Lectures On Mathematical Modelling Of Biological Systems

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GBIO 2060Lectures on Mathematical Modellingof Biological SystemsG. BastinAugust 22, 2018

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Contents1 Dynamical Modelling of Infectious Diseases1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .1.2 The basic SIR model . . . . . . . . . . . . . . . . . . . . . . . . . . .1.3 Temporary immunity : the SIRS model for endemic diseases . . . . .1.4 The SIR model with demography . . . . . . . . . . . . . . . . . . . .1.5 Variants and generalisations . . . . . . . . . . . . . . . . . . . . . . .1.5.1 SIR model with vaccination . . . . . . . . . . . . . . . . . . .1.5.2 SEIR model with a latent period . . . . . . . . . . . . . . . .1.5.3 Age structured SIR model . . . . . . . . . . . . . . . . . . . .1.5.4 SIR model with time-varying parameters and periodic forcing1.5.5 Modelling of vector-borne diseases . . . . . . . . . . . . . . .1.6 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .555811121213131417172 Quantitative Modelling of Metabolic Systems2.1 Metabolic networks . . . . . . . . . . . . . . . . . . .2.2 Elementary pathways and input-output bioreactions2.3 Metabolic flux analysis . . . . . . . . . . . . . . . . .2.4 Case study: Application to CHO cells . . . . . . . .2.5 Minimal dynamical bioreaction models . . . . . . . .2.5.1 Illustration with the case-study of CHO cells.191920212225303 Mathematical Models in Population Genetics3.1 Introduction . . . . . . . . . . . . . . . . . . . .3.2 Mendelian genetics . . . . . . . . . . . . . . . .3.3 Hardy-Weinberg equilibrium . . . . . . . . . . .3.4 Evolution dynamics . . . . . . . . . . . . . . .3.5 References . . . . . . . . . . . . . . . . . . . . .3737373839414 Modelling of within-host4.1 The basic model . . .4.2 Immune response . . .4.3 Mutants . . . . . . . .4.4 Latency . . . . . . . .4343444445HIV dynamics. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .3.

4CONTENTS4.54.64.7Antiretroviral therapies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46Resistance to antiretroviral therapies . . . . . . . . . . . . . . . . . . . . . . 46Resistance to immune response . . . . . . . . . . . . . . . . . . . . . . . . . 47

Lecture 1Dynamical Modelling of InfectiousDiseases1.1IntroductionThe aim of this lecture is to give an elementary introduction to mathematical modelsthat are used to explain epidemiologic phenomena and to assess vaccination strategies.We focus on infectious diseases, i.e. diseases where individuals are infected by pathogenmicro-organisms (like, for instance, viruses, bacteria, fungi or other microparasites). Somewell known examples of such infectious diseases are : Viral infectious diseases : AIDS, Chickenpox (Varicella), Common cold, Cytomegalovirus Infection, Dengue fever, Ebola hemorrhagic fever, Hepatitis, Influenza (Flu),Measles, Mononucleosis, Mumps, Poliomyelitis, Rubella, SARS, Smallpox (Variola),Viral meningitis, Viral pneumonia, West Nile disease, Yellow fever. Bacterial infectious diseases : Cholera, Diphtheria, Legionellosis, Leprosy, Lymedisease, Pertussis (Whooping Cough), Plague, Pneumococcal pneumonia, Salmonellosis, Scarlet Fever, Syphilis, Tetanus, Tuberculosis, Typhus. Parasitic infectious diseases : Malaria, Taeniasis, Toxoplasmosis. Prion infectious diseases : Creutzfeldt-Jakob disease.1.2The basic SIR modelA first fundamental mathematical model for epidemic diseases was formulated by Kermack and McKendrick in 1927 (see the fac-simile of their paper in Appendix). This modelapplies for epidemics having a relatively short duration (compared to life duration) thattake the form of “a sudden outbreak of a disease that infects (and possibly kills) a substantial portion of the population in a region before it disappears” (Brauer, 2005). In thismodel, the population is classified into three groups : (i) the group of individuals who are5

