Skip to main content

A dynamical model for HIV-typhoid co-infection with typhoid vaccine


Individuals living with HIV/AIDS are significantly at higher risk of infection with Salmonella Typhi. A deterministic nonlinear mathematical model that describes the co-infection dynamics of HIV and typhoid incorporating typhoid vaccine and treatment has been developed. The basic reproduction number for the co-infection model is computed. The co-infection model exhibits four steady states, namely, the disease-free, HIV alone endemic, typhoid alone endemic, and the co-infection endemic states. The local stability of the disease-free state has been investigated. The co-infection endemic state, if it exists, is found to be locally stable. The minimum vaccination rate that eliminates the typhoid infection is determined. Sensitivity analysis has been performed to ascertain model parameters that have a strong impact on the disease dynamics. It is demonstrated that the co-infection basic reproduction number can be reduced to below unity by simultaneous preventive measures, thereby eliminating both diseases. Numerical simulation of the co-infection model is carried out to examine the effects of parameters on disease spread. The study suggests that the two diseases need to be managed simultaneously for effective control of the co-infection.

This is a preview of subscription content, access via your institution.

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8


  1. Agwu, E., Ihongbe, J., Okogun, G., Inyang, N.: High incidence of co-infection with malaria and typhoid in febrile hiv infected and aids patients in ekpoma, edo state, nigeria. Braz. J. Microbiol. 40(2), 329–332 (2009)

    Article  Google Scholar 

  2. Angulo, F.J., Swerdlow, D.L.: Bacterial enteric infections in persons infected with human immunodeficiency virus. Clin. Infect. Dis. 21(Supplement–1), S84–S93 (1995)

    Article  Google Scholar 

  3. Antillón, M., Warren, J.L., Crawford, F.W., Weinberger, D.M., Kürüm, E., Pak, G.D., Marks, F., Pitzer, V.E.: The burden of typhoid fever in low-and middle-income countries: A meta-regression approach. PLoS Negl. Tropical Dis. 11(2), e0005376 (2017)

    Article  Google Scholar 

  4. Baker, S., Holt, K.E., Clements, A.C., Karkey, A., Arjyal, A., Boni, M.F., Dongol, S., Hammond, N., Koirala, S., Duy, P.T., et al.: Combined high-resolution genotyping and geospatial analysis reveals modes of endemic urban typhoid fever transmission. Open Biol. 1(2), 110008 (2011)

    Article  Google Scholar 

  5. Brachman, P.S., Abrutyn, E.: Bacterial infections of humans: epidemiology and control. Springer, New York (2009)

    Book  Google Scholar 

  6. Butler, T.: Treatment of typhoid fever in the 21st century: promises and shortcomings. Clin. Microbiol. Infect. 17(7), 959–963 (2011)

    Article  Google Scholar 

  7. Carvalho, A.R., Pinto, C.M.: A coinfection model for hiv and hcv. Biosystems 124, 46–60 (2014)

    Article  Google Scholar 

  8. Castillo-Chavez, C., Blower, S., van den Driessche, P., Kirschner, D., Yakubu, A.A.: Mathematical approaches for emerging and reemerging infectious diseases: an introduction. Springer, New York (2002)

    Book  Google Scholar 

  9. Castillo-Chavez, C., Song, B.: Dynamical models of tuberculosis and their applications. Math. Biosci. Eng. 1(2), 361–404 (2004)

    Article  MathSciNet  Google Scholar 

  10. Celum, C.L., Chaisson, R.E., Rutherford, G.W., Barnhart, J.L., Echenberg, D.F.: Incidence of salmonellosis in patients with aids. J. Infect. Dis. 156, 998–1002 (1987)

    Article  Google Scholar 

  11. Chitnis, N., Hyman, J.M., Cushing, J.M.: sensitivity analysis of a mathematical model. Bull. Math. Biol. 70(5), 1272 (2008)

    Article  MathSciNet  Google Scholar 

  12. Cooper, C.: Global, regional, and national incidence, prevalence, and years lived with disability for 354 diseases and injuries for 195 countries and territories, 1990–2017: a systematic analysis for the global burden of disease study 2017. The Lancet 392(10159), 1789–1858 (2018)

