Progression from latent infection to active disease in dynamic tuberculosis transmission models: a systematic review of the validity of modelling assumptions

Mathematical modelling is commonly used to evaluate infectious disease control policy and is influential in shaping policy and budgets. Mathematical models necessarily make assumptions about disease natural history and, if these assumptions are not valid, the results of these studies can be biased. We did a systematic review of published tuberculosis transmission models to assess the validity of assumptions about progression to active disease after initial infection (PROSPERO ID CRD42016030009). We searched PubMed, Web of Science, Embase, Biosis, and Cochrane Library, and included studies from the earliest available date (Jan 1, 1962) to Aug 31, 2017. We identified 312 studies that met inclusion criteria. Predicted tuberculosis incidence varied widely across studies for each risk factor investigated. For population groups with no individual risk factors, annual incidence varied by several orders of magnitude, and 20-year cumulative incidence ranged from close to 0% to 100%. A substantial proportion of modelled results were inconsistent with empirical evidence: for 10-year cumulative incidence, 40% of modelled results were more than double or less than half the empirical estimates. These results demonstrate substantial disagreement between modelling studies on a central feature of tuberculosis natural history. Greater attention to reproducing known features of epidemiology would strengthen future tuberculosis modelling studies, and readers of modelling studies are recommended to assess how well those studies demonstrate their validity

Structured parametric epidemic models

A stage-structured model for a theoretical epidemic process that incorporates immature, susceptible and infectious individuals in independent stages is formulated. In this analysis, an input interpreted as a birth function is considered. The structural identifiability is studied using the Markov parameters. Then, the unknown parameters are uniquely determined by the output structure corresponding to an observation of infection. Two different birth functions are considered: the linear case and the Beverton-Holt type to analyse the structured epidemic model. Some conditions on the parameters to obtain non-zero disease-free equilibrium points are given. The identifiability of the parameters allows us to determine uniquely the basic reproduction number Script capital R-0 and the stability of the model in the equilibrium is studied using Script capital R-0 in terms of the model parameters.This work has been partially supported by MTM2010-18228. The authors wish to express their thanks to the reviewers for helpful comments and suggestions.Cantó Colomina, B.; Coll, C.; Sánchez, E. (2014). Structured parametric epidemic models. International Journal of Computer Mathematics. 91(2):188-197. https://doi.org/10.1080/00207160.2013.800864188197912Allen, L. J. S., & Thrasher, D. B. (1998). The effects of vaccination in an age-dependent model for varicella and herpes zoster. IEEE Transactions on Automatic Control, 43(6), 779-789. doi:10.1109/9.679018Ben-Zvi, A., McLellan, P. J., & McAuley, K. B. (2004). Identifiability of Linear Time-Invariant Differential-Algebraic Systems. 2. The Differential-Algebraic Approach. Industrial & Engineering Chemistry Research, 43(5), 1251-1259. doi:10.1021/ie030534jBoyadjiev, C., & Dimitrova, E. (2005). An iterative method for model parameter identification. Computers & Chemical Engineering, 29(5), 941-948. doi:10.1016/j.compchemeng.2004.08.036Cantó, B., Coll, C., & Sánchez, E. (2011). Identifiability for a Class of Discretized Linear Partial Differential Algebraic Equations. Mathematical Problems in Engineering, 2011, 1-12. doi:10.1155/2011/510519Cao, H., & Zhou, Y. (2012). The discrete age-structured SEIT model with application to tuberculosis transmission in China. Mathematical and Computer Modelling, 55(3-4), 385-395. doi:10.1016/j.mcm.2011.08.017Diekmann, O., Heesterbeek, J. A. P., & Metz, J. A. J. (1990). 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(4). doi:10.1007/bf00178324Dion, J.-M., Commault, C., & van der Woude, J. (2003). Generic properties and control of linear structured systems: a survey. Automatica, 39(7), 1125-1144. doi:10.1016/s0005-1098(03)00104-3Emmert, K. E., & Allen, L. J. S. (2004). Population Persistence and Extinction in a Discrete-time, Stage-structured Epidemic Model. Journal of Difference Equations and Applications, 10(13-15), 1177-1199. doi:10.1080/10236190410001654151Farina, L., & Rinaldi, S. (2000). Positive Linear Systems. doi:10.1002/9781118033029Van den Hof, J. M. (1998). Structural identifiability of linear compartmental systems. IEEE Transactions on Automatic Control, 43(6), 800-818. doi:10.1109/9.679020T. Kailath,Linear Systems, Prentice-Hall, Englewood Cliffs, NJ, 1980.Li, C.-K., & Schneider, H. (2002). Applications of Perron-Frobenius theory to population dynamics. Journal of Mathematical Biology, 44(5), 450-462. doi:10.1007/s002850100132Li, X., & Wang, W. (2005). A discrete epidemic model with stage structure☆. Chaos, Solitons & Fractals, 26(3), 947-958. doi:10.1016/j.chaos.2005.01.063Ma, J., & Earn, D. J. D. (2006). Generality of the Final Size Formula for an Epidemic of a Newly Invading Infectious Disease. Bulletin of Mathematical Biology, 68(3), 679-702. doi:10.1007/s11538-005-9047-7Wang, W., & Zhao, X.-Q. (2004). An epidemic model in a patchy environment. Mathematical Biosciences, 190(1), 97-112. doi:10.1016/j.mbs.2002.11.00