6LECTURE 1. INFECTIOUS DISEASESuninfected and susceptible (S) of catching the disease, (ii) the group of individuals whoare infected (I) by the concerned pathogen, (iii) the group of recovered (R) individualswho have acquired a permanent immunity to the disease. The propagation of the diseaseis represented by a compartmental diagram shown in Fig.1.1. The model is derived underISRFigure 1.1: Compartmental diagramthree main assumptions : (i) a closed population (no births, no deaths, no migration), (ii)spatial homogeneity, (iii) disease transmission by contact between susceptible and infectedindividuals. The model is a system of three differential equations:(1.1)dS βIS,dt(1.2)dI βIS γI,dt(1.3)dR γI.dtIn these equations, S denotes the number of susceptibles, I the number of infected individuals and R the number of immune individuals at time t. The total populationN S I R.is constant by assumption since we havedNdS dIdR 0dtdtdtdtfrom the model equations.In the first and second model equations (1.1)-(1.2), the term βIS represents the diseasetransmission rate by contact between susceptible and infected individuals. This rate isassumed to be proportional to the sizes of both groups with a proportionality coefficientβ. In the second and third equations (1.2)-(1.3), the parameter γ is the specific rate atwhich infected individuals recover from the disease.Let us consider an epidemic outbreak in a population where, at the initial time, onlya few individuals are infected. The initial conditions areS(0) N, I(0) N S(0) 0, R(0) 0.

71.2. THE BASIC SIR MODELINImaxSγβNFigure 1.2: Epidemic trajectoryA typical trajectory of the system solution in the I-S phase plane is given in Fig.1.2.From this curve, a fundamental observation is the existence of a Threshold Effect. Themaximum value of the curve occurs at S γ/β. This means that an epidemic will startand amplify only if S(0) N is larger than γ/β or equivalently ifR0 Nβ 1.γUnder this condition, the number of infectives will increase until the number of susceptiblesis reduced to γ/β and will decrease thereafter. Thus the number R0 represents a thresholdfor an epidemic to occur. In the literature, this number is also called Basic ReproductionRatio because it represents the average number of susceptibles which are contaminated byone infective.Dividing equation (1.2) by equation (1.1), we obtain dIγ 1 .dSβSIntegrating this equation, we have(1.4)I γlog S S CβwithC N γlog N.βFrom this equation, we can compute the instantaneous maximum number of infectives(see Fig.1.2) as 1 log R0Imax N 1 R0Equation (1.4) also implies that I must vanish at some positive value of S. This means thatthe trajectory terminates on the S-axis at a positive value as shown in Fig.1.2. Therefore,

8LECTURE 1. INFECTIOUS DISEASESthe epidemic terminates before all susceptibles have become infected and some individualsescape the disease entirely. We can now determine how many susceptibles remain orequivalently the final value R( ) of the immune population size.Dividing equation (1.2) by equation (1.3), we havedSβ βR βR S S(R) S(0)e γ N e γdRγThen dR βR. γI γ(N S R) γ N N e γdtThereforet (1.5)dR 0dthi β R( ) N 1 e γ R( ) I 0 Equation (1.5) has a unique solution R( ) between 0 and N as long as R0 1. Wedenote x R( ))/N the fraction the population that has contracted the disease beforethe epidemic collapses. By solving (1.5), we have the following relation between R0 andx:R0 log(1 x).xAn interesting application of the SIR model is reported in the book of Murray [1] on thebasis of influenza epidemic data in an English boarding school published in 1978 by TheBritish Medical Journal [2]. The epidemic lasted 22nd January to 4th February 1978. Alarge fraction of the N 763 boys in the school were infected and are represented by dotsin Fig.1.3. The curves in the figure represent solutions of the SIR model fitted to the datausing least squares. The estimated parameter values areβN 1.66 per day,1.31 2.2 days,γR0 3.65.Temporary immunity : the SIRS model for endemicdiseasesIn this section, we describe how the basic Kermack-McKendrick model is modified in order to describe how a disease in a population can persist when the immunity of recoveredindividuals is temporary (and not permanent as we have assumed in the previous section).