    Article  Google Scholar 

  13. Crump, J.A.: Progress in typhoid fever epidemiology. Clin. Infect. Dis. 68(Supplement–1), S4–S9 (2019)

    Article  Google Scholar 

  14. Van den Driessche, P., Watmough, J.: Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission. Math. Biosci 180(1–2), 29–48 (2002)

    Article  MathSciNet  Google Scholar 

  15. Edward, S., Nyerere, N.: Modelling typhoid fever with education, vaccination and treatment. Eng. Math. 1(1), 44–52 (2016)

    Google Scholar 

  16. Edward, S., et al.: A deterministic mathematical model for direct and indirect transmission dynamics of typhoid fever. Open Access Libr. J. 4(05), 1 (2017)

    Google Scholar 

  17. Fischl, M.A., Dickinson, G.M., Sinave, C., Pitchenik, A.E., Cleary, T.J.: Salmonella bacteremia as manifestation of acquired immunodeficiency syndrome. Arch. Internal Med. 146(1), 113–115 (1986)

    Article  Google Scholar 

  18. González-Guzmán, J.: An epidemiological model for direct and indirect transmission of typhoid fever. Math. Biosci. 96(1), 33–46 (1989)

    Article  MathSciNet  Google Scholar 

  19. Gotuzzo, E., Frisancho, O., Sanchez, J., Liendo, G., Carrillo, C., Black, R.E., Morris, J.G.: Association between the acquired immunodeficiency syndrome and infection with salmonella typhi or salmonella paratyphi in an endemic typhoid area. Arch. Internal Med. 151(2), 381–382 (1991)

    Article  Google Scholar 

  20. Kalra, S., Naithani, N., Mehta, S., Swamy, A.: Current trends in the management of typhoid fever. Med. J. Armed Forc. India 59(2), 130 (2003)

    Article  Google Scholar 

  21. Kaur, N., Ghosh, M., Bhatia, S.: Modelling the role of awareness and screening of infectives in the transmission dynamics of hiv. World J Modelling and Simul. 12(2), 97–111 (2016)

    Google Scholar 

  22. Kgosimore, M., Kelatlhegile, G.: Mathematical analysis of typhoid infection with treatment. J. Math. Sci. Adv. Appl 40, 75–91 (2016)

    Article  Google Scholar 

  23. Kroon, F.P., van Dissel, J.T., Ravensbergen, E., Nibbering, P.H., van Furth, R.: Impaired antibody response after immunization of hiv-infected individuals with the polysaccharide vaccine against salmonella typhi (typhim-vi®). Vaccine 17(23–24), 2941–2945 (1999)

    Article  Google Scholar 

  24. LaSalle, J.P.: The stability of dynamical systems. Siam 21(3), 418–420 (1976)

    Google Scholar 

  25. Ma, S., Xia, Y.: Mathematical understanding of infectious disease dynamics. World Scientific, (2009)

  26. Marchello, C.S., Hong, C.Y., Crump, J.A.: Global typhoid fever incidence: A systematic review and meta-analysis. Clin. Infect. Dis. 68(Supplement–2), S105–S116 (2019)

    Article  Google Scholar 

  27. Mathews, J.H., Fink, K.D., et al.: Numer. Methods MATLAB. Pearson prentice hall Upper Saddle River, NJ (2004)

    Google Scholar 

  28. Mogasale, V., Maskery, B., Ochiai, R.L., Lee, J.S., Mogasale, V.V., Ramani, E., Kim, Y.E., Park, J.K., Wierzba, T.F.: Burden of typhoid fever in low-income and middle-income countries: a systematic, literature-based update with risk-factor adjustment. Lancet Global Health 2(10), e570–e580 (2014)

    Article  Google Scholar 

  29. Mushanyu, J., Nyabadza, F., Muchatibaya, G., Mafuta, P., Nhawu, G.: Assessing the potential impact of limited public health resources on the spread and control of typhoid. J Math. Biol. 77(3), 647–670 (2018)

