May  2019, 24(5): 2169-2188. doi: 10.3934/dcdsb.2019089

SIS criss-cross model of tuberculosis in heterogeneous population

1. 

Faculty of Mathematics and Computer Science, University of Warmia and Mazury in Olsztyn, Słoneczna 54, 10-710 Olsztyn, Poland

2. 

Institute of Applied Mathematics and Mechanics, Faculty of Mathematics, Informatics and Mechanics, University of Warsaw, Banacha 2, 02-097 Warsaw, Poland

* Corresponding author

Received  December 2017 Revised  January 2019 Published  May 2019 Early access  March 2019

In this paper we propose a model of tuberculosis (TB) transmission in a heterogeneous population consisting of two different subpopulations, like homeless and non-homeless people. We use the criss-cross model to describe the illness dynamics. This criss-cross model is based on the simple SIS model with constant inflow into both subpopulations and bilinear transmission function. We find conditions for the existence and local stability of stationary states (disease-free and endemic) and fit the model to epidemic data from Warmian-Masurian Province of Poland. Basic reproduction number $ \mathcal{R}_0 $ is considered as a threshold parameter for the general model. Applying local center manifold theory we show that when $ \mathcal{R}_0 = 1 $ a supercritical bifurcation occurs, and with $ \mathcal{R}_0 $ increasing above this threshold the disease-free stationary state loses stability and locally asymptotically stable endemic stationary state appears. Our analysis confirms the hypothesis that homeless individuals may be a specific reservoir of the pathogen and the disease may be transmitted from this subpopulation to the general population.

Citation: Mariusz Bodzioch, Marcin Choiński, Urszula Foryś. SIS criss-cross model of tuberculosis in heterogeneous population. Discrete and Continuous Dynamical Systems - B, 2019, 24 (5) : 2169-2188. doi: 10.3934/dcdsb.2019089
References:
[1]

T. Baxter, Low infectivity of tuberculosis, The Lancet, 342 (1993), 371. 

[2]

U. BeijerA. Wolf and S. Fazel, Prevalence of tuberculosis, hepatitis C virus, and HIV in homeless people: A systematic review and meta–analysis, Lancet Infectious Diseases, 12 (2012), 859-870.  doi: 10.1016/S1473–3099(12)70177–9.

[3]

S. Bowong and A. A. Alaoui, Optimal intervention strategies for tuberculosis, Communications in Nonlinear Science and Numerical Simulation, 18 (2013), 1441-1453.  doi: 10.1016/j.cnsns.2012.08.001.

[4]

T. F. Brewer and S. J. Heymann, To control and beyond: Moving towards eliminating the global tuberculosis threat, Journal of Epidemiology and Community Health, 58 (2004), 822-825.  doi: 10.1136/jech.2003.008664.

[5]

C. Castillo–Chavez and B. Song, Dynamical models of tuberculosis and their applications, Mathematical Biosciences and Engineering, 1 (2004), 361-404.  doi: 10.3934/mbe.2004.1.361.

[6]

M. Choiński, M. Bodzioch and U. Foryś, Analysis of a criss–cross model of tuberculosis for homeless and non–homeless subpopulations, Communications in Nonlinear Science and Numerical Simulation, (2018), under review.

[7]

Central statistical office of Poland, Statistical yearbooks, (2017), accessed 2018 April 30, http://stat.gov.pl/en/topics/statistical–yearbooks/

[8]

A. B. CurtisR. RidzonL. F. NovickJ. DriscollD. BlairM. OxtobyM. McGarryB. HiscoxC. FaulknerH. TaberS. Valway and I. M. Onorato, Analysis of Mycobacterium tuberculosis transmission patterns in a homeless shelter outbreak, International Journal of Tuberculosis and Lung Disease, 4 (2000), 308-313. 

[9]

O. DiekmannJ. A.P. Heesterbeek and J. A. J. Metz, On the definition and the computation of the basic reproduction ratio $R_0$ in models for infectious diseases in heterogeneous populations, Journal of Mathematical Biology, 28 (1990), 365-382.  doi: 10.1007/BF00178324.

