where N (t) = S(t) + I(t) + R(t). If we add the three differential equations together, we obtain a differential equation for N (t),
dN
dt = (b − a)N.
If b = a, then the population remains constant. This is sometimes a useful assumption to make provided the time-scale over which the model is applied is sufficiently short so that a constant population is a reasonable assumption (but also long enough that births create a sufficient source of new susceptibles).
A numerical solution of equations (5.9) is given in Figure 5.4. We have assumed a population size of 1 million, with an initial number of infectives, i0= 10, initial number of
susceptibles, s0= 105and the initial number of people who have previously had the disease
and are immune, r0= N − i0− s0. These are typical values for a disease such as measles
in a medium sized city and where many of the population are immune through vaccination or previous exposure, but a significant proportion (10%) are not. For parameter values we have used γ = 52, years (γ−1= 1week = 1/52 years, and a life expectancy, a−1, of 80 years.
For β we have assumed R0= 10, typical for measles. This leads to a transmission coefficient
β = 5.2 × 10−4 since here R0= βN/(γ + a).
The number of infectives initially has similar dynamics to that in Figure 5.2, but after some time the number of infectives rebounds instead of dying out. Recurrent outbreaks occur every two to three years. Initially there are sufficient susceptibles for the infectious disease to spread in the population (i.e., R0> 1). After a time, the number of susceptibles
falls so that there are insufficient susceptibles available to sustain the increase (i.e., a single infective cannot infect more than one susceptible while they are still infectious), and the number of infectives begins to fall. However, due to births of new susceptibles, there will come a time when the number of susceptibles again reaches a critical value so that the number of infectives begins to rise again, thus causing a new outbreak. In this model we obtain damped oscillations, and the number of infectives tends to a steady state.
In practice, the transmission coefficient β can be seasonally dependent and this can cause sustained oscillations rather than damped oscillations. Additional oscillatory forcing of this type often produces chaotic behaviour. This is discussed in Keeling and Rohani (2008), and see also Roberts and Tobias (2000) for an example of the same behaviour for recurrent epidemics of measles in New Zealand.
5.2 Model for an influenza outbreak 107 0 5 10 15 20 0 500 1000 time t (years) in fe ct iv es I (t )
Figure 5.4: An endemic disease such as measles. Number of infectives from a numerical solution of the differential equations (5.9) for an infectious disease in a hypothetical population of N = 106. Parameter
values used are β = 5.2∗10−4years−1persons, γ = 52 years−1, b = a = 1/80 years−1with initial populations
s0= 105, i0= 10 and r0= N − i0− s0.
Frequency-dependent and density-dependent transmission
Recall that the force of infection is the product of the contact rate and the probability a contact leads to an infection, given the contact is a susceptible with an infective. The contact rate (number of contacts each individual makes per unit time) can depend on the population size or density. We might assume the rate of contacts is independent of the population density or, alternatively, we might assume it is proportional to the population size or density, that is, more crowded populations have more frequent contacts.
When the total population N is constant, as it was with the influenza model considered here, the contact rate is constant so the two systems of equations (5.5) and (5.6) are exactly the same with β = βf/N . However, there are many circumstances where the population
is not constant. These include endemic models over a long time scale where the per-capita birth rate is greater than the per-capita death rate and diseases that result in a significant number of deaths can cause a decreasing total population. For this situation it becomes important to focus on the type of transmission. There are two main types: frequency dependent transmission and density-dependent transmission.
For frequency-dependent transmission, we assume the contact rate is constant, c(N ) = c0,
and the force of infection is then λ(t) = pc0
I(t) N (t) = βf
I(t)
N (t), βf = pc0.
The terminology ‘frequency-dependent’ comes from the assumption that the contact rate depends only on the frequency of contacts. For density-dependent transmission, c(N ) is assumed proportional to N , c(N ) = κN , and the force of infection is now
λ(t) = pκN (t)I(t)
N (t)= βI(t), β = pκ.
It is called density-dependent transmission because the contact rate depends on the popu- lation size (or density).
Frequency-dependent and density-dependent transmission represent two extremes. Typi- cally, frequency-dependent transmission is more appropriate for human populations (where humans usually have a fixed circle of social contacts mostly independent of population size). On the other hand, density-dependent transmission can be more appropriate for some an- imal populations where numbers of contacts per unit time are higher with more crowded populations due to more likely chance encounters.