    Article  MathSciNet  Google Scholar 

  30. Mushayabasa, S.: Modeling the impact of optimal screening on typhoid dynamics. Int. J. Dyn. and Control 4(3), 330–338 (2016)

    Article  MathSciNet  Google Scholar 

  31. Mushayabasa, S., Bhunu, C.P., Mhlanga, N.A.: Modeling the transmission dynamics of typhoid in malaria endemic settings. Applications & Applied Mathematics 9(1), (2014)

  32. Mutua, J.M., Wang, F.B., Vaidya, N.K.: Modeling malaria and typhoid fever co-infection dynamics. Math. Biosci. 264, 128–144 (2015)

    Article  MathSciNet  Google Scholar 

  33. Nthiiri, J., Lawi, G., Akinyi, C., Oganga, D., Muriuki, W., Musyoka, M., Otieno, P., Koech, L.: Mathematical modelling of typhoid fever disease incorporating protection against infection. Journal of Advances in Mathematics and Computer Science pp. 1–10 (2016)

  34. Pitzer, V.E., Bowles, C.C., Baker, S., Kang, G., Balaji, V., Farrar, J.J., Grenfell, B.T.: Predicting the impact of vaccination on the transmission dynamics of typhoid in south asia: a mathematical modeling study. PLoS Negl. Trop. Dis. 8(1), e2642 (2014)

    Article  Google Scholar 

  35. Rizvi, F.: Mathematical modeling of two-dose vaccines. Ph.D. thesis, The Ohio State University (2016)

  36. Saha, S., Samanta, G.: Modelling and optimal control of hiv/aids prevention through prep and limited treatment. Phys. A: Stat. Mech. Appl. 516, 280–307 (2019)

    Article  MathSciNet  Google Scholar 

  37. Sperber, S.J., Schleupner, C.J.: Salmonellosis during infection with human immunodeficiency virus. Rev. Infect. Dis. 9(5), 925–934 (1987)

    Article  Google Scholar 

  38. Tian, J.P., Wang, J.: Global stability for cholera epidemic models. Math. Biosci. 232(1), 31–41 (2011)

    Article  MathSciNet  Google Scholar 

  39. Tilahun, G.T., Makinde, O.D., Malonza, D.: Modelling and optimal control of typhoid fever disease with cost-effective strategies. Comput. Math. Methods Med. 2017, 1–16 (2017)

    Article  MathSciNet  Google Scholar 

  40. Tilahun, G.T., Makinde, O.D., Malonza, D.: Co-dynamics of pneumonia and typhoid fever diseases with cost effective optimal control analysis. Appl. Math. Comput. 316, 438–459 (2018)

    MathSciNet  MATH  Google Scholar 

  41. UNAIDS: Global hiv & aids statistics-2019 fact sheet (2019)

Download references


The first author is grateful to the Indian Council for Cultural Relations (ICCR) for financial support during his Ph.D. work. The authors are thankful to the anonymous reviewers for their valuable comments and suggestions, which helped us to improve the quality of our original manuscript.

Author information

Authors and Affiliations


Corresponding author

Correspondence to Tsegaye Kebede Irena.

Ethics declarations

Conflict of interest

The authors declare that they have no conflict of interest.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.


Proof of Theorem 1


The right hand side of the co-infection model system (2.2) is continuous and satisfies locally Lipschitz condition on the space of continuous functions. Thus, its solution \((S(t), I_H(t), I_T(t), I_{HT}(t), I_A(t), I_{AT}(t), R(t), B(t))\) exists and is unique over [0, T), where \(0<T\le \infty \).