[10]

K. Dietz, Models for vector–borne parasitic diseases, Lecture Notes in Biomathematics, 39 (1980), 264-277.  doi: 10.1007/978–3–642–93161–1_15.

[11]

P. van den Driessche and J. Watmough, Reproduction numbers and sub–threshold endemic equilibria for compartmental models of disease transmission, Mathematical Biosciences, 180 (2002), 29-48.  doi: 10.1016/S0025–5564(02)00108–6.

[12]

J. DushoffW. Huang and C. Castillo–Chavez, Backwards bifurcations and catastrophe in simple models of fatal diseases, Journal of Mathematical Biology, 36 (1998), 227-248.  doi: 10.1007/s002850050099.

[13]

H. W. Hethcote, An immunization model for a heterogeneous population, Theoretical Population Biology, 14 (1978), 338-349.  doi: 10.1016/0040–5809(78)90011–4.

[14]

H. W. Hethcote and J. W. van Ark, Epidemiological models for heterogeneous populations: Proportionate mixing, parameter estimation, and immunization programs, Mathematical Biosciences, 84 (1987), 85-118.  doi: 10.1016/0025–5564(87)90044–7.

[15]

H. HethcoteM. Zhien and L. Shengbing, Effects of quarantine in six endemic models for infectious diseases, Mathematical Biosciences, 180 (2002), 141-160.  doi: 10.1016/S0025–5564(02)00111–6.

[16]

S. A. Knopf, Tuberculosis as a cause and result of poverty, Journal of the American Medical Association, 63 (1914), 1720-1725.  doi: 10.1001/jama.1914.02570200014004.

[17]

A. Lajmanovich and J. A. Yorke, A deterministic model for gonorrhea in a nonhomogeneous population, Mathematical Biosciences, 28 (1976), 221-236.  doi: 10.1016/0025–5564(76)90125–5.

[18]

J. LukacsV. TubakJ. MesterS. DávidZ. BártfaiT. KubicaS. Niemann and A. Somoskövi, Conventional and molecular epidemiology of tuberculosis in homeless patients in Budapest, Hungary, Journal of Clinical Microbiology, 42 (2004), 5931-5934.  doi: 10.1128/JCM.42.12.5931–5934.2004.

[19]

Marshall office, Regional Center for Social Policy, Olsztyn, Poland, Information about homelessness, (2017), accessed 2018 April 30, http://warmia.mazury.pl/images/Departamenty/Regionalny_Osrodek_Polityki_Spolecznej/bezdomnosc-raport-2017/Bezdomno%C5%9B%C4%87__2016.doc

[20]

B. M. MurphyB. H. SingerS. Anderson and D. Kirschner, Comparing epidemic tuberculosis in demographically distinct heterogeneous populations., Mathematical Biosciences, 180 (2002), 161-185.  doi: 10.1016/S0025–5564(02)00133–5.

[21]

J. D. Murray, Mathematical Biology: Ⅰ. An Introduction, Springer, 2002.

[22]

C. OzcaglarA. ShabbeerS. L. VandenbergB. Yener and B. Bennet, Epidemiological models of Mycobacterium tuberculosis complex infections, Mathematical Biosciences, 236 (2012), 77-96.  doi: 10.1016/j.mbs.2012.02.003.

[23]

J. RomaszkoA. SiemaszkoM. BodziochA. Buciński and A. Doboszyńska, Active case finding among homeless people as a means of reducing the incidence of pulmonary tuberculosis in general population, Advances in Experimental Medicine and Biology, 911 (2016), 67-76.  doi: 10.1007/5584_2016_225.

[24]

S. P. N. SinghN. K. MehraH. B. DingleyJ. N. Pande and M. C. Vaidya, Human leukocyte antigen (HLA)–linked control of susceptibility to pulmonary tuberculosis and association with HLA–DR types, The Journal of Infectious Diseases, 148 (1983), 676-681.  doi: 10.1093/infdis/148.4.676.