1.3. TEMPORARY IMMUNITY : THE SIRS MODEL FOR ENDEMIC DISEASES 9800700 SusceptiblesNumber of Boys600500400300200Infectives 1000051015Time (Days)Figure 1.3: An influenza epidemic in an English boarding school in 1978 (reprinted from[3], see also re 1.4: SIRS compartmental diagramThis is illustrated with the SIRS compartmental diagram of Fig.1.4. An additional parameter δ is introduced in order to represent the specific rate of immunity loss. The SIRSmodel is as follows.(1.6)dS βIS δR,dt(1.7)dI βIS γI,dt(1.8)dR γI δR.dtAs above the total population N S I R is constant (dN/dt 0).

10LECTURE 1. INFECTIOUS DISEASESThis system has two equilibria which are the two possible constant solutions of equations (1.6)-(1.7)-(1.8). The first one is the disease free equilibrium :S N,I 0,R 0.The second one is the endemic equilibrium : γγN N γββ γ , R .S , I δβ1 1 δγObviously, the endemic equilibrium exists only if the numerators are strictly positive. Thisimplies that the following condition must hold :R0 Nβ 1.γHence the condition for the existence of an endemic equilibrium in the SIRS model is inagreement with the condition for an epidemic to occur in the SIR model.In order to analyse the system trajectories and the equilibrium stability, we considerthe second order system obtained from equations (1.6)-(1.7) by substituting R N S I:dS βIS δ(N S I),dtdI βIS γI.dtThe Jacobian matrix of this system, evaluated at an equilibrium point (S , I ) is (βI δ) (βS δ).(1.9)J βI (βS γ)An equilibrium is asymptotically stable if trace(J) 0 and det(J) 0. Otherwise it isunstable.For the disease free equilibrium (S N, I 0) we havetrace(J) βN γ δ,det(J) δ(βN γ)Consequentlyif R0 1, then trace(J) 0, det(J) 0 and the disease free equilibrium is stable,if R0 1, then det(J) 0 and the disease free equilibrium is unstable.For the endemic equilibrium which exists only if R0 1, we havetrace(J) (βI δ) 0,det(J) βI (βS δ) βI (γ δ) 0Consequently the endemic equilibrium is necessarily stable when it exists.

111.4. THE SIR MODEL WITH DEMOGRAPHY1.4The SIR model with demographyWe now reconsider the basic SIR model of Section 1.2 in the case where demographic effectsare taken into account. Therefore, as it is illustrated with the compartmental diagram ofFig.1.5, births (or immigration) at the rate ν as well as deaths (or emigration) at the rateµ are introduced in the model:(1.10)dS νN βIS µS,dt(1.11)dI βIS γI µI,dt(1.12)dR γI µR.dtνNSIRµSµIµRFigure 1.5: Compartmental diagram of the SIR model with birth (or immigration) andmortality (or emigration).In order to have a constant total population N S I R (dN/dt 0), we assumethat µ ν. The system has two equilibria and we shall see that the analysis is quitesimilar to the previous case. The first equilibrium is the disease free equilibrium :S N,I 0,R 0.The second equilibrium is the endemic equilibrium :S γ µS γ µ1 ,βNβNR0I µ(N S )µ(R0 1). βS βObviously, the endemic equilibrium exists only if S N and I 0 which means that,as in the previous cases, the Basic Reproduction Ratio R0 must be greater than 1:R0 Nβ 1.γ µ