Let us first establish that \(S(t)>0\), \(\forall t\in [0,T)\). Suppose this statement is false. Then \(\exists t_1\in [0,T)\) such that \(S(t_1)=0\), \(\frac{dS(t_1)}{dt}\le 0\) and \(S(t)>0\), \(\forall t\in [0,t_1)\). Then there must be \(I_H(t)>0\), \(I_T(t)>0\) and \(R(t)>0\), \(\forall t\in [0,t_1)\). Suppose not. Then \(\exists t_2\in [0,t_1)\) such that

$$\begin{aligned}&(i)~~I_H(t_2)=0,~ \frac{dI_H(t_2)}{dt}\le 0 ~~and~~ I_H(t)>0, \forall t\in [0,t_2) \nonumber \\&(ii)~~I_T(t_2)=0, ~\frac{dI_T(t_2)}{dt}\le 0 ~~and~~ I_T(t)>0, \forall t\in [0,t_2) \nonumber \\&(iii)~~R(t_2)=0, ~\frac{dR(t_2)}{dt}\le 0 ~~and~~ R(t)>0, \forall t\in [0,t_2). \end{aligned}$$

Then there must have \(I_{HT}(t)>0\), \(I_A(t)>0\) and \(B(t)>0\), \(\forall t\in [0,t_2)\). Suppose this is not true. Then \(\exists t_3\in [0,t_2)\) such that

$$\begin{aligned}&(i)~~I_{HT}(t_3)=0,~ \frac{dI_{HT}(t_3)}{dt}\le 0~~ and~~I_{HT}(t)>0, \forall t\in [0,t_3)\nonumber \\&(ii)~~I_A(t_3)=0,~ \frac{dI_A(t_3)}{dt}\le 0~~ and~~ I_A( t)>0, \forall t\in [0,t_3) \nonumber \\&(iii)~~B(t_3)=0,~ \frac{dB(t_3)}{dt}\le 0 ~~and~~ B(t)>0, \forall t\in [0,t_3). \end{aligned}$$

Our claim is \(I_{AT}(t)>0\), \(\forall t\in [0,t_3)\). If this is not true, \(\exists t_4\in [0,t_3)\) such that \(I_{AT}(t_4)=0\), \(\frac{dI_{AT}(t_4)}{dt}\le 0\) and \(I_{AT}( t)>0\), \(\forall t\in [0,t_4)\). From the sixth equation of system (2.2)

$$\begin{aligned} \frac{dI_{AT}(t_4)}{dt}=\phi I_{HT}(t_4)+\epsilon \lambda _T(t_4)I_A(t_4)>0 \end{aligned}$$

which contradicts \(\frac{dI_{AT}(t_4)}{dt}\le 0\). Therefore, \(I_{AT}(t)>0\), \(\forall t\in [0,t_3)\).

So the fourth, fifth, and eighth equations of system (2.2) give

$$\begin{aligned} \frac{dI_{HT}(t_3)}{dt}= & {} \delta _2I_{AT}(t_3)>0,~~\frac{dI_A(t_3)}{dt}=\phi I_H(t_3)+\tau _3I_{AT}(t_3)>0,\\ \frac{dB(t_3)}{dt}= & {} \alpha _1I_T(t_3)+\alpha _3I_{AT}(t_3)>0 \end{aligned}$$

which contradicts \(\frac{dI_{HT}(t_3)}{dt}\le 0\), \(\frac{dI_A(t_3)}{dt}\le 0\) and \(\frac{dB(t_3)}{dt}\le 0\), respectively (See Eq. (A.2)). Hence, \(I_{HT}(t)>0\), \(I_A(t)>0\) and \(B(t)>0\), \(\forall t\in [0,t_2)\). Similarly, \(I_{AT}>0\), \(\forall t\in [0,t_2)\).

Now from the second, third, and seventh equations of system (2.2)

$$\begin{aligned} \frac{dI_H(t_2)}{dt}= & {} \lambda _H(t_2)S(t_2)+(\tau _2+(1-p)\xi )I_{HT}(t_2)+\delta _1I_A(t_2)>0,\\ \frac{dI_T(t_2)}{dt}= & {} \lambda _T(t_2)S(t_2)>0,~~\frac{dR(t_2)}{dt}=\theta S(t_2)>0 \end{aligned}$$

which is a contradiction to \(\frac{dI_H(t_2)}{dt}\le 0\), \(\frac{dI_T(t_2)}{dt}\le 0\) and \(\frac{dR(t_2)}{dt}\le 0\), respectively (See Eq. (A.1)). Hence, \(I_H(t)>0\), \(I_T(t)>0\) and \(R(t)>0\), \(\forall t\in [0,t_1)\). This implies that \(I_{HT}>0\), \(I_A(t)>0\), \(I_{AT}>0\) and \(B(t)>0\), \(\forall t\in [0,t_1)\). It follows from the first equation of system (2.2)

$$\begin{aligned} \frac{dS(t_1)}{dt}=\varLambda +\omega R(t_1)>0 \end{aligned}$$

which contradicts \(\frac{dS(t_1)}{dt}\le 0\). Hence, \(S(t)>0\), \(\forall t\in [0,T)\).