In practice, contact rates often show linear dependence on population size for smaller population sizes, and constant dependence for large population sizes, and can be modelled by a suitable Michaelis–Menten type function, e.g. c(N ) = κN/(1 + ǫN ). Roberts and Heesterbeek (1993) and Begon et al. (2002) provide a good discussion of these issues.
Discussion
One question we might ask is whether a rapid increase in the number of infectives is always followed by a decrease? Also, by adjusting any of the parameters, could we limit the increase or even prevent it? Changing parameters could correspond to, for example, adopting certain vaccination strategies.
To answer these questions it is useful to gain more qualitative information about this infectious disease model. We would like to be able to say what happens for any values of the parameters. An exact solution of the simultaneous equations, however, is not easily obtained because the differential equations are nonlinear. An alternative approach is to use the chain rule to eliminate time and reduce the pair of differential equations to a single first-order differential equation, from which some insight can be gained. This analysis is covered in the next chapter.
There are a number of extensions of the basic SIR infectious disease model. Some of these are developed in the exercises. One extension is to incorporate a latent period, which is neglected in the basic SIR model. A latent period is the time from contact to when an individual is infectious to others. The simplest way to model the latent period is to include an additional compartment, with population size E(t), consisting of those exposed who are infected but not yet infectious. The exposed then become infectious at a constant per-capita rate. This leads to an additional differential equation in the system. Such models are known as SEIR models.
Further extensions include continuous vaccination, where susceptibles move into a vacci- nated compartment; sexually transmitted diseases; and disease spread by an animal agent, such as malaria spread by mosquitoes.
A classic reference in the field of mathematical epidemiology is Anderson and May (1991). Murray (1990) discusses the same model developed above. Also considered are the Black Plague, and rabies in foxes, among other examples. Braun (1979) outlines an extension to the model for sexually transmitted diseases. For an introduction to stochastic approaches to modelling human epidemics, see Daley and Gani (1999), Allen (2003) and Keeling and Rohani (2008).
For some further extensions of the basic models, see Grenfell and Dobson (1995). Keeling and Rohani (2008) give a comprehensive treatment of the modelling of infectious diseases in human and animal populations, including a chapter on vaccination and other means of con- trolling infectious diseases and a discussion of density-dependent transmission coefficients (where the rate of contacts between individuals is proportional to population size) and frequency-dependent transmission coefficients (where the rate of contacts is independent of the size of the population). These are particularly significant when the total population size changes with time. Diekmann et al. (2012) discuss many fundamental ideas in infectious disease modelling and, in particular, how to compute the basic reproduction number for more complicated models that involve populations structured into several groups, such as age groups or social groups.
5.3 Case Study: Cholera 109
Summary of skills developed here:
• Formulate differential equations for variations on the two models presented here, such as diseases with a latent period, continuous vaccination and diseases without immunity.
• Obtain a numerical solution for a system of differential equations for the SIR model and its variations.
• Calculate the basic reproduction number for the SIR model • Understand the different assumptions underlying the SIR model
5.3
Case Study: Cholera
Cholera is a particularly dangerous disease. Modelling can provide an understanding of circumstances under which an outbreak can occur. Here we formulate a model that includes interacting susceptible and infectious populations. However, what is different from the usual approach is that it involves transmission from the environment. The case study is based on Code¸co (2001) and Grad et al. (2012).
Cholera is a serious water-borne gastrointestinal disease that is contracted through the ingestion of contaminated water or food. In severe cases, and without treatment, it can kill victims through dehydration within hours of infection. Infection occurs from water contaminated with untreated sewerage, where the infectious agent responsible for cholera (bacterium Vibrio cholerae) forms a disease reservoir in the water supply.
Cholera poses a real and serious public health problem in communities with poor sanita- tion infrastructure, and one of the reasons special attention is paid to clean drinking water in camps setup to house refugees from war-torn areas is to minimise the risk of cholera outbreaks. Outbreaks of cholera can also occur after natural disasters, when infrastructure fails and water supplies become contaminated, such as in Haiti after the 2010 earthquake.
The statistical study of cholera began with the work of physician John Snow in the suburb of Soho in London, UK, in 1854. By mapping sites of infections, Snow traced the cause of cholera back to a certain water pump used by most residents. He managed to have the contaminated pump disabled, but it was a very controversial decision as it occurred well before the discovery of bacteria as a cause of disease. Snow’s study is considered the beginning of the science of epidemiology.
Governing equations. Because cholera has a short latent period, the variables needed to describe the prevalence of cholera in the population are S(t), susceptibles, and I(t), infectives, where t is time. Individuals who recover from cholera have immunity from reinfection lasting approximately two years. As long as the time scale of interest in the model is less than two years, it is reasonable to assume that infected individuals recover without becoming susceptible again.