[25]

J. Tan de BibianaC. RossiP. RivestA. ZwerlingL. ThibertF. McIntoshM. A. BehrD. Menzies and K. Schwartzman, Tuberculosis and homelessness in Montreal: A retrospective cohort study, BMC Public Health, 11 (2011), 833.  doi: 10.1186/1471–2458–11–833.

[26]

X. ZhouX. Shi and H. Cheng, Modelling and stability analysis for a tuberculosis model with healthy education and treatment, Computational and Applied Mathematics, 32 (2013), 245-260.  doi: 10.1007/s40314–013–0008–8.

show all references

References:
[1]

T. Baxter, Low infectivity of tuberculosis, The Lancet, 342 (1993), 371. 

[2]

U. BeijerA. Wolf and S. Fazel, Prevalence of tuberculosis, hepatitis C virus, and HIV in homeless people: A systematic review and meta–analysis, Lancet Infectious Diseases, 12 (2012), 859-870.  doi: 10.1016/S1473–3099(12)70177–9.

[3]

S. Bowong and A. A. Alaoui, Optimal intervention strategies for tuberculosis, Communications in Nonlinear Science and Numerical Simulation, 18 (2013), 1441-1453.  doi: 10.1016/j.cnsns.2012.08.001.

[4]

T. F. Brewer and S. J. Heymann, To control and beyond: Moving towards eliminating the global tuberculosis threat, Journal of Epidemiology and Community Health, 58 (2004), 822-825.  doi: 10.1136/jech.2003.008664.

[5]

C. Castillo–Chavez and B. Song, Dynamical models of tuberculosis and their applications, Mathematical Biosciences and Engineering, 1 (2004), 361-404.  doi: 10.3934/mbe.2004.1.361.

[6]

M. Choiński, M. Bodzioch and U. Foryś, Analysis of a criss–cross model of tuberculosis for homeless and non–homeless subpopulations, Communications in Nonlinear Science and Numerical Simulation, (2018), under review.

[7]

Central statistical office of Poland, Statistical yearbooks, (2017), accessed 2018 April 30, http://stat.gov.pl/en/topics/statistical–yearbooks/

[8]

A. B. CurtisR. RidzonL. F. NovickJ. DriscollD. BlairM. OxtobyM. McGarryB. HiscoxC. FaulknerH. TaberS. Valway and I. M. Onorato, Analysis of Mycobacterium tuberculosis transmission patterns in a homeless shelter outbreak, International Journal of Tuberculosis and Lung Disease, 4 (2000), 308-313. 

[9]

O. DiekmannJ. A.P. Heesterbeek and J. A. J. Metz, On the definition and the computation of the basic reproduction ratio $R_0$ in models for infectious diseases in heterogeneous populations, Journal of Mathematical Biology, 28 (1990), 365-382.  doi: 10.1007/BF00178324.

[10]

K. Dietz, Models for vector–borne parasitic diseases, Lecture Notes in Biomathematics, 39 (1980), 264-277.  doi: 10.1007/978–3–642–93161–1_15.

[11]

P. van den Driessche and J. Watmough, Reproduction numbers and sub–threshold endemic equilibria for compartmental models of disease transmission, Mathematical Biosciences, 180 (2002), 29-48.  doi: 10.1016/S0025–5564(02)00108–6.

[12]

J. DushoffW. Huang and C. Castillo–Chavez, Backwards bifurcations and catastrophe in simple models of fatal diseases, Journal of Mathematical Biology, 36 (1998), 227-248.  doi: 10.1007/s002850050099.

[13]

H. W. Hethcote, An immunization model for a heterogeneous population, Theoretical Population Biology, 14 (1978), 338-349.  doi: 10.1016/0040–5809(78)90011–4.

[14]

H. W. Hethcote and J. W. van Ark, Epidemiological models for heterogeneous populations: Proportionate mixing, parameter estimation, and immunization programs, Mathematical Biosciences, 84 (1987), 85-118.  doi: 10.1016/0025–5564(87)90044–7.

