Asymptotic Theory of an Infectious Disease Model

Abstract

In this paper, we present asymptotic theory as a viable alternative solution method for infectious disease models. We consider a particular model of a pathogen attacking a host whose immune system responds defensively, that has been studied previously [Mohtashemi and Levins in J. Math. Biol. 43: 446-470 (2001)]. On rendering this model dimensionless, we can reduce the number of parameters to two and note that one of them has a large value that suggests an asymptotic analysis. On doing this analysis, we obtain a satisfying qualitative description of the dynamic evolution of each population, together with simple analytic expressions for their main features, from which we can compute accurate quantitative values.

On The Stability Analysis of The General Mathematical Modeling of Bacterial Infection

“…In addition that, the most common form of resistance acquisition to antibiotic is the conjugation including the transfer of genes between susceptible and resistant bacteria [13,14]. Since this transfer occurs between adjacent bacteria in a well mixed population [15,16], we have represented that this interaction through mass action kinetics with a conjugation rate, , being proportional to the levels of susceptible and resistant bacteria to antibiotic in the population [17,18]. Moreover, bacteria have per capita rates of death due to immune cells response of host, and so this rate in (1) is .…”

“…Jacobian matrix in(18) for Z is Z eigenvalues are founded as Z,U = − X , Z,Z = − Z , Z,\ = ; 1 − \ and Z,X = − . Due to (6), Z,U , Z,Z and Z,X are negative.…”

Lokal Bakteriyel enfeksiyon durumunda çoklu antibiyotik tedavisine karşı bakteriyel direncin kesirsel mertebeden matematiksel modellemesi

“…While the former is generally used for planning, prevention and control strategies, the latter can be influence in the therapy and intervention programs for treating the individuals exposed to the specific pathogen. Understanding the early dynamics of acute infections and foreseeing the time of occurrence and magnitude of the maximum load of the bacteria can be critical in choosing effective intervention schemes [16].…”

Mathematical modelling of bacterial resistance to multiple antibiotics and immune system response

“…While the former is usually used for planning, prevention and control scenarios, the latter can be active in therapy/intervention programs for treating the individuals exposed to the particular pathogen. In this respect, understanding and anticipating the time of occurrence and magnitude of the maximum load of the bacteria and immune system cells by mathematical modelling can be crucial in selecting effective intervention strategies (Whitman and Ashrafiuon 2006 ).…”

“…Consequently, results on reproduction of sensitive and resistant bacteria to antibiotics are obtained in Austin et al ( 1997 ), Bonten et al ( 2001 ), Esteva et al ( 2011 ), Wiesch et al ( 2011 ), Zhang ( 2009 ); definitions of factors responsible for resistance prevalence are studied in Austin and Anderson ( 1999 ), Linares and Martinez ( 2005 ), Opatowski et al ( 2011 ), Rodrigues et al ( 2007 ); bacteria behavior under different antibiotic treatments is examined in Alavez et al ( 2006 ), Bonhoeffer et al ( 1997 ), Bootsma et al ( 2012 ), D’Agata et al ( 2007 ), Sun et al ( 2010 ); optimization results and design of control measures are investigated in Bonten et al ( 2001 ), Haber et al ( 2010 ), Massad et al ( 2008 ), Sotto and Lavigne ( 2012 ); biological cost and persistence of antibiotic resistance are analyzed in Andersson et al ( 2001 ), Andersson and Levin ( 1999 ), Antia et al ( 1996 ), Johnson and Levin ( 2013 ), Mondragón et al ( 2014 ); dynamics between pathogens and immune response are given in André and Gandon ( 2006 ), Carvalho et al ( 2012 ), D’Onofrio ( 2005 ), Gilchrist and Coombs ( 2006 ), Gilchrist and Sasaki ( 2002 ), Kostova ( 2007 ), Mohtashemi and Levins ( 2001 ), Nowak and May ( 2000 ), Whitman and Ashrafiuon ( 2006 ), respectively.…”