12LECTURE 1. INFECTIOUS DISEASESThe Jacobian matrix of the sub-system (1.10)-(1.11), evaluated at an equilibrium point(S , I ) is (βI µ) βS (1.13)J .βI βS γ µIt is readily checked that the disease free equilibrium (S N, I 0) is stable if R0 6 1and unstable if R0 1.For the endemic equilibrium which exists only if R0 1, we havetrace(J) (βI µ) µR0 0,det(J) β 2 I S µ(γ µ)(R0 1) 0Consequently the endemic equilibrium is necessarily stable when it exists. Moreover, theendemic equilibrium is a focus if the following inequality holds:4det(J) (trace(J))2 4µ(γ µ)(R0 1) µ2 R02 .and the trajectories exhibit oscillations as shown in Fig.1.51.5.1Variants and generalisationsSIR model with vaccinationIn this section, we explore how the SIR model of the previous section can be modified inorder to explain how epidemic diseases can be eradicated by vaccination. A new parameterσ is introduced in the model which represents the specific vaccination rate of the newborns.The model is now written as follows.(1.14)dS (µ σ)N βIS µS,dt(1.15)dI βIS γI µI,dt(1.16)dR γI µR.dtThe disease free equilibrium is:S (1 σ)N (1 p)N,µI 0where p σ/µ is the fraction of the newborn population which is vaccinated. The endemicequilibrium isS γ µ,βI µ(N S ) σN.βS

131.5. VARIANTS AND GENERALISATIONSTable 1.1: Estimates of R0 and of the corresponding vaccinated fraction p of the populationto achieve eradication.DiseaseSmallpoxMeaslesWhooping coughChicken poxDiphteriaPoliomyelitisEstimate ofThreshold R0413131056Minimal p (%)for eradication759292908083The disease is eradicated if the disease free equilibrium is the only possible stable equilibrium and if the endemic equilibrium does not exist. This is achieved if the followingcondition holds:p σ1 (1 ).µR0It is remarkable that not everybody has to be vaccinated in order to prevent an endemicdisease. This is called herd immunity. Typical examples of R0 values and the corresponding critical level of vaccination are given in Table 1.1. It can be seen that the criticallevel of vaccination is about 80% only for severe diseases like smallpox and polyomelithiswhich have been eradicated in developped countries, while it is much higher for childhooddiseases like measles and whooping cough (about 92%) which are fortunately much lesssevere.1.5.2SEIR model with a latent periodA latent period is a phase of the disease where individuals are already infected by thepathogens but not yet infectious (i.e. cannot yet transmit the disease to other people).The compartmental diagram is extended with an additional E compartment representingthe ”exposed” fraction of the population during the latent period as shown in Fig.1.6. Anadditionnal parameter η is introduced which represents the specific transfer rate from Eto I. The derivation of the model equations is left as an exercise. For this model, thethreshold parameter isR0 1.5.3βNη.γ µη µAge structured SIR modelA very relevant issue is obviously to extend the basic SIR model to heterogeneous populations. There are infectious diseases (e.g. sexually transmitted diseases) where considering

14LECTURE 1. INFECTIOUS DISEASESµNSEIRµSµEµIµRFigure 1.6: Compartmental diagram of the SEIR model with a latent period.all susceptible and infected individuals as a single homogeneous group cannot sufficientlycapture the dynamics of disease propagation. Individuals may differ in characteristicsthat are epidemiologically relevant. Traits as age, sex or genetic composition can influence susceptibility and infectivity of individuals. In order to account for such individualcharacteristics, it is useful to extend the dimension of the system, either by consideringhigher order systems of ordinary differential equations, or, in a more radical way, systemsof partial differential equations (PDEs). As a matter of illustration, we present here theexample of a PDE model which is derived under the assumption that individuals agingmatters.In this type of models, the dependent variables S, I and R are functions of bothtime and age. More precisely, the variable S(t, a) represents the age distribution of thepopulation of susceptible individuals at time t. This means thatZ a2S(t, a)daa1is the number of susceptible individuals with ages between a1 and a2 . Similar definitionsare introduced for the age distributions I(t, a) of Infected individuals and R(t, a) of recovered individuals. Then a natural extension of equation (1.10) is to describe the dynamicsof S(t, a) by the following hyperbolic partial differential equation: S S µ(a)S φ(t, a)S t aZ φ(t, a) β(a, b)I(t, b)db0where β(a, b) denotes the transmission coefficient by contact between a susceptible havingage a and an infected having age b.1.5.4SIR model with time-varying parameters and periodic forcingVarious childhood diseases exhibit sustained periodic oscillations with pluriennal epidemiccycles which are not explained by the models that we have presented so far. Typical