A Mathematical Model to Control the Prevalence of a Directly and Indirectly Transmitted Disease

[EN] In this paper, a mathematical model to describe the spread of an infectious disease on a farm is developed. To analyze the evolution of the infection, the direct transmission from infected individuals and the indirect transmission from the bacteria accumulated in the enclosure are considered.A threshold value of population is obtained to assure the extinction of the disease. When this size of population is exceeded, two control procedures to apply at each time are proposed. For each of them, a maximum number of steps without control and reducing the prevalence of disease is obtained. In addition, a criterion to choose between both procedures is established. Finally, the results are numerically simulated for a hypothetical outbreak on a farm.Cantó Colomina, B.; Coll, C.; Pagán Moreno, MJ.; Poveda, J.; Sánchez, E. (2021). A Mathematical Model to Control the Prevalence of a Directly and Indirectly Transmitted Disease. Mathematics. 9(20):1-15. https://doi.org/10.3390/math9202562S11592

A study on vaccination models for a seasonal epidemic process

In this paper seasonal epidemiological processes are considered and a strategy of periodic vaccination is proposed. The invariant formulations associated with an N-periodic system and the reproduction numbers associated with them are considered. A collection of measures to study the stability of the system is introduced. Moreover, the collection of s-basic reproduction number at time j help us to establish conditions on the periodic vaccination rates in the vaccination program. Finally, an SIR model is showed and a comparison between the results obtained using constant or periodic vaccination program is analyzed. (C) 2014 Elsevier Inc. All rights reserved.The authors wish to express their thanks to the reviewers for helpful comments and suggestions. This paper is supported by Grant MTM2010-18228.Cantó Colomina, B.; Coll, C.; Sánchez, E. (2014). A study on vaccination models for a seasonal epidemic process. Applied Mathematics and Computation. 243:152-160. https://doi.org/10.1016/j.amc.2014.05.104S15216024

Modelling and controlling infectious diseases

The financial support by IDRC has made it much easier to put together network activities involving scientists in both countries, a special example is the large presence of the Chinese students in the 2012 Summer School on Mathematics for Public Health the Canadian group organized in Edmonton in May of 2012.Infectious disease control is a major challenge in China due to China’s fast growing economy, changing social networks and evolving health service infrastructures. The success of disease control in China has a profound impact beyond its borders. In support of better disease control, this five year research program was designed to enhance China’s national capacity for analyzing, modeling and predicting transmission dynamics of infectious diseases through joint research, training young scientists, and building collaborative relationships. This successful program was led by the National Center for AIDS/STD Control and Prevention (Chinese Centre for Disease Control and Prevention, China) and the Centre for Disease Modeling (York University, Canada), and involved a number of Canadian and Chinese universities in various areas of infectious disease modelling and control. The bilateral collaboration also trained numerous highly qualified personnel and built a network for sustaining collaboration. This capacity building was facilitated by joint projects and bilateral annual meetings in major cities in China and Canada. The research activities on modeling major public health threats of infectious diseases focused on major diseases in China and/or issues of global public health concern including HIV transmission and prevention among high risk population, HIV treatment and drug resistance, influenza, schistosomiasis, mutation and stemma of SIV and HIV, latent and active tuberculosis infection, HBV control and vaccination. The outputs of the project were reported through peer-reviewed publications and modelling– based and science-informed public policy recommendations