Another important variable is the concentration of cholera bacteria in the water supply. This influences how easily cholera is spread to susceptibles as they make contact with the water through food preparation or drinking. We use the variable B(t) to represent the bacterial concentration measured as a cell count per ml in the water supply, also called the bacterial count. This will change with time as more bacteria enter the water supply
through ongoing sewerage contamination, which then increases with an increasing number of infectives shedding cholera bacteria.
The differential equations for the model are dS dt = −λ(t)S, dI dt = λ(t)S − γI, dB dt = eI + (nb− mb)B,
where differentiation is with respect to time t, λ(t) is the force of infection,discussed below, γ is the recovery rate and (nb−mb) is the net per-capita growth rate of bacteria in the water
supply. Normally the bacteria population will become extinct if not for the introduction of new bacteria by infected individuals, so nb− mb < 0. The parameter e represents the
rate of excretion of bacteria into the water supply from a single infection, so that eI(t) is the total rate of increase of bacteria (per unit volume of water per unit time). We do not include deaths due to cholera, but this could be easily incorporated.
Following Code¸co (2001), we assume that cholera is only contracted through contact with the environment and not through person-to-person contact (generally not important) or by food contact. The force of infection λ(t) is the probability per unit time of a susceptible being infected. This is the contact rate (c contacts with the water supply per day) multiplied by the probability of infection, which depends on the bacterial concentration B(t). While we could assume the probability is proportional to B(t), it is more realistic to assume that it is linear for small B(t), tending to one as B(t) becomes large — that is, for large bacteria concentrations a contact with the water supply always results in infection. A suitable functional form for the probability of getting infected, given contact with the water supply, is p(B) = B/(k50+ B), where the constant k50 represents the bacterial concentration that
leads to a 50% chance of becoming infected. We therefore assume a force of infection λ(t) = cp(B) = c B
k50+ B.
Substituting for λ(t) we obtain the governing equations for the model dS dt = −c B k50+ B S, dI dt = c B k50+ BS − γI, dB dt = eI + (nb− mb)B. (5.10)
To examine the system graphically, appropriate parameter values are required. These have been taken from Code¸co (2001) and are given in Table 5.1. We specify initial conditions of S(0) = 10,000 (the size of a a small town), I(0) = 1 (one infective individual introduced) and B(0) = 0 (initially the water supply is not contaminated with cholera bacteria).
The result of running this model is shown in Figure 5.5, where it is evident that with this parameter combination an outbreak occurs. The outbreak lasts for about 200 days with the number of infectives increasing from 1 to about 400 people, and peaking at around 140 days after the introduction of the single infective.
The bacterial count is also plotted, and it also peaks around the 140-day mark. However, careful inspection of the graph shows that the maximum of the bacterial count occurs about 2.7 days after the maximum in the number of infections occurs.
5.3 Case Study: Cholera 111
Table 5.1: Parameters used in the model with units. The units for the parameter e are cells ml−1day−1person−1.
Symbol Parameter Value Units c Rate of contact with water supply 1 day−1
k50 Bacterial concentration for 50% chance of infection 106 cells ml−1
γ Recovery rate for infected person 0.2 day−1
e Excretion rate 10 ∗
(nb− mb) Net per-capita growth rate of bacteria in water 0.33 day−1 ∗The units for e are cells ml−1day−1person−1
0 100 200 300 0 100 200 300 400 500 in fe ct ed I (t ) time t (days) 0 100 200 300 0 5000 10000 15000 b a ct er ia B (t ) time t (days)
Figure 5.5: Result of running the cholera model (5.10) with parameter values given in Table 5.1, and initial conditions S(0) = 104, I(0) = 1 and B(0) = 0.
Basic reproduction number. For diseases, it is of particular interest to establish interventions that could prevent an outbreak. The measure R0, the basic reproduction
number, predicts that an outbreak could occur when R0 > 1, but will not occur when
R0< 1.
The definition of R0 is the number of new infections produced directly from a single in-
fective introduced into a fully susceptible population. For this model, the time the infective is infectious is given by 1/exit-rate from the I compartment (i.e. 1/γ) and the rate of new infections at the start is the term cBs0/(K + B), where s0= S(0) is the initial number of
susceptibles.
Suppose we introduce I = 1 infective into a susceptible population of size s0. This one
infective will shed a number of bacteria into the water supply determined by substituting I = 1 into the equilibrium equation for B:
0 = e × 1 + (nb− mb)B,
which gives
B = e mb− nb
bacteria. This will be positive only if the bacteria death rate mbis greater than the bacteria
birth rate nb.
We now calculate the rate of new infections from an initial number of susceptibles s0 as
cBs0/(k50+ B), but we also linearise this for small B to cBs0/k50. The number of new
infectives produced during the time 1/γ for which the introduced infective is infectious is given by R0= cs0 k50× e mb− nb × 1 γ = ces0 γk50(mb− nb) . This is the same as the value derived in Code¸co (2001).
From the formula for R0, increasing contact with the water-supply, c, or increasing the
rate of excreted bacteria contamination of the water supply, e, or increasing the initial popu- lation, s0, all contribute to an increased R0. Increasing the recovery rate, γ, (i.e. decreasing
the length of time infected) or increasing the net per-capita death rate of bacteria, mb, each
lead to a decreased R0. These interpretations are as expected. With the parameters given
in Table 5.1 the calculated value of R0 is R0 ≃ 1.5. Since R0 > 1, this means that one
introduced infected individual produces more than one new infection. Thus an outbreak will occur.
Since R0 depends on the initial number of susceptibles, we can set R0 = 1 to find a
critical town size, Sc, below which an outbreak will not occur. We obtain
Sc=
γk50(mb− nb)
ce .
With the parameter values given in Table 5.1 the critical town size is Sc = 6,600. The value
of s0= 10,000 used in Figure 5.5 is above this critical value and an outbreak does occur.
We can use the formula for Scto investigate the impact of possible interventions to prevent
cholera outbreaks. Figure 5.6 illustrates values for e (the unit rate bacteria enters the water source) for each initial number of susceptibles (s0), such that R0= 1 or equivalently s0= Sc.
This diagram illustrates the impact of improved sanitation (modelled by reducing the value of the parameter e for contamination of water with cholera bacteria). For each s0, if the
point (e, s0) is above the curve an outbreak occurs, while if the point (e, s0) is below the
curve then there is no outbreak.
0 2 4 6 8 10 0 2 4 6 8 10x 10 4
contamination rate e
in
it
ia
l
p
o
p
u
la
ti
o
n
s
0 c = 1 c = 0.5Figure 5.6: Plot of critical town size against excretion rate e for model (5.10) and two values for the contact rate c, c = 1 and c = 0.5; each curve represents R0 = 1. Points (e, s0) above the curve predict an
outbreak and points below the curve predict no outbreak. Parameter values used are k50= 106, γ = 0.2,
mb− nb= 0.33.
This analysis illustrates that as the population increases, only a narrow range of e (very low rates of bacteria entering the water source) can prevent an outbreak. And only for relatively small populations is this not the case. While the result is intuitive, Figure 5.6 quantifies the nonlinear nature of the relationship, and can inform an understanding of risks and the design of control programs.
The above model is quite simple, and an obvious drawback is that births and deaths in the population are not included. Code¸co (2001) considers births and deaths for susceptibles only and omits deaths of infected and recovered, while all are included by Fung (2014).
5.4 Predators and prey 113 Grad et al. (2012) review further developments of the Code¸co (2001) model. Some of these include allowing for waning immunity, asymptomatic cases, vaccines and spatial aspects. The latter models allow more detailed investigations of interventions and control, and show that cholera can remain endemic within populations with outbreaks possibly triggered by weather events. Nevertheless, the methodology developed for the simple model above is also relevant for these more complex models and the provision of information for disease control.
5.4
Predators and prey
In this section, we develop a simple predator-prey model for carnivores using the growth of a population of small insect pests that interact with another population of beetle predators. An example of a model for herbivores is examined in a case-study in Section 8.6, while models for parasitic interactions or cannibalism will be simple to derive from these given examples.
Background
There are several types of predator-prey interactions: that of herbivores, which eat plant species, that of carnivores, which eat animal species, that of parasites, which live on or in another species (the host), and that of cannibals, which eat their own species and which is often an interaction between the adults and young.
One interesting example of a predator-prey interaction occurred in the late nineteenth century when the American citrus industry was almost destroyed by the accidental intro- duction from Australia of the cottony cushion scale insect. To combat this pest, its natural predator, the Australian ladybird beetle, was also imported, but this did not solve the prob- lem and finally DDT was used to kill both predator and prey in a bid to eradicate the pest. Surprisingly, application of DDT to the orchards led to an increase in the scale insects, the