Using simple mathematical transmission models of infectious diseases, one can create and investigate dozens of epidemics in an afternoon, and nobody becomes ill and nobody dies, a feature that makes this an informative and rewarding line of epidemiologic research. Simulation programs on personal computers quickly draw pictures of epidemics and allow rapid explorations of the interactions of populations of hosts and parasites. Even simple host-parasite systems have complex dynamic behavior which initially may appear counterintuitive, but with mathematical models it is possible to educate the intuition and learn about the general behavior of an infectious agent in a particular population. Using these systems one can explore the dynamic behavior of hosts and parasites that is an inherent characteristic of the system.
In particular one can learn to anticipate particularly good or particularly unfortunate behavior of the system for human health. How might the system respond to changes in nature or acts of man? What might be the short and long term effects of interventions of various types at different times? Initially, one needs to learn to avoid an action that inadvertently may cause a perverse outcome, such as provoking an epidemic. Then one can explore the possible beneficial effects of different interventions, and compare their applicability, acceptability, costs and possible adverse effects.
There has been a curious dichotomy in the acceptability of mathematical models in sciences such as physics and engineering, where the use of such models is universal, and infectious disease epidemiology, where mathematical models have only recently been used. Newton's laws of motion are simple differential equation models that are easily tested, and every student in an introductory course in physics in secondary school verifies one of Newton's laws as a first laboratory exercise. Similarly, these mathematical models, expressed as Newton's three body problem, were essential in planning our explorations of the moon.
Scientists make predictions on the basis of theories expressed in mathematical models, and as quickly as possible seek to verify these predictions with experiments in the real world. Mathematical methods were first applied to infectious disease epidemiology by the great Daniel Bernoulli, who was also the author of the Bernoulli trial in probability theory and of the Bernoulli principle in physics. The transmission of infectious agents (parasites) in populations of hosts was modeled beginning much more recently, after the development of the germ theory of disease. Until the book of Anderson and May that became an instant classic, there was little effort to gather the vast body of observational data on the occurrence of infectious diseases and epidemics and the mathematical models that might help explain them. Indeed, there is no explanation why, in most areas of science, theories expressed in mathematical models are tested against real data as soon as possible, while in the infectious disease arena such empirical testing has only recently been conducted. In this chapter we will provide readily available modern references that contain the citations to a number of the older original papers.
It is the purpose of this chapter to illustrate the use of mathematical models in understanding and controlling vector borne diseases. This chapter is intended for biologists and field practitioners who have no special training in mathematics. We will use malaria and the main example and we will begin with the simplest models, and add more realistic features in a stepwise fashion so the reader can understand how these models evolved over time, and begin to understand the literature.
Microbiologists have not used the words “microparasite” and “macroparasite” as they are used in modeling (in the Anderson and May sense), so these terms will be described here. A microparasite is not a type of creature. Rather, microparasites are whole categories of organisms, usually a bacteria or viruses, that have direct reproduction in the host, usually at high rates. Hosts are either infected or not, but a parasite burden usually has no meaning for microparasites. Microparasites generally are small, have short generation times, and usually produce long lasting immunity against reinfection, as in measles. The duration of infection with microparasites is usually short compared with the expected life span of the host, so the host sees the infection as transient.
Macroparasites such as worms and one celled organisms like malaria have no direct reproduction in the definitive host. They are larger, and have longer generation times, which may be a substantial fraction of the life expectancy of the host. When an immune response is elicited by a macroparasite, it is usually transient, and will rapidly disappear when the parasite is removed, as with chemotherapy. These infections are usually persistent, with hosts being continually reinfected, as in malaria.
By direct transmission modelers mean that the infection moves from person to person directly, with no environmental source, intermediate vector or host. To a modeler direct transmission may take place by contact between mucous membranes as for sexually transmitted diseases, or by droplets aerosolized by a cough or sneeze, as for colds or measles. This may seem very imprecise to a biologist, but what is implied is that for directly transmitted infections one need only model the behavior of the parasites in people.
In contrast, transmission of malaria by mosquitoes would be an example of indirect transmission. The fundamental difference is that if one is modeling malaria transmission, one has to include equations for the behavior of parasites in populations of mosquitoes and also in populations of humans. As a result, modeling indirect transmission is fundamentally more complex.
Modelers use one other term that seems odd to an epidemiologist. In epidemiology we commonly use the word density in a particular way, as in probability density or incidence density. When a modeler uses the concept of density-dependent functions it means number dependent. If a modeler says that the occurrence of an epidemic is density-dependent, that means it depends, for example, on the actual number of susceptibles present in the population.
A concept from ecology that is central to thinking about the transmission of infectious diseases is the basic reproductive number, R0. R0 represents the average number of secondary infections produced when a single infectious individual is introduced into a host population in which every individual is susceptible. The time implied is the entire period of infectiousness for the infected case.
For directly transmitted microparasites one is considering a system that includes infectious and susceptible humans, but for indirectly transmitted macroparasites such as malaria, one must consider a system that includes infectious and susceptible mosquitoes as well as infectious and susceptible human populations. For indirectly transmitted infections like malaria, the value of R0 is for human to human transmission via mosquitoes.
If this reproductive number, R0, is less than unity (one) then the infection will eventually die out and not persist in that community. There may be some secondary cases, but these will decrease with time, and eventually the infection will be come extinct. If this reproductive number is exactly unity, then the infection just barely succeeds in reproducing itself, and there will be a similar number of cases at any later time. If this reproductive number is larger than unity, then the number of cases will increase with time, at least initially, and there may be an epidemic. The nature of the parasite, the nature of the host(s), and the behavior of host(s) all help determine the value for R0 for a particular infectious disease and community.
When there are some already infected or immune or resistant individuals in the population, then not everybody is susceptible. At that point there is a value of R, the reproductive number for the system at that point, but it is not R0. R0 is the upper bound for the value of R, which is usually less than R0, and the value of R may vary widely during the course of an infectious disease through a population. Remember, R0 is a characteristic of the system assuming that everybody is susceptible, while R is the value of the quantity at a particular moment, when some or possibly even most individuals are already infected, immune or resistant.
Timing is a crucial aspect of the study of the epidemiology of infectious diseases, but symptomatology is less so. Transmission can only take place during the period when a host is infectious. There may be no symptoms associated with infection, and when symptoms do occur, they may be apparent in no particular relation to the period of transmissibility. Individuals infected with Human Immunodeficiency Virus (HIV) are asymptomatic but infectious for an average of about ten years before they become clinically ill. In contrast, most infected with tuberculosis (TB) organisms may remain non-infectious for their entire lifetimes, while a few will develop clinical pulmonary TB after a period of months or years. Individuals infected with TB only become infectious to others when they begin to cough. These extreme differences in the behavior of HIV and TB underline the need to separate the latent and infectious periods of a disease from the incubation period and the period of clinical disease. Infectiousness may have little to do with symptoms; an individual newly infected with falciparum malaria will become symptomatic after 7 – 10 days, but will not become infectious to vector mosquitoes for three weeks. Timing determines if transmission of an infectious disease will take place at all, and if it does, timing determines the nature of transmission. While the clinician treats the symptomatic patient, the epidemiologist seeks the infectious individual, who may not be symptomatic, or may be symptomatic at a time when he is not particularly infectious.
The incubation period is the interval from the time a person is infected until he develops clinical disease. The period of clinical disease is the period of symptoms. An infected person may never develop clinical disease, and the period of infectiousness may not correspond very well with the period of symptoms. For many childhood infections, for example, the period of greatest infectiousness is just prior to the appearance of clinical symptoms. This has important implications for control.
The latent period is the interval from the time a person is infected until he becomes infectious to others. The infectious period is the interval during which an individual can transmit an infection. As was noted above, the latent and infectious periods are variably related to clinical symptomatology, but are crucial in the study of the epidemiology and transmissibility of infectious diseases.
In this chapter we will explore mathematical models of disease transmission, as a model can represent aspects of human behavior as well as measurable demographics. As must be evident, mathematical transmission models are totally dependent on knowledge of the latent and infectious period for an infectious disease. The first model we will investigate is the classic SEIR (Susceptible, Latently infected, Infectious, Resistant or immune) model made up of four differential equations. There are different systems of notation used in modeling but in this chapter we will use the most common and point out confusing and conflicting notation (3 – 5). (FIRST CONFUSING NOTATION WARNING: the R and R0 used to represent reproductive numbers are distinct from the R used to indicate the immune or recovered state for a host.)
A differential equation is an algebraic equation that includes a derivative, which is simply a slope. A slope can do one of three things: it can go up, in which case it is positive; it can go down, in which case it is negative, or it can do neither or stay the same, in which case it is zero (no change). With modern simulation programs we can always look at pictures of the performance of differential equations, which translates into pictures of slopes, which in turn means one repeatedly has to answer the question, is this going up, down, or straight sideways?
As those who have studied differential equations know, writing a differential equation is easy; it is the solution that is difficult. Most interesting differential equations remain analytically insoluble for the amateur. What has made mathematical modeling readily accessible in the last decade is the existence of the personal computer with a simulation program for numerical solutions to differential equations. The program does not actually solve the equations, it just presents a picture of what they do, which is what we wanted to know anyway.
One solves an algebraic equation by solving for x in terms of y and z, and then one can draw the picture or graph. One solves a differential equation by integration in order to produce an algebraic equation which one then solves for x in terms of y and z, and then one can draw the picture or graph. The simulation program goes from the differential equations directly to the picture without any intermediate stops. We will present the four differential equations of the SEIR model below to describe the movement of individuals from the Susceptible state to the Latent state to the Infectious state to the Resistant or immune state. We have used this model and the SEIR notation because it is the oldest in the literature and the most commonly used (5), although Anderson and May have used X Y Z instead of S I R throughout their book. In parallel we will also consider the SEIS model as well, a model for an infection with no long lasting immunity, that will become part of our malaria model later. In the SEIS model individuals that have recovered from the infectious stage do not become immune, and revert back to being susceptible again. That is, after recovering from infection they move back into the S compartment again.
The classic SEIR model uses four derivatives or slopes with respect to time (t);
dS/dt = the change in the numbers of Susceptibles (S) over time,
dE/dt = the change in the numbers of Latents (E) over time,
dI/dt = the change in the numbers of Infectious (I) over time, and
dR/dt = the change in the numbers of Resistant (R) or immune over time.
Other quantities are used as well;
b = the probability of transmission of infection per unit time,
G = the duration of the latent state, in units of time,
g = 1/G or the rate of leaving the latent state per unit time
D = the duration of infectiousness of this disease, in units of time,
d = 1/D or the rate of leaving the infectious state per unit time,
m = birth rate or death rate or rate of entering or leaving life per unit of time.
The movement of individuals from state to state is illustrated in the accompanying compartmental diagram (Fig. 1A). Those who leave one compartment must progress into the next. Entries are (+), departures are (–), and the total number N = S + E + I + R remains constant. In the first model there are no births, no deaths and there is no migration. The susceptibles become latently infected, the latently infected become infectious, the infectious recover and become immune. In an epidemic the number of susceptibles will decrease as they become infected, the number of latents will increase (initially) and the infectious will follow, and the number of immune will increase as the infected recover.
dS/dt = – b x S x I
dE/dt = + b x S x I – E x g
dI/dt = + E x g – I x d
dR/dt = + I x d
In simple algebra this set of differential equations for the SEIR model can be written as,
dS/dt = – bSI
dE/dt = bSI – gE
dI/dt = gE – dI
dR/dt = dI
If this were a disease that produced no long term immunity, then there would be no R state, and those who recovered from the infectious or R state would reenter the S or susceptible state. The SEIS model is given below. Here those recovering leave the infectious state as –dI, and reenter the susceptible state as +dI.
dS/dt = – bSI + dI
dE/dt = bSI – gE
dI/dt = gE – dI
In order to add births and deaths (vital dynamics) to the SEIR model, one would add all of the births to the susceptible state, as mN ( the total population), and then subtract deaths from each state. Those dying in the S state would be – mS, those dying in the E state would be – mE, those dying in the I state would be – mI, and those dying in the R state would be – mR.
dS/dt = mN – bSI – mS
dE/dt = bSI – gE – mE
dI/dt = gE – dI – mI
dR/dt = dI – mR
Similarly one could add vital dynamics to the SEIS model. When exact timing is not important it is common to drop the equation relating to the latent period, and to describe epidemics in terms of SIR and SIS models. In these models without latent periods, the newly infected proceed directly from the S state to the I state. The initial malaria model created by Ross did not use latent periods, but as the need for realism increased the latent period for malaria in mosquitoes was added by MacDonald and the latent period in people was added by Anderson and May.
An essential concept for modeling is the rates at which subjects enter and leave various compartments or states. If D is the duration of infection or the duration in the state I for example, and D is 7.0 days, then there is one complete turnover in the I compartment every seven days. On average, one seventh of those in the compartment must come out each day. That is, if the average stay in the compartment is 7.0 days, then the daily rate of recovery from infection must be 1/7.0 In general the parameter for leaving that state is 1/the average duration in that state. For a state with a duration D, on average 1/D individuals leave per unit time.
The last change in convention is that we will call the rate of leaving, d = 1/D. This leads to the mortality rate m = 1/average age at death, or 1/„age at leaving life. In a stationary population when births equal deaths them m is also the birth rate. Also we will have g = 1/average duration of latency, and d = 1/average duration of infectiousness. All must be in the same units of time. If we model in units of years, then all parameters must be in years. For example, 7 days is 7/365 or 0.02 years, and the transition parameter is 1/0.02 or 50 per year.
If the average age at death is 74 years, then 1/74 of the living will leave the living state and die in one year.
If the average age at infection for a disease like measles is 5 years, then 1/5 of the uninfected will leave the uninfected state and become infected in one year.
If the average latent period for falciparum malaria in people is 21 days, then 1/21 of people in the latent state will leave this state (and become infectious) in one day.
If the average duration of infectiousness for untreated TB is 5 years (60 months) then 1/5 of the untreated will leave the infectious state by recovering or dying each year, and 1/60 will do so in one month. With appropriate treatment the duration of infectiousness can be reduced to 2 months, so that 1/2 will leave the infectious state in one month. If the average duration of infectiousness for untreated falciparum malaria is nine months, 1/9 will leave the infectious state in one month. With appropriate treatment the duration of infectiousness can be reduced to one month, and all will leave the infectious state in a month. For infectious diseases chemotherapy can shorten the duration of infectiousness. This is also an example of how treatment may also be prevention for an infectious disease.
The above models are based on the law of mass action from chemistry, in that we assume that any individual in a population is equally likely to bump into (and infect) any other individual, like gas molecules moving about in a balloon. Also, we have used S and E and I and R to represent absolute numbers of individuals rather than proportions, because this is most common in the literature, and leads to the evaluation of thresholds, or the minimum number of individuals in a population that could support an epidemic, a topic that is beyond the scope of this chapter.
All of the models in this chapter are deterministic models, meaning that they will do the same thing every time they run. Models that include random variation are called stochastic models, but stochastic models are more complex and are beyond the scope of this chapter. Stochastic models are important when populations are small.
Using algebra it is possible to show that for the SIR or SIS models without births and deaths, R0 is,
R0= bN/d,
from the steady-state SIR or SIS model with vital dynamics we have;
R0 = bN/(m + d),
and from the steady-state SEIR or SEIS model we have;
R0 = bgN : (d + m) (g + m)
When both the duration of the latent state, 1/g, and the duration of the infectious state, 1/d, are small (a few days) compared to the length of life or 1/m (50 or 70 years), then all three expressions for R0 can be approximated as;
R0 ∼ bN/d.
Epidemiologists who deal with acute or short term outbreaks tend to think of these epidemics as occurring in closed populations, because few individuals are born or die, or move into or out of a community in a matter of weeks. We are faced with differentiating two fundamentally different types of epidemics in closed populations; propagated epidemics, and, point source epidemics.
Propagated epidemics must always result from some self-reproducing agent such as an infectious agent, while point source epidemics may be either of infectious or non-infectious etiology. Epidemics of measles are propagated epidemics as each infected individual acquires the measles virus from a person in the infectious stage who was infected in the previous generation of infection. An outbreak of salmonellosis from eating contaminated turkey at a hospital party would produce a point source epidemic with an infectious agent that all of the exposed acquired within a period of a few minutes (eating the main course). In fact, parents with salmonellosis from a point source epidemic may then go home and begin a propagated epidemic among the children in their own families.
In contrast, poisonings must all be point source epidemics, as a toxin cannot reproduce itself. This is true of bacterial toxins (staphylococcal food poisoning) as well as chemical toxins (pesticides) not of microbial origin. Staphylococcal food poisoning often takes place in the absence of living organisms because the toxin is heat stable while the bacteria are not. Cooking may well kill the bacteria and effectively sterilize the food while leaving the toxin unchanged.
This distinction between propagated and point source epidemics was first formulated by Ross (of malaria fame), who described propagated epidemics as “dependent happenings” because the number affected per unit time depended on the number already affected. In contrast, the number affected per unit time during an episode of poisoning was independent of the number of individuals already affected, so these Ross termed “independent happenings.”
Consider malaria as an example of an indirectly transmitted disease, an infection transmitted to humans by a mosquito vector. Both humans and mosquitoes are considered to be born uninfected. An uninfected female mosquito has a blood meal from an infected human and becomes infected with malaria herself. After a suitable latent period she becomes infectious and has another blood meal, this time on an uninfected human, and can transmit malaria to the previously uninfected human. The human infects the mosquito, then the mosquito infects the human. Humans do not infect other humans (except by blood transfusion), and mosquitoes do not infect mosquitoes.
The mosquito is the definitive host for the malaria parasite. That is, sexual reproduction takes place in the mosquito. Only asexual reproduction takes place in humans. Humans can be thought of as warm, friendly, wet reservoirs in which the malaria parasite can survive during hard times for adult mosquitoes. Tropical climates have a wet season and a dry season, and during the dry season adult mosquito populations are greatly reduced, and may disappear altogether. In more temperate climates there is also substantial temperature variation and the cold season is similarly hard on adult mosquitoes. Human reservoirs are essential to tide malaria parasites over until the next season of abundance for adult mosquitoes.
Malaria parasites persist in humans waiting for those wonderful mosquitoes to return, so that the parasites can get back to sexual reproduction. In an evolutionary sense, malaria parasites are just treading water with asexual reproduction in the human. However, natural selection pressure is exerted on humans, so the gene frequency that is sampled by mosquitoes is that in surviving humans. Meanwhile, mosquitoes survive in the form of fertilized eggs, waiting to hatch into larvae when water and warmth return to their part of the earth. Both the human and the mosquito part of the cycle are essential, but for different reasons. Both offer opportunity for intervention. Mosquitoes are seasonal in most places, so that there is usually a six month dry season when there are few mosquitoes and little or no malaria transmission. There are, however, some places where malaria transmission occurs throughout the year without respite.
The form of the malaria parasite that is infectious for the mosquito is the gametocyte in man. For falciparum malaria, gametocytes only appear in the human bloodstream about 21 days after infection, so there is a relatively long latent period from infection to infectiousness in the human. The incubation period, or time until clinical disease in a naive human is about seven to ten days for P. falciparum malaria, so the infected human may become severely clinically ill and die a week or more before becoming infectious to mosquitoes.
The form of the malaria parasite that is infectious for man is the sporozoite, which is delivered from the salivary gland of the female mosquito. After a female mosquito has a blood meal on an infectious human, sexual reproduction between ingested gametocytes takes place in the gut of the mosquito. When a human has been infected by multiple mosquitoes (a frequent happening in endemic areas) then multiple different broods of parasites are circulating in a single human and gametocytes from different broods are taken by a mosquito in a single blood meal. After recombination, sexual reproduction then results in sporozoites with different combinations of genes from those in either of the parent broods. There is an approximate 10 day latent period in the mosquito between the time the mosquito has an infectious blood meal from a human, and the time (after sexual reproduction) when infectious sporozoites appear in the salivary glands of the mosquito. Since this latent period may be longer than the average length of life for a mosquito, it is a relatively old and rare (lucky) mosquito that is able to infect a human.
Most malaria mosquitoes are shy, night biting creatures that go relatively unnoticed by their human prey. While feeding, a female mosquito loads up like a tank truck, and can barely fly to the nearest vertical surface, where she spends some hours diuresing most of the fluid she has taken in so that she can move more freely and go about her business and lay eggs. It is during this period that a mosquito is most vulnerable to control measures. This is the time when residual insecticides, such as DDT, are most effective.
To model malaria, Ross used two differential equations: one showing the infection of the human, and the other showing the infection of the mosquito. The simple Ross two differential equation model appears in textbooks, and illustrates some of the important features of malaria. At this point we have to change our conventions and our constant names to make them consistent with the malaria literature.
For SEIR, and SEIS processes described above we have used the absolute numbers of individuals in models so that we could consider eradication and numerical thresholds. With growing or shrinking populations, however, proportions are easier to manage, and for vector borne diseases there are usually so many vectors that human thresholds do not assume much importance. Furthermore, the mosquito vector has elaborate mechanisms to locate prey, thus the name vector, so that the law of mass action is not operative for mosquito-human interactions. As you will recall, if you are enclosed in a tent or a bedroom at night with a hungry female mosquito, she will definitely find you before morning. Chance and the law of mass action are not operating here.
In the Ross model the proportion of infected humans is H, so that the proportion of uninfected humans is 1 – H. The proportion of infected mosquitoes is M, so the proportion of uninfected mosquitoes is 1 – M.
Absolute numbers of people and mosquitoes do not enter this equation, only the ratio, m, between the numbers of female mosquitoes and the numbers of people (CONFUSING NOTATION WARNING: unfortunately, this is another use for the letter m, which was used for the death rate before.) The initial value of m = 40, which would be reasonable for the rainy season.
The man biting rate, a, is the number of bites by a female mosquito delivered on humans per day. This is determined by how often the mosquito needs to feed (the gonotrophic cycle), and whether the mosquito feeds on a human or on another mammal, like a cow or a pig. A common value is 0.25 human bites/day, or one meal on a person every four days.
The proportion of bites by infected mosquitoes on susceptible humans that produce infection in the human is b, that has been measured at 0.09.
The human recovery time is the duration of disease in a human or the time during which an infected human can infect a susceptible mosquito. For falciparum malaria this is in the range of 9.5 months or 285 days. The recovery rate is thus 1/„285 or 0.0035/day.
Not all bites on infected humans produce infection in the mosquito. The fraction that do is c, or 0.47.
Although humans spontaneously recover from malaria, mosquitoes do not, and the only way mosquitoes leave the infected pool is by dying. Since an average mosquito lifetime is about eight days, the rate of leaving life for a mosquito is pm (daily probability of dying for a mosquito) so pm = 0.12/day.
The Ross model consists of two equations, one for humans and one for mosquitoes. Each equation is conceptually similar to the dI/dt equations for SIR models in that each describes entries and departures from the infectious stage. dH/dt is for infection in humans and dM/dt is for infections in mosquitoes. Also in direct analogy to the dI/dt equations, the first part of each equation describes those humans or mosquitoes becoming infectious, and the last part of each equation describes those humans or mosquitoes becoming uninfectious because they recover (humans) or die (mosquitoes). The Ross model does not include latent periods so the equation for human infections represents an SIS model as humans who have recovered return to the susceptible pool, while the equation for mosquito infections represents an SI model as all infected mosquitoes die in that state. As you look at the models you will see that these are models of mosquito-human interaction (Fig. 2). The key difference between malaria models of indirect (vector) transmission and our previous models of direct transmission is that with malaria humans infect mosquitoes and mosquitoes infect humans, but humans do not infect humans nor mosquitoes infect mosquitoes.
dH/dt = a x b x m x M x (1 – H) – g x H
dM/dt = a x c x H x (1 – M) – pm x M
The Ross model is presented below in simple algebra,
dH/dt = abmM(1 – H) – gH
(This is an SIS model for people)
dM/dt = acH(1 – M) – pmM
(This is an SI model for mosquitoes).
For the Ross model,
R0 = mao2bc/gpm
Notice that the female mosquito has to bite twice to complete the cycle, so that the a term is squared. The simple Ross model outlines the basic features of malaria, but does not consider the approximate 10 day latent period in mosquitoes nor the exponential survival of mosquitoes. As a result, the Ross model predicts a too rapid progress for a malaria epidemic in people, and much too high an equilibrium prevalence of infectious mosquitoes. The results of the Ross Model of the progress of P. falciparum infection when one infectious person is introduced into a community of 100 susceptible individuals is presented in Fig. 3A.
Mosquitoes that can transmit malaria have a survival pattern that is represented almost perfectly by the exponential distribution in continuous terms, or the geometric distribution in discrete terms. Models built on both distributions are common in the literature, and will be explained in parallel below. In general terms, a fixed proportion of mosquitoes survive each day (or die each day), so that a minority, even a tiny minority, survive as long as the latent period and have the potential to become infectious for humans. To transmit malaria a mosquito must have a first blood meal on an infected human, become infected herself, and then survive as long as the latent period to become infectious, and then have a second blood meal on an uninfected human. The Anderson and May formulations of the malaria models include the latent period for mosquitoes and the death rate for mosquitoes using continuous distribution.
In continuous terms, if pm is the constant mosquito death rate per day and taum (greek letter tau, m for mosquito) is the latent period for mosquitoes, then the proportion of mosquitoes that is infectious follows the exponential distribution and is approximately,
exp(–pmtaum),
and this is the multiplier for the expression for the number of infectious mosquitoes.
The discrete counterpart to the exponential distribution is the geometric distribution, where p would be the probability of dying per day and q = 1 – p, is the probability of surviving, which would usually be described in probability terms as the distribution of
qtaum, where q is the probability of surviving per day.
In the discrete model terminology MacDonald has used p for q and n for taum, and –ln(p) is the daily mortality for mosquitoes, similar to pm in the continuous model. Thus the proportion of infectious mosquitoes in the discrete model as formulated by MacDonald becomes
pn .
The equivalent continuous and discrete expressions for R0 for malaria models are,
R0 = {ma2bc[exp(–pmtaum)]}/gpm, and
R0 = ma2bcpn/–gln(p).
They are comparable, with the daily mosquito mortality rate, pm or –ln(p) in the denominator, and the proportion of mosquitoes surviving the latent period, exp(–pmtaum) or pn, in the numerator.
In the MacDonald model the lags are for the latent period in mosquitoes, as it is the mosquitoes which have survived the latent period which are infectious now, and they, in turn were infected one latent period ago. For the description below, the first line over an equation is the usual word model, and the second line is an attempted analogy to the familiar dE/dt and dI/dt equations of the SEIR model. dH/dt is analogous to dI/dt for humans, dML/dt is analogous to dE/dt for mosquitoes, and dMI/dt is analogous to dI/dt for mosquitoes. This is an SIS model for humans and an SEI model for mosquitoes. The results of the MacDonald Model of the progress of P. falciparum infection when one infectious person is introduced into a community of 100 susceptible individuals is presented in Fig. 3B.
dH/dt = + a*b*m*MI*(1 – H) – g*H
dML/dt = + a*c*H*(1 – ML – MI)
dMI/dt = + a*c*lag_H_taum*(1 – lag_ML_taum – lag_MI_taum)*exp(–pm*taum) – pm*MI
To review, the MacDonald model does include the latent period in mosquitoes and the known exponential survival of mosquitoes during the latent period. MacDonald's original form of the model was based on the discrete form as the geometric distribution, pn, where p was the daily probability of survival for the female mosquito, n was the latent period or time before infectious sporozoites appear in the salivary glands of the infected mosquito, and –ln(p) was the daily mortality for mosquitoes, similar to pm in the Ross model. Measured values of p range from 0.76 to 0.95, and measured values of –ln(p) range from 0.05 to 0.28. For falciparum malaria, n is about 10.
In contrast, the continuous form of the MacDonald model as presented in Anderson and May is based on the exponential distribution. The latent period in mosquitoes of 10 days is represented by the Greek letter spelled out as tau, and the fact that it is for mosquitoes is indicated by the suffix m. Thus, taum is the 10 day latent period in mosquitoes. One way to insert the 10 day difference in time is to lag a variable, so that, for example, lag_H_taum is the proportion, H, from 10 days ago. However, the discrete and continuous forms give similar results for the basic reproductive number. The results of the Anderson and May Model of the progress of P. falciparum infection when one infectious person is introduced into a community of 100 susceptible individuals is presented in Fig. 3C. Note that in Fig. 3 each successive model adds a latent period, and the apparent progress of the infection through the community is slower. The Ross model (Fig. 3A) contains no latent periods, the MacDonald Model includes a 10 day latent period for mosquitoes (Fig. 3B, and the Anderson and May Model includes a 10 day latent period for mosquitoes and a 21 day latent period for humans.
These systems come to equilibrium because there is a continuous supply of susceptible humans and susceptible mosquitoes. Infected humans recover and reenter the susceptible pool, and new broods of uninfected adult female mosquitoes continue to hatch.
Setting the derivatives equal to zero, in the steady state it is possible to find the equilibrium proportions for infected humans (H*) and infected mosquitoes (M*). For the Ross model;
H* = (R0 – 1)/[R0 + (ac/pm)].
The equilibrium proportion of infected mosquitoes predicted by the Ross model is much too high, so the relation from the MacDonald model is given below;
M* = [(R0 – 1)/R0][(ac/pm)/(1 + ac/pm)] exp(–pmtaum)
MacDonald defined the quantity ac/pm, the number of bites on humans per day that produced infection in the mosquito, as the stability index. High levels of ac/pm, in the range of 2 – 4, indicate that mosquitoes bite man often and have relatively long lifespans, and produce continuous endemic malaria. Macdonald called this stable malaria. Where ac/pm is low, in the range of 0.5, malaria tends to occur in repeated outbreaks. MacDonald called this unstable malaria (3, 6). One needs also to appreciate that vector density changes orders of magnitude with the seasons, so that malaria occurs in annual epidemics in the rainy season when the mosquito density is high.
Below is a model of malaria modified from that published by Anderson and May that deals with the complexities of humans as well as mosquitoes. In addition to the 10 day latent period in mosquitoes and mosquito mortality, the complex model includes the 21 day latent period in humans (until the appearance of infectious gametocytes), the recovery of humans from both latent and infectious stages, and the death of humans in both latent and infectious stages. This appears to be a reasonably realistic model. The modification of the Anderson and May model was to allow infected humans to recover from the latent stage before they became infectious to mosquitoes. This happens if medical treatment is readily available and individuals who become newly symptomatic at the end of the incubation period (7 – 10 days) are treated before they pass through the latent period (21 days) and become infectious to mosquitoes. This is an SEIS model for human infection and an SEI model for mosquitoes.
dHL/dt = a*b*m*MI*(1 – HL – HI) – a*b*m*lag_MI_tauh*(1 – lag_HL_tauh – lag_HI_tauh) * exp((–ph –pg)*tauh)) – pg*HL – ph*HL
dHI/dt = a*b*m*lag_MI_tauh*(1 – lag_HL_tauh – lag_HI_tauh) * exp((–ph – pg)*tauh)) – pg*HI – ph*HI
dML/dt = a*c*HI*(1 – ML – MI) – a*c*lag_HI_taum*(1 – lag_ML_taum – lag_MI_taum)*exp(–pm*taum) – pm*ML
dMI/dt = a*c*lag_HI_taum*(1 – lag_ML_taum – lag_MI_taum) * exp(–pm*taum) – pm*MI
For this malaria model the basic reproductive number is
R0 = {ma2bc[exp(–phtauh–pmtaum)]}/pgpm
(continuous)
Malaria does produce some immunity, and malaria models including immunity have been developed, but are beyond the scope of this chapter.
A number of interventions have been developed to limit the transmission of malaria, and it is useful to review how these will appear in the Anderson and May model. A reduction in the number of larvae that will hatch into adult mosquitoes will reduce m, the number of adult female mosquitoes per person. This can be accomplished by eliminating standing water, killing larvae, or putting larva eating fish into ponds that breed mosquitoes. A reduction in the human biting rate, a, can be effected by screening windows and doors, using a plain bednet, using insect repellents, or introducing alternative animals like cows or pigs on which hungry mosquitoes will feed (zooprophylaxis). The duration of life of an adult mosquito can be shortened by increasing the daily mortality of adult mosquitoes (pm) through the use of residual insecticides on vertical indoor walls. Use of an insecticide-impregnated bednet will combine the last two and both lower a and increase pm. Chemotherapy, or treating infectious people, will shorten the duration of the infectious period and increase the rate of human recovery, or pg. Chemoprophylaxis, the taking of drugs (by short stay residents like tourists) to prevent malaria infection reduces b, the probability of infection in a susceptible human from an infectious bite. Note that the dangerous mosquito is the female mosquito who has fed at least once on an infectious human, survived for the 10 day latent period, and has now become infectious herself when she feeds on a susceptible human. Preventing second bites by these infectious mosquitoes would interrupt transmission.
One reason to run this complex model is to observe the dynamics of the two host parasite relationships and appreciate rapidity with which a vector borne disease can run through a population, and the level of infection in humans at which it is stable. Another reason is to try out the effects of various interventions and observe the results on the progress of the disease through the community. In Fig. 4 we present the results of four different interventions, one at a time, so that the effects of each can be observed independently, and the relative efficacy observed.
In Fig. 4A we present the baseline Anderson and May Model, identical to Fig. 3C, but here the results of infections in mosquitoes are not visible, and a third line for human infection is added, the sum of both latent and infectious people. In this baseline it is evident that in a six month rainy season almost the entire population will become infected.
In Fig. 4B we present the result of an intervention on the model in Fig. 4A that reduced m, the number of adult mosquitoes per human by half, from 40 to 20. Reducing m by half resulted in about a 20% decrease in the final prevalence of human infection at the end of the rainy season in that community.
In Fig. 4C we present the result of an intervention on the model in Fig. 4A that reduced a, the human biting rate of adult mosquitoes by half, from 0.25 to 0.125. Reducing m by half resulted in about an 80% decrease in the final prevalence of human infection at the end of the rainy season in that community.
In Fig. 4D we present the result of an intervention on the model in Fig. 4A that doubled pm, the mosquito mortality per day from 0.12 to 0.24. This reduces the survival of adult at the end of the latent period. Doubling pm resulted in about a 90% decrease in the final prevalence of human infection at the end of the rainy season in that community.
Comparing the relative effects of different interventions aimed at reducing one or another factor by a similar amount did not reduce the prevalence of malaria equally. Reducing m was least effective, reducing a was substantially more effective, and reducing the length of life of adult mosquitoes (increasing daily mortality) was most effective. Note that this model is non-linear, and that the relative effectiveness of these different interventions is not necessarily intuitively apparent without examination of model results.
Remember that the dangerous mosquito is the female mosquito who has fed at least once on an infectious human, survived for the 10 day latent period, and has now become infectious herself when she feeds on a susceptible human. Preventing second bites by these infectious mosquitoes interrupts transmission. Insecticide-impregnated bednets work well because they affect both a and pm, but, ironically, kill the fewest mosquitoes. Impregnated bednets simply kill or exclude the most dangerous mosquitoes. This is the sort of insight that can be gained from looking at mathematical models of malaria.
The relative effects of various interventions can also be appreciated by examination of the expression for the basic reproductive number, or R0.
R0 = {ma2bc[exp(–phtauh–pmtaum)]}/pgpm.
Values for R0 for malaria in endemic areas where transmission is intense are commonly in the range of 100 but may be lower where malaria is unstable and occurs in the form of periodic epidemics.
From the above, it is evident that R0 is linear in m, but varies as a2, so any decrease in a will have a larger effect than a similar sized decrease in m. Further, it is clear that pm appears as an exponent, so that a change in pm of similar magnitude will be more effective still. These observations have focused attention on spraying indoor vertical walls with insecticide and dipping bed nets in insecticide in order to increase pm and to the use of screens, bednets and repellents to decrease a.
Also, note that R0 is linear in pg, the human recovery rate, so that chemotherapy for individual infected and sick people, while important and life-saving, is not as effective as a community intervention as some of the entomological actions mentioned above. Because malaria transmission can only take place when there are adult mosquitoes feeding, there is a six month hiatus in transmission in locations where a dry season intervenes. This temporary halt in transmission can serve to allow a health care system to catch up and use chemotherapy as a community intervention during the dry season. Seasonality in transmission thus allows chemotherapy to be used as an intervention in some communities where transmission is not continuous.
Eliminating water sources where the larvae of malaria mosquitoes hatch has been effective in eradicating malaria in locations as diverse as Italy, Greece, Spain and parts of the US. Impregnated bednets have reduced mortality in areas like the Gambia, where transmission is most intense. Human chemotherapy saves lives and has been effective in interrupting the spread of malaria where there is a dry season during which there are no mosquitoes and transmission naturally stops.
Naive meddling with malaria can be dangerous, however. In endemic areas infants are born with maternal antibodies against disease (not infection) and they begin being bitten and infected immediately. Continuous infection in the individual produces continuous protection from severe clinical disease. If that cycle of continuous infection is broken by an attempt at eradication that drives the prevalence of infection in humans to low levels (but not to zero), a cohort of individuals is created that is susceptible to severe clinical disease. If the eradication program is abandoned and malaria again becomes endemic, the susceptible individuals (now adults) experience severe clinical disease with much excess avoidable mortality. Boom and bust cycles must be avoided in malaria control. The key concept here is that interventions must be sustainable, and once implemented, must never be stopped if they will leave populations of older individuals susceptible to severe clinical disease.
We have spent a considerable amount of time looking at the basic reproductive number for the propagation of infectious diseases in humans, or R0. It is fitting to end with the same idea from the vector's point of view, vectorial capacity, or Vc. The vectorial capacity is the daily rate at which new infections will occur in humans from a single currently infected human. This depends entirely upon the vector and is simply the basic reproductive number without the duration of infection in the human. For the discrete form of the MacDonald model below, R0 is the number of humans that will become infected from a single infected human during the entire infectious period for the human, and Vc is the number that will become infected by the vectors in a single day.
Remember here we have switched back to the discrete system where p is the probability of survival and –ln(p) is mortality;
R0 = ma2bcpn/–gln(p) and
Vc = ma2bcpn/–ln(p)
If the average case of falciparum malaria is infectious for 285 days, then the recovery rate or g = 1/285 days = 0.0035/day. Thus the magnitudes of R0 and Vc for this disease differ by a factor of 285, and the vectorial capacity may be smaller than unity while the basic reproductive number is high and virtually all persons in the community are continually infected. In the above example, if R0 was 100 then Vc would be 0.35, and R0/Vc = 100/0.35 = 285.
Antibodies induced by malaria infection have relatively short term effects and primarily protect infected persons from becoming physically ill. In an area where malaria is endemic neonates are born with antibodies from their mothers that keep them from becoming ill with malaria. These neonates begin being bitten and infected as soon as they are born, and they develop their own antibodies from their own infections as the antibodies from their mothers decline. As these individuals age they are repeatedly infected, remain perpetually parasitemic, but rarely become seriously ill with malaria. An adult that has been infected all of her life is at little risk for illness in an endemic area, while a naive adult would become severely ill very quickly. Malaria mortality is highest among naive young children and women during their first pregnancy.
In 1957 Senators John F. Kennedy and Hubert Humphrey attached worldwide malaria eradication to the Mutual Security Act. This provided substantial funds to produce zero prevalence of malaria in five years, that is, by 1961. In Sri Lanka the prevalence went from approximately one million cases in 1957 to 100 by 1961 and 18 in 1963. Then DDT resistance began to appear, and DDT use in Sri Lanka was dramatically reduced from the initial rate of two million pounds per year. Malaria rates began to creep up and in 1968 there was a huge outbreak involving about half a million cases, and the prevalence kept climbing until it was back to a million cases in 1994.
Before 1957 when malaria was endemic children were infected at birth and most of the population was perpetually infected and relatively asymptomatic. By 1961 there was virtually no malaria, and all of the previously infected adults had lost their protective antibodies. As was mentioned, malaria infection in a naive adult produces severe illness, so when the huge outbreak began in 1968, most of the half million cases were seriously symptomatic. This produced a devastating effect on the economy, and there were hundreds of excess deaths among the adults who had lost their protective antibodies.
Malaria interventions must be carefully thought out so that one does not cause epidemics. Devastating epidemics have occurred as the result of well-intentioned failures. Minimalist interventions cause no epidemics. Malaria containment involves ignoring the mean prevalence but eliminating outbreaks. Reduce the variance about the mean and prevent new cases in naive adults. Malaria suppression involves attempts to lower the prevalence. These interventions must be sustainable in the community without outside help, otherwise, when the aid expires, malaria epidemics will occur again in naive adults. Creation of boom and bust behavior introduces chaos.
African trypanosomiasis or sleeping sickness is another protozoan parasite transmitted by an insect vector, the Glossina or tsetse fly. There are two features of this disease that make control difficult. There are non-human animal reservoirs for the parasite. Also, the parasite populations with an individual human undergo cyclic antigenic variation. The parasite is able to express something on the order of 100 different variable surface antigens (VATs). Every time the host develops antibodies to control one VAT, a different VAT emerges.
Leishmania species are protozoans transmitted by sand flies, and are thought of as New World trypanosomes. There are important non human animal reservoirs such as dogs.
Arboviruses are viruses transmitted by insect vectors, and almost 100 arboviruses are known to infect man. Two of the most important are yellow fever and dengue.
Yellow fever is transmitted either person to person by mosquitoes, or primate to person, and tends to occur in epidemics. It is still a considerable problem in parts of Asia, tropical Africa and South America. Immunity in humans is lifelong, suggesting control by vaccination, although herd immunity is defeated by primate reservoirs and vertical transmission within mosquito vectors.
Transmission of dengue is similar to yellow fever, except that there are four major types of dengue that all occur together so that a vaccine must be effective against all four types. Infection with one type of dengue is uncomfortable, but infection with a second type after recovery from the first is twenty times as likely to produce the syndrome of Dengue Hemorrhagic Fever that may be fatal.
A Dengue model (or a yellow fever model) would be structurally similar to a malaria model, but since humans infected with Dengue develop long lasting immunity to that strain of virus there would be an extra equation for infectious humans who recover and enter the immune or R state. In addition to the latent period in mosquitoes and mosquito mortality, the complex model includes the latent period in humans (until the appearance of viremia), the recovery of humans from both latent and infectious stages, and the death of humans in both latent and infectious stages.
The basic dengue model can also be used for eastern equine encephalitis with birds replacing people in the dengue model. The behavior of the system depends on whether the birds die with the infection or develop long term immunity, whether immune birds return to the same roost repeatedly, and the numbers of birds in a roost.
We have shown how basic reasoning was developed beginning with the SIS, SIR SEIS and SEIR compartment models, how the reasoning evolved to include vectors in the malaria models without immunity, and ultimately how the same logic can be extended to other vector borne diseases. The value of mathematical models is that they can educate the public health practitioner about quantitative aspects of host parasite interactions in populations and help guide the choice and application of effective interventions.
