem 8.
The endemic equilibriumℰs is globally asymptotically stable whenever conditions outlined on (B.5) are satisfied.


## 3.3.2. Drug-Resistant Strain Endemic Equilibrium

In the absence of drug-sensitive strain in the community, that is,(Ic=I=0), system (1) becomes
(8)S′=Λ-λS-μS,Icr′=pλrS-(μ+ϕ+ω)Icr,I′=(1-p)λrS+ϕIcr-(μ+ν)I.
For system (8), it can be shown that the region
(9)Δ={(S,Icr,Ir)∈ℝ+3:N≤Λμ}
is invariant and attracting. Thus, the dynamics of typhoid sensitive strain will be considered in Δ. System (8) has an endemic equilibrium ℰr given by
(10)ℰr={S*=Λ(μ+λr*),Icr*=pλr*Λ(μ+λr*)k3,Ir*=λr*Λ(pϕ+(1-p)k3)(μ+λr*)k3k4,withλr*=(βrIcr*+θrIr*),andk3,k4areasdefinedearlieron(4).
Substituting Icr* and Ir* (from (10)) into λr*=βrIcr+θrIr, one gets
(11)λr*(1-Λ(θr(1-p)k3+p(αθr+βrk4))(λr*+μ)k3k4)=0.
Equation (11) is a quadratic equation, with one of the solutions λr*=0 (which defines the existence of disease-free equilibrium). The second solution is given by
(12)λr*=μ(Λ(θr(1-p)k3+p(αθr+βrk4))μk3k4-1)=μ(ℛr-1),
which defines the existence of the endemic equilibrium ℰr, whenever ℛr>1. We have the following Theorem.Theorem 9.
Ifℛr>1, then the endemic equilibrium point ℰr exists.By Theorem9 if ℛr>1, the infection persists. Regarding the fashion in which the infection persists, we establish in this section that when ℛr>1, 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 [37] (see Appendix C for the proof).Theorem 10.
Ifℛr>1, then ℰr 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)
(13)ℰrs=(S*,Ic*,I*,Icr*,Ir*).
The following symmetric conditions will be used to determine the stability: (a)
ℛs<1 (typhoid-sensitive strain dies out) and ℛr>1 (typhoid drug-resistant strain persists),(b)
ℛs>1 (typhoid-sensitive strain persists) and ℛr<1 (typhoid-resistant strain dies out),(c)
ℛh>1 (typhoid-sensitive strain persists) and ℛc>1 (typhoid drug-resistant strain persists). The symmetric conditions above may be summarized diagrammatically as shown in Figure 2.Figure 2
Symmetric conditions forℛ0>1, n∈ℵ, (n>1).Using the above symmetric conditions and results deduced on Theorems6, 7, and 9, we deduce that, for ℛ0>1, the endemic equilibrium will be globally asymptotically stable.


## 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 ℛs and ℛr we observe that Λ and μ 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:
(14)∂ℛs∂ℛr=∂ℛs/∂Λ∂ℛr/∂Λ+∂ℛs/∂μ∂ℛr/∂μ=[θ(1-p)k1+p(αθ+βk2)]k3k4[θr(1-p)k3+p(αθr+βrk4)]k1k2+(+pk3(ϕθr(μ+k4)+βrk42]k1k2)-1([(1-p)(μ+k2)θk12+(αθ+βk2)pμk2+pk1(αθ(μ+k2)+βk22)]k3k4)×([(1-p)(μ+k4)θrk32+(ϕθr+βrk4)pμk4+pk3(ϕθr(μ+k4)+βrk42)]k1k2)-1)>0.
Equation (14) suggests that an increase of drug-resistant cases may increase the prevalence of typhoid in the community.


## 4. Numerical Results

In order to illustrate the results of the foregoing analysis, we have simulated model system (1) using the parameters in Table 1.Table 1
Model parameters and their interpretations.
Parameter definition
Symbol
Units
Pointestimate
Range
Source
Disease-induced mortality forIc class
d
/year
0.01
0.01–0.3
[31]
Disease-induced mortality forI class
δ
/year
0.012
0.01–0.3
[31]
Disease-induced mortality forIcr class
ω
/year
0.015
0.01–0.3
[31]
Disease-induced mortality forIr class
ν
/year
0.02
0.01–0.3
[31]
Typhoid transmission forIc class
β
—
0.01
0.00001–0.025
Assumed
Typhoid transmission forIcr class
β
r
—
0.015
0.00001–0.025
Assumed
Typhoid transmission forIc class
θ
—
0.02
0.00001–0.025
Assumed
Typhoid transmission forIr class
θ
r
—
0.025
0.00001–0.025
Assumed
Natural mortality rate
μ
/year
0.0142
0.01–0.02
[38]
Rate of becoming symptomatic infectious forIc class
α
/year
0.05
0.05–0.1
[31]
Rate of becoming symptomatic infectious forIcr class
ϕ
/year
0.09
0.05–0.1
[31]
Proportion of individuals who joinIc
p
—
0.5
0.0–1.0
Assumed
Recruitment rate
Λ
People/year
1000000
—
[29, 31]
Recovery rate
γ
/year
0.150
0.0–0.2
Assumed