The above step-by-step discussion follows that \(I_H(t)>0\), \(I_T(t)>0\), \(I_{HT}(t)>0\), \(I_A(t)>0\), \(I_{AT}(t)>0\), \(R(t)>0\) and \(B(t)>0\) for all \(t\in [0,T)\). Thus, the theorem is proved. \(\square \)

Proof of Theorem 2


The dynamics of the total human population is

$$\begin{aligned} \frac{dN}{dt}= & {} \varLambda +(1-p)\xi (I_H+I_{HT})-\mu N-d_T(I_T+I_{AT})-d_{HT} I_{HT}-d_A(I_A+I_{AT}).\\&\Rightarrow ~~~\frac{dN}{dt}\le \varLambda -(\mu -(1-p)\xi )N. \end{aligned}$$


$$\begin{aligned} 0\le N(t)\le \frac{\varLambda }{\mu -(1-p)\xi }+\left( N(0)-\frac{\varLambda }{\mu -(1-p)\xi }\right) e^{-(\mu -(1-p)\xi )t}. \end{aligned}$$


$$\begin{aligned} 0\le N(t)\le \frac{\varLambda }{\mu -(1-p)\xi }~~as ~~t\rightarrow \infty . \end{aligned}$$

Thus, the total human population N(t) is bounded. Hence each of its coordinates \((S(t), I_H(t), I_T(t), I_{HT}(t), I_A(t), I_{AT}(t), R(t))\) is bounded.

Further, the dynamics of Salmonella Typhi bacteria in the environment is given by

$$\begin{aligned} \frac{dB}{dt}= & {} \alpha _1 I_T+\alpha _2 I_{HT}+\alpha _3 I_{AT}-\bar{\mu }B.\\&\Rightarrow ~~~\frac{dB}{dt}+\bar{\mu }B \le (\alpha _1+\alpha _2+\alpha _3)\frac{\varLambda }{\mu -(1-p)\xi }. \end{aligned}$$

This gives

$$\begin{aligned} 0\le B(t)\le \frac{\varLambda \left( \alpha _1+\alpha _2+\alpha _3\right) }{\bar{\mu }(\mu -(1-p)\xi )}+\left( B(0)-\frac{\varLambda \left( \alpha _1+\alpha _2+\alpha _3\right) }{\bar{\mu }(\mu -(1-p)\xi )}\right) e^{-\bar{\mu }t}. \end{aligned}$$


$$\begin{aligned} 0\le B(t)\le \frac{\varLambda \left( \alpha _1+\alpha _2+\alpha _3\right) }{\bar{\mu }(\mu -(1-p)\xi )}~~as ~~t\rightarrow \infty . \end{aligned}$$

Hence, B(t) is also bounded.

Therefore, every solution of the model system (2.2) with initial conditions in \(\mathbb {R}_+^{8}\) will enter and remains in region

$$\begin{aligned} \varOmega= & {} \left\{ (S,I_H,I_T,I_{HT},I_A,I_{AT},R,B)\in \mathbb {R}_+^8:0\le N(t)\le \frac{\varLambda }{Q};0\le B(t)\le \frac{\varLambda \left( \alpha _1+\alpha _2+\alpha _3\right) }{\bar{\mu }Q}\right\} ; \\ Q= & {} \mu -(1-p)\xi>0 \quad and \quad \bar{\mu }=\mu _b-r>0. \end{aligned}$$

\(\square \)

Rights and permissions

Reprints and Permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Irena, T.K., Gakkhar, S. A dynamical model for HIV-typhoid co-infection with typhoid vaccine. J. Appl. Math. Comput. 67, 641–670 (2021).

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: