TÍTULO TERCERO De la posesión

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−1_day−1_person−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}−1_day−1_person−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

Figure 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