### 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 ℛs and the model parameters which define it.Figure3 illustrates the PRCCs using ℛs 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 ℛs. We also noted that an increase on treatment level has the greatest influence on reducing the magnitude of ℛs; hence typhoid treatment should be encouraged in order to control cumulative typhoid cases.Figure 3
Partial rank correlation coefficients showing the effects of parameter variation onℛs using ranges in Table 1. Parameters with positive PRCCs will increase ℛs when they are increased, whereas parameters with negative PRCCs will decrease ℛs when they are increased.Results on Figure4 suggest that an increase in human recruitment and typhoid transmission by nonsymptomatic individuals has more influence on increasing the magnitude of ℛr, 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.Figure 4
Partial rank correlation coefficients showing the effects of parameter variation onℛr using ranges in Table 1. Parameters with positive PRCCs will increase ℛr when they are increased, whereas parameters with negative PRCCs will decrease ℛr when they are increased.The effects of increasingp (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 p 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 p does not give a more significant change as observed on Figure 5(a). We also note that an increase on p 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.Simulations of model system (1) showing the effects of increasing p (the proportion of individuals who become infectious carriers upon infection) on the cumulative new infections. The rest of parameters are fixed on their baseline values from Table 1, with the following initial conditions: S=10000, Ic=I=500, and Icr=Ir=150. It is worth noting that (a) and (b) are not reflecting the disease-free equilibrium; the convergence is simply due to scale.
(a)
(b)
(c)
(d)In Figure6, 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.Simulations of model system (1) showing the effects of different initial conditions (levels of typhoid outbreaks cases) on the prevalence of typhoid: (a) Ic=I=500, Icr=Ir=150, (b) Ic=I=50, Icr=Ir=150, and (c) Ic=I=50, Icr=Ir=50. The rest of parameters are fixed on their baseline values from Table 1. It is worth noting that (a) and (b) are not reflecting the disease-free equilibrium; the convergence is simply due to scale.
(a)
(b)
(c)Simulation in Figure7 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.Simulations of model system (1) showing the effects of treatment rate (γ) on cumulative typhoid cases. The following initial conditions were used: (a) Ic=I=500, Icr=Ir=150, (b) Ic=I=50, Icr=Ir=150, and (c) Ic=I=50, Icr=Ir=50, and the rest of parameters are fixed on their baseline values from Table 1.
(a)
(b)
(c)
(d)


## 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 ℛs and the model parameters which define it.Figure3 illustrates the PRCCs using ℛs 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 ℛs. We also noted that an increase on treatment level has the greatest influence on reducing the magnitude of ℛs; hence typhoid treatment should be encouraged in order to control cumulative typhoid cases.Figure 3
Partial rank correlation coefficients showing the effects of parameter variation onℛs using ranges in Table 1. Parameters with positive PRCCs will increase ℛs when they are increased, whereas parameters with negative PRCCs will decrease ℛs when they are increased.Results on Figure4 suggest that an increase in human recruitment and typhoid transmission by nonsymptomatic individuals has more influence on increasing the magnitude of ℛr, 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.Figure 4
Partial rank correlation coefficients showing the effects of parameter variation onℛr using ranges in Table 1. Parameters with positive PRCCs will increase ℛr when they are increased, whereas parameters with negative PRCCs will decrease ℛr when they are increased.The effects of increasingp (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 p 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 p does not give a more significant change as observed on Figure 5(a). We also note that an increase on p 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.Simulations of model system (1) showing the effects of increasing p (the proportion of individuals who become infectious carriers upon infection) on the cumulative new infections. The rest of parameters are fixed on their baseline values from Table 1, with the following initial conditions: S=10000, Ic=I=500, and Icr=Ir=150. It is worth noting that (a) and (b) are not reflecting the disease-free equilibrium; the convergence is simply due to scale.
(a)
(b)
(c)
(d)In Figure6, 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.Simulations of model system (1) showing the effects of different initial conditions (levels of typhoid outbreaks cases) on the prevalence of typhoid: (a) Ic=I=500, Icr=Ir=150, (b) Ic=I=50, Icr=Ir=150, and (c) Ic=I=50, Icr=Ir=50. The rest of parameters are fixed on their baseline values from Table 1. It is worth noting that (a) and (b) are not reflecting the disease-free equilibrium; the convergence is simply due to scale.
(a)
(b)
(c)Simulation in Figure7 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.Simulations of model system (1) showing the effects of treatment rate (γ) on cumulative typhoid cases. The following initial conditions were used: (a) Ic=I=500, Icr=Ir=150, (b) Ic=I=50, Icr=Ir=150, and (c) Ic=I=50, Icr=Ir=50, and the rest of parameters are fixed on their baseline values from Table 1.
(a)
(b)
(c)
(d)


## 5. Discussion

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 [3]. 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.



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*Source: 303645-2013-06-06.xml*