[15]

H. HethcoteM. Zhien and L. Shengbing, Effects of quarantine in six endemic models for infectious diseases, Mathematical Biosciences, 180 (2002), 141-160.  doi: 10.1016/S0025–5564(02)00111–6.

[16]

S. A. Knopf, Tuberculosis as a cause and result of poverty, Journal of the American Medical Association, 63 (1914), 1720-1725.  doi: 10.1001/jama.1914.02570200014004.

[17]

A. Lajmanovich and J. A. Yorke, A deterministic model for gonorrhea in a nonhomogeneous population, Mathematical Biosciences, 28 (1976), 221-236.  doi: 10.1016/0025–5564(76)90125–5.

[18]

J. LukacsV. TubakJ. MesterS. DávidZ. BártfaiT. KubicaS. Niemann and A. Somoskövi, Conventional and molecular epidemiology of tuberculosis in homeless patients in Budapest, Hungary, Journal of Clinical Microbiology, 42 (2004), 5931-5934.  doi: 10.1128/JCM.42.12.5931–5934.2004.

[19]

Marshall office, Regional Center for Social Policy, Olsztyn, Poland, Information about homelessness, (2017), accessed 2018 April 30, http://warmia.mazury.pl/images/Departamenty/Regionalny_Osrodek_Polityki_Spolecznej/bezdomnosc-raport-2017/Bezdomno%C5%9B%C4%87__2016.doc

[20]

B. M. MurphyB. H. SingerS. Anderson and D. Kirschner, Comparing epidemic tuberculosis in demographically distinct heterogeneous populations., Mathematical Biosciences, 180 (2002), 161-185.  doi: 10.1016/S0025–5564(02)00133–5.

[21]

J. D. Murray, Mathematical Biology: Ⅰ. An Introduction, Springer, 2002.

[22]

C. OzcaglarA. ShabbeerS. L. VandenbergB. Yener and B. Bennet, Epidemiological models of Mycobacterium tuberculosis complex infections, Mathematical Biosciences, 236 (2012), 77-96.  doi: 10.1016/j.mbs.2012.02.003.

[23]

J. RomaszkoA. SiemaszkoM. BodziochA. Buciński and A. Doboszyńska, Active case finding among homeless people as a means of reducing the incidence of pulmonary tuberculosis in general population, Advances in Experimental Medicine and Biology, 911 (2016), 67-76.  doi: 10.1007/5584_2016_225.

[24]

S. P. N. SinghN. K. MehraH. B. DingleyJ. N. Pande and M. C. Vaidya, Human leukocyte antigen (HLA)–linked control of susceptibility to pulmonary tuberculosis and association with HLA–DR types, The Journal of Infectious Diseases, 148 (1983), 676-681.  doi: 10.1093/infdis/148.4.676.

[25]

J. Tan de BibianaC. RossiP. RivestA. ZwerlingL. ThibertF. McIntoshM. A. BehrD. Menzies and K. Schwartzman, Tuberculosis and homelessness in Montreal: A retrospective cohort study, BMC Public Health, 11 (2011), 833.  doi: 10.1186/1471–2458–11–833.

[26]

X. ZhouX. Shi and H. Cheng, Modelling and stability analysis for a tuberculosis model with healthy education and treatment, Computational and Applied Mathematics, 32 (2013), 245-260.  doi: 10.1007/s40314–013–0008–8.