On stability and reachability of perturbed positive systems

This paper deals mainly with the structural properties of positive reachability and stability. We focus our attention on positive discrete-time systems and analyze the behavior of these systems subject to some perturbation. The effects of permutation and similar transformations are discussed in order to determine the structure of the perturbation such that the closed-loop system is positively reachable and stable. Finally, the results are applied to Leslie’s population model. The structure of the perturbation is shown such that the properties of the original system remain and an explicit expression of its set of positively reachable populations is given.The authors would like to thank the referee and the associate editor for their very helpful suggestions. This work has been partially supported by Spanish Grant MTM2013 43678 P.Cantó Colomina, B.; Coll, C.; Sánchez, E. (2014). On stability and reachability of perturbed positive systems. Advances in Difference Equations. 296(1):1-11. https://doi.org/10.1186/1687-1847-2014-296S1112961Cantó B, Coll C, Sánchez E: Parameter identification of a class of economical models. Discrete Dyn. Nat. Soc. 2010., 2010: Article ID 408346Cao H, Zhou Y: The discrete age-structured SEIT model with application to tuberculosis transmission in China. Math. Comput. Model. 2012, 55: 385-395. 10.1016/j.mcm.2011.08.017Coll C, Herrero A, Sánchez E, Thome N: A dynamic model for a study of diabetes. Math. Comput. Model. 2009, 50: 713-716. 10.1016/j.mcm.2008.12.027Emmert HE, Allen LSJ: Population persistence and extinction in a discrete-time, stage-structured epidemic model. J. Differ. Equ. Appl. 2004, 10: 1177-1199. 10.1080/10236190410001654151Li CK, Schneider H: Applications of Perron-Frobenius theory to population dynamics. J. Math. Biol. 2002, 44: 450-462. 10.1007/s002850100132Li X, Wang W: A discrete epidemic model with stage structure. Chaos Solitons Fractals 2006, 26: 947-958.De la Sen M, Alonso-Quesada S: Some equilibrium, stability, instability and oscillatory results for an extended discrete epidemic model with evolution memory. Adv. Differ. Equ. 2013., 2013: Article ID 234Caccetta L, Rumchev VG: A survey of reachability and controllability for positive linear systems. Ann. Oper. Res. 2000, 98: 101-122. 10.1023/A:1019244121533Berman A, Plemons RJ: Nonnegative Matrices in Mathematical Science. SIAM, Philadelphia; 1994.Diblík J, Khusainov D, Ruzicková M: Controllability of linear discrete systems with constant coefficients and pure delay. SIAM J. Control Optim. 2008, 47: 1140-1149. 10.1137/070689085Diblík J, Feckan M, Pospísil M: On the new control functions for linear discrete delay systems. SIAM J. Control Optim. 2014, 52: 1745-1760. 10.1137/140953654Bru R, Romero S, Sánchez E: Canonical forms for positive discrete-time linear systems. Linear Algebra Appl. 2000, 310: 49-71. 10.1016/S0024-3795(00)00044-6Farina L, Rinaldi S: Positive Linear Systems. Wiley, New York; 2000.Bru R, Coll C, Romero S, Sánchez E: Reachability indices of positive linear systems. Electron. J. Linear Algebra 2004, 11: 88-102.Kajin M, Almeida PJAL, Vieira MV, Cerqueira R: The state of the art of population projection models: from the Leslie matrix to evolutionary demography. Oecol. Aust. 2012, 16(1):13-22. 10.4257/oeco.2012.1601.02Leslie PH: Some further notes on the use of matrices in population mathematics. Biometrika 1948, 35: 213-245. 10.1093/biomet/35.3-4.213Muratori S, Rinaldi S: Equilibria, stability and reachability of Leslie systems with nonnegative inputs. IEEE Trans. Autom. Control 1990, 35: 1065-1068. 10.1109/9.58539Caswell H: Matrix Population Models: Construction, Analysis and Interpretation. Sinauer, Sunderland; 2001