1.5. VARIANTS AND GENERALISATIONS15examples are shown in Fig.1.7 for measles in Rekyavik (with a 4 year cycle) and chickenpox in New York (with an annual cycle). During the last twenty years, numerous scientificFigure 1.7: Observed notifications of measles in Rekyavik and chicken-pox in New-Yorkbefore the vaccination era (Reprinted from [6])studies have demonstrated that seasonal variations in disease transmission rates appearto be a major factor to explain sustained epidemic cycles. This effect of seasonality isillustrated in Fig.1.8 with the trajectory of a simple SIRD model where the transmissioncoefficient β is an annual sinusoidal forcing function β(t) β0 1 β1 cos(2πt)with the time t in years and β1 0.2, (i.e. a 20% seasonal variation). It can be seen,in this example, that the annual forcing induces biennal epidemics. In fact, by changingthe value of the birth-mortality rate quadriennal epidemics that are reminiscent to theobserved data for Rekyavik in Fig.1.7 could be induced as well.Now, if in addition to seasonal variations of transmission rate, time varying birth ratesµ(t) are introduced in the model according to the real-life demography, then very realisticsimulations are achieved as illustrated in Fig.1.9. After 1950, we see sustained biennal

16LECTURE 1. INFECTIOUS DISEASES(a)(b)Figure 1.8: Impact of seasonal forcing on the SIRD model. (a) time evolution. (b) phaseplane trajectory (the square dot is the unforced equilibrium) (reprinted from [5]).Figure 1.9: Observed measles data in London (circles) and corresponding simulation ofan SIR model (solid line) with annual forcing of β(t) and time varying birth rate µ(t)(reprinted from [4]).cycles. In contrast, the baby boom in the period 1945-1950 increases the recruitment ofsusceptibles and induces annual epidemics (see [4] for more details).

171.6. REFERENCES1.5.5Modelling of vector-borne diseasesVector-borne diseases constitute another clas of diseases where it is relevant to extend thebasic SIR model. A typical example is mosquito-borne diseases such as malaria, yellowfever, dengue or chikungunya. In such diseases, susceptible humans are infected by theparasite when they are bitten by infectious mosquitoes while the susceptible mosquitoesbecome themselves infected when they bite infectious humans. Furthermore, the immunityof recovered humans is generally temporary. It is therefore natural to consider two coupledSIR-type models for the humans and for the mosquitoes as shown in Fig.1.10 where thedotted lines represent the reciprocal infections between humans and mosquitoes. Thederivation of the corresponding equations is left as an exercice (an interesting reference igure 1.10: Compartmental diagram for mosquito-borne diseases.1.6References[1] J.D. Murray, Mathematical Biology, 3rd edition, Springer, 2007 (Volume 1, Chapter 10).[2] Anonymous contribution to the British Medical Journal The Lancet, March 1978, p.587.

18LECTURE 1. INFECTIOUS DISEASES[3] M. J. Keeling and P. Rohani, Modeling Inf

Lectures on Mathematical Modelling of Biological Systems G. Bastin August 22, 2018. 2. Contents . 1.2 The basic SIR model A rst fundamental mathematical model for epidemic diseases was formulated by Ker-mack and McKendrick in 1927 (see the fac-simile of their paper in Appendix). This model

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