Figure 1.  Nullclines $C - xy + y - \mu x = 0$, $x - k = 0$ and phase portraits for system (2), for $\frac{C}{\mu} <k$ (A) and $\frac{C}{\mu}>k$ (B)
Figure 2.  Graphic representation of solutions of system (10) when (A) $C_1 >\mu_1 k_1$ and $C_2 >\mu_2 k_2$, (B) $C_1 <\mu_1 k_1$ and $C_2 >\mu_2 k_2$, (C) $C_1 > \mu_1 k_1$ and $C_2 <\mu_2 k_2$, (D) $C_1 = \mu_1 k_1$ and $C_2 = \mu_2 k_2$. Red curves represent the graph of the function $y_2(y_1)$, blue ones represent the graph of $y_1(y_2)$. Dotted lines bound the region defined by $\left(\xi_1,\tfrac{C_1}{\mu_1\kappa_1}\right) \times \left(\xi_2,\tfrac{C_2}{\mu_2\kappa_2}\right)$
Figure 3.  Graphic representation of solutions of system (10) for $C_i <\mu_i k_i$, $i = 1,\ 2$, if condition (11) holds (A) and does not hold (B). Red curves represent the graph of the function $y_2(y_1)$, blue ones represent the graph of $y_1(y_2)$. Red line is the tangent line to the function $y_2(y_1)$ at zero, blue one is the tangent line to $y_1(y_2)$ at zero. Dotted lines bound the region defined by $\left(\xi_1,\tfrac{C_1}{\mu_1\kappa_1}\right) \times \left(\xi_2,\tfrac{C_2}{\mu_2\kappa_2}\right)$
Figure 4.  Tuberculosis in the Warmian-Masurian province over the years 2001-2016 (number of infected non-homeless individuals). Comparison between the actual data and the model
Figure 5.  Phase portraits for system (5) in the phase planes $(S_1,I_1)$ (A) and $(S_2,I_2)$ (B) with fitted values of parameters summarized in Table 1
Figure 6.  Bifurcation diagram for system (5). The solid line depicts the graph of $\big(\mu_1(\gamma_1 + \alpha_1 + \mu_1)-C_1\beta_{11}\big)\big(\mu_2(\gamma_2 + \alpha_2 + \mu_2)-C_2\beta_{22}\big)- \beta_{12}\beta_{21}C_1C_2 = 0$. The dotted black lines depict $C_1 = \tfrac{\mu_1(\gamma_1 + \alpha_1 + \mu_1)}{\beta_{1}}$ and $C_2 = \tfrac{\mu_2(\gamma_2 + \alpha_2 + \mu_2)}{\beta_{2}}$. The red point depicts the point $(C_1,C_2)$ for the values of $C_1$ and $C_2$ taken from Table 1
Figure 7.  Bifurcation diagram for system (6) with $\beta_{1},\ \beta_{2}$ tending to zero. The solid black lines depict $C_1 = \mu_1 k_1$ and $C_2 = \mu_2 k_2$
Table 1.  Parameters for the model described by system (5)
Name Definition Value
$\alpha_1$, $\alpha_2$ Disease-related death rates $0.09$
$\gamma_1$, $\gamma_2$ Recovery coefficients $0.9$
$\mu_1$, $\mu_2$ Natural death rate $0.009$
$C_1$ Constant inflow of humans into the subpopulation of the non-homeless $11 000$
$C_2$ Constant inflow of humans into the subpopulation of the homeless $60$
$\beta_{11}$ Transmission coefficient $0.57566\cdot 10^{-6}$ (estimated)
$\beta_{12}$ Transmission coefficient $11.276\cdot 10^{-6}$ (estimated)
$\beta_{21}$ Transmission coefficient $3.7679\cdot 10^{-6}$ (estimated)
$\beta_{22}$ Transmission coefficient $98.249\cdot 10^{-6}$ (estimated)
Name Definition Value
$\alpha_1$, $\alpha_2$ Disease-related death rates $0.09$
$\gamma_1$, $\gamma_2$ Recovery coefficients $0.9$
$\mu_1$, $\mu_2$ Natural death rate $0.009$
$C_1$ Constant inflow of humans into the subpopulation of the non-homeless $11 000$
$C_2$ Constant inflow of humans into the subpopulation of the homeless $60$
$\beta_{11}$ Transmission coefficient $0.57566\cdot 10^{-6}$ (estimated)
$\beta_{12}$ Transmission coefficient $11.276\cdot 10^{-6}$ (estimated)
$\beta_{21}$ Transmission coefficient $3.7679\cdot 10^{-6}$ (estimated)
$\beta_{22}$ Transmission coefficient $98.249\cdot 10^{-6}$ (estimated)
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